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b/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ba38f1b4e8366c3f2bb5cf8cbc4143e8843cdab6 Binary files /dev/null and b/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf differ diff --git a/0dAzT4oBgHgl3EQftv1g/content/tmp_files/2301.01680v1.pdf.txt b/0dAzT4oBgHgl3EQftv1g/content/tmp_files/2301.01680v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc12feb47245d408f3b55a6df9a4ef7e3d3fb514 --- /dev/null +++ b/0dAzT4oBgHgl3EQftv1g/content/tmp_files/2301.01680v1.pdf.txt @@ -0,0 +1,150 @@ +arXiv:2301.01680v1 [math.NT] 3 Jan 2023 +CM ELLIPTIC CURVES AND VERTICALLY ENTANGLED 2-ADIC GROUPS +NATHAN JONES +Abstract. Consider the elliptic curve E given by the Weierstrass equation y2 = x3 − 11x − 14, which has +complex multiplication by the order of conductor 2 inside Z[i]. It was recently observed in a paper of Daniels +and Lozano-Robledo that, for each n ≥ 2, Q(µ2n+1) ⊆ Q(E[2n]). In this note, we prove that this (a priori +surprising) “tower of vertical entanglements” is actually more a feature than a bug: it holds for any elliptic +curve E over Q with complex multiplication by any order of even discriminant. +1. Main result and proof +Let E be an elliptic curve over Q with complex multiplication by the order OK,f ⊆ OK of conductor f +inside the imaginary quadratic field K. Since every endomorphism of E defined over Q commutes with the +action of Gal(Q/Q), it follows that the image of the Galois representation +ρE,m : Gal(Q/Q) −→ Aut(E[m]) ≃ GL2(Z/mZ), +(which is defined by letting Gal(Q/Q) act on E[m], the m-torsion subgroup of E, and fixing a Z/mZ-basis +thereof) lies inside a certain subgroup Nδ,φ(m) ⊆ GL2(Z/mZ), which we now specify, following [2]. First, +let us set +φ = φ(OK,f, m) := +� +0 +if ∆Kf 2 ≡ 0 mod 4 or if m is odd, +f +if ∆Kf 2 ≡ 1 mod 4 and m is even, +δ = δ(OK,f, m) := +� +∆Kf 2/4 +if ∆Kf 2 ≡ 0 mod 4 or if m is odd, +(∆K − 1)f 2/4 +if ∆Kf 2 ≡ 1 mod 4 and m is even. +Next, we define the associated Cartan subgroup Cδ,φ(m) by +Cδ,φ(m) := +�� +a + bφ +b +bδ +a +� +: a, b ∈ Z/mZ, a2 + φab − δb2 ∈ (Z/mZ)× +� +. +(1) +Finally, we define Nδ,φ(m) ⊆ GL2(Z/mZ) by +Nδ,φ(m) := +�� +−1 +0 +φ +1 +� +, Cδ,φ(m) +� +. +(2) +If E is any elliptic curve over Q with CM by OK,f, then, for an appropriate choice of Z/mZ-basis of E[m], +we have ρE,m(GQ) ⊆ Nδ,φ(m). For more details, see [2]. +Let E−16 be the elliptic curve defined by the Weierstrass equation y2 = x3 − 11x − 14 (i.e. the elliptic +curve with Cremona label 32a3). The curve E−16 has CM by the order O := Z + 2iZ of conductor 2 inside +the field Q(i). Furthermore, as observed in [1, Theorem 1.5], we have +n ∈ N≥2 =⇒ Q(ζ2n+1) ⊆ Q(E−16[2n]). +(3) +The authors also observed that the elliptic curves E−4,1 and E−4,2, given, respectively, by the Weierstrass +equations y2 = x2 + x and y2 = x3 + 2x satisfy +Q(E−4,1[2]) = Q(ζ4) +and +Q(ζ8) ⊆ Q(E−4,2[4]). +The purpose of this note is to show that the two elliptic curves E−4,1 and E−4,2 also satisfy (3). More +generally, we will prove the following theorem. +Theorem 1.1. Let E be any elliptic curve over Q with complex multiplication by an order OK,f in an +imaginary quadratic field K. Assuming that the discriminant ∆Kf 2 of OK,f is even, we have that, for each +n ∈ N≥2, Q(ζ2n+1) ⊆ Q(E[2n]). +1 + +Proof. Let us denote by G(2n) := ρE,2n(GQ) ⊆ Nδ,φ(2n) the mod 2n image associated to E. +We will +establish that, for each n ∈ N≥2, there is a surjective homomorphism δ : G(2n) ։ (Z/2n+1Z)× for which +det |G(2n+1) = δ ◦ π, where π : G(2n+1) → G(2n) denotes the projection map. In other words, the following +diagram will commute: +G(2n+1) +(Z/2n+1Z)× +G(2n) +(Z/2nZ)×. +det +π +det +δ +(4) +Once established, it will follow that +Q(ζ2n+1) = Q(E[2n+1])ker det = Q(E[2n+1])π−1(ker δ) = Q(E[2n])ker δ ⊆ Q(E[2n]). +(5) +The key observation is that, since ∆Kf 2 is assumed even, it follows that φ = 0. Considering (1) and (2), we +may then see that +n ∈ N≥2 =⇒ ker +� +Nδ,φ(2n+1) → Nδ,φ(2n) +� +⊆ SL2(Z/2n+1Z). +(6) +(The reason we require n > 1 is that otherwise we do not have ker +� +Nδ,φ(2n+1) → Nδ,φ(2n) +� +⊆ Cδ,φ(2n+1), +since +�−1 +0 +φ +1 +� +≡ I mod 2; the consequent of (6) is false for n = 1.) We now define the map δ by +δ(g) := det(g′), +where g′ ∈ π−1(g). +By virtue of (6), this is independent of the choice of g′ ∈ π−1(g), and thus defines a map δ : G(2n) → +(Z/2n+1Z)×. +It is surjective since det : G(2n+1) → (Z/2n+1Z)× is, and the diagram (4) commutes by +definition of δ. Thus, by (5), we deduce that +∀n ∈ N≥2, +Q(ζ2n+1) ⊆ Q(E[2n]), +as asserted. +□ +The proof of Theorem 1.1 applies to a more general situation, as follows. Given an algebraic group G and +a Galois representation ρ : Gal(Q/Q) → G(ˆZ), let us denote by ρm : Gal(Q/Q) → G(Z/mZ) the composition +of ρ with the natural projection map G(ˆZ) → G(Z/mZ) and define Q(E(ρ[m])) := Q +ker ρm. +Definition 1.2. Suppose G is any algebraic group that admits a homomorphism δ : G → Gm to the mul- +tiplicative group. We say that a Galois representation ρ : Gal(Q/Q) → G(ˆZ) extends the cyclotomic +character if δˆZ ◦ ρ : Gal(Q/Q) → G(ˆZ) → ˆZ× agrees with the cyclotomic character. For any prime number +p, we say that G(Zp) form a vertically entangled p-adic group if there exists n0 ∈ N so that, for each +n ∈ N≥n0, we have ker +� +G(Z/pn+1Z) → G(Z/pnZ) +� +⊆ ker δpn+1, where δpn+1 : G(Z/pn+1Z) → (Z/pn+1Z)× +denotes the group homomorphism associated to δ on the mod pn+1 points of G. +Remark 1.3. Let ρ : Gal(Q/Q) → G(ˆZ) be any Galois representation that extends the cyclotomic character +and suppose that, for some prime number p, the group G(Zp) is a vertically entangled p-adic group. Then +the proof of Theorem 1.1 shows that, in this more general context, we have +∀n ∈ N≥n0, +Q(µpn+1) ⊆ Q(ρ[pn]). +2. Acknowledgement +The author gratefully acknowledges Harris Daniels for bringing the phenomenon (3) to his attention, and +also Ken McMurdy for subsequent stimulating conversations. +References +[1] H. Daniels and A. Lozano-Robledo, Coincidences of division fields, preprint. To appear in Ann. Inst. Fourier. Available at +https://arxiv.org/abs/1912.05618 +[2] A. Lozano-Robledo, Galois representations attached to elliptic curves with complex multiplication, preprint. To appear in +Algebra and Number Theory. Available at https://arxiv.org/abs/1809.02584 +Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S Morgan +St, 322 SEO, Chicago, 60607, IL, USA +Email address: ncjones@uic.edu +2 + diff --git a/0dAzT4oBgHgl3EQftv1g/content/tmp_files/load_file.txt b/0dAzT4oBgHgl3EQftv1g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8652741bdb3f6c8049e352374c4b9acac45442fd --- /dev/null +++ b/0dAzT4oBgHgl3EQftv1g/content/tmp_files/load_file.txt @@ -0,0 +1,71 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf,len=70 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='01680v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='NT] 3 Jan 2023 CM ELLIPTIC CURVES AND VERTICALLY ENTANGLED 2-ADIC GROUPS NATHAN JONES Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Consider the elliptic curve E given by the Weierstrass equation y2 = x3 − 11x − 14, which has complex multiplication by the order of conductor 2 inside Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' It was recently observed in a paper of Daniels and Lozano-Robledo that, for each n ≥ 2, Q(µ2n+1) ⊆ Q(E[2n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' In this note, we prove that this (a priori surprising) “tower of vertical entanglements” is actually more a feature than a bug: it holds for any elliptic curve E over Q with complex multiplication by any order of even discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Main result and proof Let E be an elliptic curve over Q with complex multiplication by the order OK,f ⊆ OK of conductor f inside the imaginary quadratic field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Since every endomorphism of E defined over Q commutes with the action of Gal(Q/Q), it follows that the image of the Galois representation ρE,m : Gal(Q/Q) −→ Aut(E[m]) ≃ GL2(Z/mZ), (which is defined by letting Gal(Q/Q) act on E[m], the m-torsion subgroup of E, and fixing a Z/mZ-basis thereof) lies inside a certain subgroup Nδ,φ(m) ⊆ GL2(Z/mZ), which we now specify, following [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' First, let us set φ = φ(OK,f, m) := � 0 if ∆Kf 2 ≡ 0 mod 4 or if m is odd, f if ∆Kf 2 ≡ 1 mod 4 and m is even, δ = δ(OK,f, m) := � ∆Kf 2/4 if ∆Kf 2 ≡ 0 mod 4 or if m is odd, (∆K − 1)f 2/4 if ∆Kf 2 ≡ 1 mod 4 and m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Next, we define the associated Cartan subgroup Cδ,φ(m) by Cδ,φ(m) := �� a + bφ b bδ a � : a, b ∈ Z/mZ, a2 + φab − δb2 ∈ (Z/mZ)× � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' (1) Finally, we define Nδ,φ(m) ⊆ GL2(Z/mZ) by Nδ,φ(m) := �� −1 0 φ 1 � , Cδ,φ(m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' (2) If E is any elliptic curve over Q with CM by OK,f, then, for an appropriate choice of Z/mZ-basis of E[m], we have ρE,m(GQ) ⊆ Nδ,φ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' For more details, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Let E−16 be the elliptic curve defined by the Weierstrass equation y2 = x3 − 11x − 14 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' the elliptic curve with Cremona label 32a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' The curve E−16 has CM by the order O := Z + 2iZ of conductor 2 inside the field Q(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Furthermore, as observed in [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='5], we have n ∈ N≥2 =⇒ Q(ζ2n+1) ⊆ Q(E−16[2n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' (3) The authors also observed that the elliptic curves E−4,1 and E−4,2, given, respectively, by the Weierstrass equations y2 = x2 + x and y2 = x3 + 2x satisfy Q(E−4,1[2]) = Q(ζ4) and Q(ζ8) ⊆ Q(E−4,2[4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' The purpose of this note is to show that the two elliptic curves E−4,1 and E−4,2 also satisfy (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' More generally, we will prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Let E be any elliptic curve over Q with complex multiplication by an order OK,f in an imaginary quadratic field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Assuming that the discriminant ∆Kf 2 of OK,f is even, we have that, for each n ∈ N≥2, Q(ζ2n+1) ⊆ Q(E[2n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Let us denote by G(2n) := ρE,2n(GQ) ⊆ Nδ,φ(2n) the mod 2n image associated to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' We will establish that, for each n ∈ N≥2, there is a surjective homomorphism δ : G(2n) ։ (Z/2n+1Z)× for which det |G(2n+1) = δ ◦ π, where π : G(2n+1) → G(2n) denotes the projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' In other words, the following diagram will commute: G(2n+1) (Z/2n+1Z)× G(2n) (Z/2nZ)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' det π det δ (4) Once established, it will follow that Q(ζ2n+1) = Q(E[2n+1])ker det = Q(E[2n+1])π−1(ker δ) = Q(E[2n])ker δ ⊆ Q(E[2n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' (5) The key observation is that, since ∆Kf 2 is assumed even, it follows that φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Considering (1) and (2), we may then see that n ∈ N≥2 =⇒ ker � Nδ,φ(2n+1) → Nδ,φ(2n) � ⊆ SL2(Z/2n+1Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' (6) (The reason we require n > 1 is that otherwise we do not have ker � Nδ,φ(2n+1) → Nδ,φ(2n) � ⊆ Cδ,φ(2n+1), since �−1 0 φ 1 � ≡ I mod 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' the consequent of (6) is false for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=') We now define the map δ by δ(g) := det(g′), where g′ ∈ π−1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' By virtue of (6), this is independent of the choice of g′ ∈ π−1(g), and thus defines a map δ : G(2n) → (Z/2n+1Z)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' It is surjective since det : G(2n+1) → (Z/2n+1Z)× is, and the diagram (4) commutes by definition of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Thus, by (5), we deduce that ∀n ∈ N≥2, Q(ζ2n+1) ⊆ Q(E[2n]), as asserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' □ The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='1 applies to a more general situation, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Given an algebraic group G and a Galois representation ρ : Gal(Q/Q) → G(ˆZ), let us denote by ρm : Gal(Q/Q) → G(Z/mZ) the composition of ρ with the natural projection map G(ˆZ) → G(Z/mZ) and define Q(E(ρ[m])) := Q ker ρm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Suppose G is any algebraic group that admits a homomorphism δ : G → Gm to the mul- tiplicative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' We say that a Galois representation ρ : Gal(Q/Q) → G(ˆZ) extends the cyclotomic character if δˆZ ◦ ρ : Gal(Q/Q) → G(ˆZ) → ˆZ× agrees with the cyclotomic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' For any prime number p, we say that G(Zp) form a vertically entangled p-adic group if there exists n0 ∈ N so that, for each n ∈ N≥n0, we have ker � G(Z/pn+1Z) → G(Z/pnZ) � ⊆ ker δpn+1, where δpn+1 : G(Z/pn+1Z) → (Z/pn+1Z)× denotes the group homomorphism associated to δ on the mod pn+1 points of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Let ρ : Gal(Q/Q) → G(ˆZ) be any Galois representation that extends the cyclotomic character and suppose that, for some prime number p, the group G(Zp) is a vertically entangled p-adic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Then the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='1 shows that, in this more general context, we have ∀n ∈ N≥n0, Q(µpn+1) ⊆ Q(ρ[pn]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Acknowledgement The author gratefully acknowledges Harris Daniels for bringing the phenomenon (3) to his attention, and also Ken McMurdy for subsequent stimulating conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Daniels and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Lozano-Robledo, Coincidences of division fields, preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' To appear in Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Fourier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Available at https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='org/abs/1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='05618 [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Lozano-Robledo, Galois representations attached to elliptic curves with complex multiplication, preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' To appear in Algebra and Number Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content=' Available at https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='org/abs/1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='02584 Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S Morgan St, 322 SEO, Chicago, 60607, IL, USA Email address: ncjones@uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} +page_content='edu 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'} diff --git a/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf b/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0b13793cb3a395ccc6800b4bccdd67863de513db --- /dev/null +++ b/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:beb232cd9678b443059fe49f7afbb94a60cf7b964dda28ae0cd31a2a89099a79 +size 1714361 diff --git a/2NE1T4oBgHgl3EQfAAJE/vector_store/index.faiss b/2NE1T4oBgHgl3EQfAAJE/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..497f024ddf52cf372881f72d17ff89d67a4675f8 --- /dev/null +++ b/2NE1T4oBgHgl3EQfAAJE/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5d62a86e55aa4e288bdec5b45d09a5e0e061ca69ccfbb484c2bd62fd6fd9784a +size 4128813 diff --git a/49FAT4oBgHgl3EQfmR1P/content/tmp_files/2301.08622v1.pdf.txt b/49FAT4oBgHgl3EQfmR1P/content/tmp_files/2301.08622v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cdcd6f0ddf877bb61f6759d51e99b2d92ce34412 --- /dev/null +++ b/49FAT4oBgHgl3EQfmR1P/content/tmp_files/2301.08622v1.pdf.txt @@ -0,0 +1,570 @@ +MNRAS 000, 1–5 (2018) +Preprint 23 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The Effect of the Peculiar Motions of the Lens, Source and the Observer on +the Gravitational Lensing Time Delay +Gihan Weerasekara,1★ Thulsi Wickramasinghe,2 Chandana Jayaratne1 +1Department of Physics, University of Colombo, Sri Lanka +2Department of Physics, The College of New Jersey, Ewing, NJ 08628, USA +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +An intervening galaxy acts as a gravitational lens and produces multiple images of a single source such as a remote galaxy. +Galaxies have peculiar speeds in addition to the bulk motion arising due to the expansion of the universe. There is a difference in +light arrival times between lensed images. We calculate more realistic time delays between lensed images when galaxy peculiar +motions, that is the motion of the Lens, the Source and the Observer are taken into consideration neglecting the gravitomagnetic +effects. +Key words: gravitational lensing: strong – galaxies: peculiar +1 INTRODUCTION +A remote galaxy S at redshift 𝑧𝑠 (Shown in Figure 1) is lensed by an +intervening galaxy L at redshift 𝑧𝑑. A light ray from S bends by an +angle 𝛼 before arriving at the observer O. The image I of S forms at +an angle 𝜃 while S is at 𝛽. The distances 𝐷𝑑, 𝐷𝑠 and 𝐷𝑑𝑠 shown are +the angular diameter distances. Walsh (1979), Chen (1995) +From the theory of lensing, we can derive the angular positions 𝜃1 +and 𝜃2 of the two lensed images formed due to a single point lens. +There is a delay Δ𝜏 of light arrival times from these two images. +This delay is arising due to both geometrical path difference and the +fact that two light rays are traveling in two different potential wells +on either side of the lens. The total time delay is given by, Schneider +(1992), Bradt (2008) +Δ𝜏 = +𝐷 𝑓 +𝑐 (1 + 𝑧𝑑) +� 1 +2 (𝜃2 +1 − 𝜃2 +2) + |𝜃1𝜃2| ln +���� +𝜃1 +𝜃2 +���� +� +(1) +where, +𝐷 𝑓 = 𝐷𝑑𝐷𝑠 +𝐷𝑑𝑠 +(2) +We calculate analytically a more realistic time delay between the +two images when the peculiar speeds of the lens, the source and the +observer are considered. These peculiar speeds are random speeds +With respect to the cosmic microwave background radiation - Hubble +flow. +But as we already know a point mass lens is a highly idealized +and less practical lensing model for a real lensing system, in the next +part of the paper we will be considering a more practical Singular +Isothermal Sphere (SIS) lensing model to calculate the time delay +difference when the peculiar speeds of the objects are considered. +★ E-mail: contactgihan@gmail.com +Figure 1. Gravitational Lensing Diagram. The peculiar speed 𝑣 of the lens L +is measured with respect to a freely falling observer with the Hubble flow at +the location of the lens. The angle 𝜖 is measured from the optic axis OL. +© 2018 The Authors +arXiv:2301.08622v1 [astro-ph.CO] 20 Jan 2023 + +10 +S +Dds +So +α +Ds +β; +PO +;02 +Weerasekara et al. +2 THEORY +The angular diameter distance D of a source having no peculiar +motion at a red shift 𝑧 is given by, Weinberg (1972), Hobson (2006) +𝐷(𝑧, ΩΛ,0) = 𝑐 +𝐻0 +1 +1 + 𝑧 +1 +∫ +1 +1+𝑧 +𝑑𝑥 +√︃ +𝑥4 ΩΛ,0 + 𝑥 Ωm,0 + Ωr,0 +(3) +where Ωi,0 is the density parameter of the substance 𝑖 of the cosmic +fluid measured at the present time 𝑡0. We assume a flat universe +(𝑘 = 0) for which Perlmutter (1999), +Ωm,0 + Ωr,0 + ΩΛ,0 = 1 +(4) +The red shift 𝑧𝑑𝑠 of S as measured by L is given by, +1 + 𝑧𝑠 = (1 + 𝑧𝑑)(1 + 𝑧𝑑𝑠) +(5) +Thus, from the equations (3), (4) and (5), neglecting Ωr,0 and elimi- +nating Ωm,0 and expressing everything with the dark energy, we can +derive the value of 𝐷𝑑𝑠, the angular diameter distance of the source +as measured by an observer on the lens as, +𝐷𝑑𝑠 +�𝑧𝑑, 𝑧𝑠, ΩΛ,0 +� = +𝑐 +𝐻0 +1 +√︁ΩΛ,0 +1 + 𝑧𝑑 +1 + 𝑧𝑠 +1 +∫ +1+𝑧𝑑 +1+𝑧𝑠 +𝑑𝑥 +√︂ +𝑥4 + 𝑥 +� +1 +ΩΛ,0 − 1 +� +(1 + 𝑧𝑑)3 +(6) +By evaluating the integral analytically, the value of 𝐷𝑑𝑠 can be +written as +𝐷𝑑𝑠 +�𝑧𝑑, 𝑧𝑠, ΩΛ,0 +� = 𝑐 +𝐻0 +1 +1 + 𝑧𝑠 +� +Ψ �𝑧𝑠, ΩΛ,0 +� − Ψ �𝑧𝑑, ΩΛ,0 +�� +(7) +where in terms of hypergeometric function 2𝐹1 +Ψ �𝑧, ΩΛ,0 +� = +1 + 𝑧 +√︁ΩΛ,0 +2𝐹1 +� 1 +3, 1 +2; 4 +3; +� +1 − +1 +ΩΛ,0 +� +(1 + 𝑧)3 +� +(8) +In the theory of lensing, the source S, lens L, and the observer O in +Fig. 1 are all freely falling with the smooth expansion of the universe; +that is, experiencing no peculiar motions. The angular diameter dis- +tances 𝐷𝑠, 𝐷𝑑 and 𝐷𝑑𝑠 are then measured between these objects +which are freely falling with the Hubble flow. Thus, the redshifts +entering Eq (8) should be associated with the freely falling objects. +However, all galaxies are subjected to peculiar or random motions, +for an example in the scenario given here the Source S, the Lens L +and the Observer O are having peculiar motions. Thus, the redshift +of the lens we measure includes this peculiar motion. Therefore, the +redshifts entering Eq (7), which should be the redshifts of freely +falling objects, must be corrected for random peculiar motions. For +this, consider initially the random motion of L neglecting the random +motions of S and O. This is similar to OS axis being fixed and L +having a peculiar motion with respect to this axis. An observer freely +falling with the Hubble flow at the location of L will see a Doppler +shift of L arising due to the random (peculiar) speed 𝜈. In addition +to this shift, we have the cosmological redshift of that freely falling +observer arising due to the bulk expanding motion of the universe. +Thus, the redshift z of the freely falling observer, from special theory +of relativity, becomes (see Figure. 1) +1 + 𝑧 = +√︁ +1 − 𝛽2 +1 − 𝛽 cos 𝜖 (1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑) +(9) +where 𝑣 = 𝛽𝑐 is the peculiar speed of the object as seen by the freely +falling observer and 𝜖 is the angle between the peculiar velocity vector +and the line-of-sight to L (see Fig. 1). It is this redshift 𝑧 (Eq. 9) that +should enter in (7) for the angular diameter distance calculation. If +𝜖 = 0, L is approaching a freely falling observer and if 𝜖 = 𝜋 it is +receding. Inserting (9) in (8) and expanding to first order in 𝛽 we get, +Ψ �𝑧, ΩΛ,0 +� ∼ 1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑 +√︁ΩΛ,0 +× +2𝐹1 +� +1 + +� +1 + 3 +8 +� +1 − +1 +ΩΛ,0 +� � +1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑�3� +𝛽 cos 𝜖 +� +(10) +where the hypergeometric function is the one appearing in (8) with +𝑧 = 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑. Now that we have an expression to account for the +peculiar motion of L, we can employ the same in our code to calculate +the time delay taking all the peculiar motions into consideration. That +is including the peculiar motions of S, L and O. while doing so, we +find that the other higher order terms are very small and the time +delay is linear to first order in 𝛽. Then the form of the observed time +delay becomes, +Δ𝜏 ≈ Δ𝜏0 (1 + 𝜅 𝛽 cos 𝜖) +(11) +where Δ𝜏0 is when the peculiar motions are neglected. +As we now have an equation for the gravitational time delay differ- +ence when the peculiar speeds are considered for a point mass lens +model, let us now proceed to the Singular Isothermal Sphere lensing +model and derive the time delay difference equation for that. +According to the theory of lensing the time delay difference for a +SIS model is given by the equation, Schneider (1992) +𝑐Δ𝜏 = +� +4𝜋 +� 𝜎𝑣 +𝑐 +�2�2 𝐷𝑑𝐷𝑑𝑠 +𝐷𝑠 +(1 + 𝑧𝑑)2𝑦 +(12) +further by making use of the following equations, +𝑦 = 𝜂 +𝜂0 +(13) +𝜉0 = 4𝜋 +� 𝜎𝑣 +𝑐 +�2 𝐷𝑑𝐷𝑑𝑠 +𝐷𝑠 +(14) +we can arrive at the following equation that gives us the required +time delay. +Δ𝜏 = 4𝜋 +𝑐 +� 𝜎𝑣 +𝑐 +�2 +𝐷𝑑(1 + 𝑧𝑑)2𝛽 +(15) +we do a realistic assumption for 𝛽 by making use of the point mass +lens model as, +𝛽 = 𝜃1 + 𝜃2 +(16) +In this equation when we consider the peculiar speeds of the ob- +jects, we have to use 𝑧 = 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑 in accordance with (9) similar +to the calculation we have carried out with the point mass lens. +MNRAS 000, 1–5 (2018) + +Gravitational Lensing Time Delay with Peculiar Motions +3 +Figure 2. Lensing image. The Optical and Radio delay for this system has +been measured. Koopmans (1998) +3 RESULTS AND DISCUSSION +The example we have used is the lensing system illustrated in the +Figure 2. Koopmans (1998) This lens is referred to as B1600+434 +and it has the following characteristics. +Optical time delay += 51 ± 2 Days +𝑧𝑠 += 1.59 +𝜃1 += +1.14" +𝑧𝑑 += 0.42 +𝜃2 += -0.25" +According to the given set of angular distances and angles assum- +ing the non-realistic assumption that the lens is a point mass, we +can calculate a theoretical lensing delay time of 73.92 days for the +WMAP cosmological parameters. When we compare the theoretical +time delay and the observed time delays it is clear that they are not +matching. We believe that the discrepancy is arising due to the lens +point-mass assumption and that we have not taken peculiar speeds +into account. However we would like to illustrate the effect of the +peculiar motions on the time delay assuming initially a point-mass +lens here. +We simulated 1000 scenarios with the above given particular set +of lensing parameters (𝑧𝑠 = 1.59, 𝑧𝑑 = 0.42, 𝜃1 = +1.14" and 𝜃2 = +-0.25" ). For each scenario the lens and the observer have random +peculiar speeds in random directions with respect to the back ground +radiation. In the simulations of Figure 3/4/5. the peciliar speeds are +non relativistic and they range from 0 to 0.01𝑐. +for this lensing system Eq (11) can be written as, +Δ𝜏 ≈ 73.92 (1 + 4.69 𝛽 cos 𝜖) +(17) +The observer, that is the Milky Way has an estimated peculiar +speed of 600𝑘𝑚𝑠−1 Kogut (1993) with respect to the back ground +radiation. The directions of the peculiar motions are taken to be +random in relation to the OL axis. We have taken ΩΛ,0 = 0.73. +The simulated time delays as shown in Figure 3. are showing a time +delay range of 8 days with the contribution of the peculiar motions +Figure 3. Point Mass lens. The Source, the Lens, and the Observer all are +having peculiar speeds in the range of 0 to 0.01c in any random direction +Figure 4. Point Mass lens. The Source and the Observer are having peculiar +speeds in the range of 0 to 0.01c in any random direction. The Lens is +stationary +while no peculiar motion time delay being 73.9 days. Therefore the +maximum time delay when all three objects are moving is nearly 4 +days and it is a significant value. Therefore the peculiar motions will +give rise to a measurable and significant difference in the gravitational +lensing time delay. +In the second simulation given in Figure 4 we have excluded only +the peculiar motion of the Lens. In this case it is seen that the maxi- +mum time delay difference is about 1 day. From this result it is clear +that the peculiar motions of the source and the Observer alone when +the lens is not moving is not creating a significant gravitational lens- +ing time delay. To further enhance this fact we have taken another +simulation with only the Lens having peculiar motions and the ob- +server and the source are stationary. That result is given in the Figure +5. +MNRAS 000, 1–5 (2018) + +1000TimeDelaysinDays +NopeculiarmotionDelay=73.9days +TheSource,theLensandtheObserverhavePeculiarSpeeds +140 +120 +100 +ber +Num +80 +40F +20 +70 +72 +74 +78 +78 +TimeDelayinDays1000TimeDelavsinDays +Nopeculiar.motionDelay=73.9days +OnlySourceandObserverhavePeculiarSpeeds,Lensisnotmoving +140 +120 +100 +ber +unN +80 +60 +40 +20 +73.0 +73.5 +74.0 +74.5 +75.0 +TimeDelayinDays4 +Weerasekara et al. +Figure 5. Point Mass lens. The Lens is having peculiar speeds in the range of +0 to 0.01c in any random direction. The source and the observer are stationary +The result we have obtained in Figure 5 is almost identical to the +result we have obtained in the Figure 3. +From these results it is clear that the gravitational lensing time +delay is highly sensitive to the peculiar speeds of the lens. An- +other interesting result of the simulation is the peculiar speeds of +the observer and the source is not having a significant effect on the +gravitational lensing time delay. +As we have figured out by now, the gravitational lensing time delay +is mostly affected by the peculiar motions of the Lens. Thus we can +neglect the peculiar motions of the Observer and the Source. +In the next simulation given in Figure 6, we have taken a lensing +system with only the lens moving. In that we have taken the speed +and the direction of the lens separately. The lens in the simulation +is having speeds from 0 to 0.005𝑐 and the direction is 0 (The lens +is approaching the observer) to 𝜋 (The lens is receding from the +observer). If the 𝜖 is 𝜋/2 then the Lens is moving in a transverse +direction. +From Figure 6, we can identify that when the lens is moving +towards the observer the gravitational lensing time delay is increasing +and it is attaining larger values directly in proportion with the peculiar +speed of the lens. That is, when the lens is having larger approaching +peculiar speeds the gravitational lensing time delay is also larger. +In contrast to that when the lens is receding from the observer the +gravitation lensing time delay is decreasing. It can be also seen that +when the receding peculiar speed is becoming larger the gravitational +lensing time delay is becoming smaller. +If the lens is moving in a transverse direction then there is no +measurable effect in the gravitational lensing time delay as the effect +is in second order. +The lenses we have considered so far are having small velocities. +But if we consider lenses having relativistic speeds then the effect +become more prominent. That is the measurable gravitational lensing +time delay becomes much larger. Results are illustrated in the Figure +7, where the peculiar speeds of the lens are relativistic. +In the example we have taken, the Lens B1400+434 is having an +measured optical time delay of 51 days and a theoretical time delay +of 73.92 days, assuming a point-mass lens. From our results we can +account for the difference of this time delay. That is we can have +this particular observed optical time delay difference if the lens is +Figure 6. Point Mass lens. The lens is having different peculiar speeds in +different directions +Figure 7. Point Mass lens. The Lens is having relativistic peculiar speeds +having a relativistic peculiar speed in the range of 0.05𝑐 to 0.06𝑐 in +a receding direction from us provided that we model the lens as a +point mass, which is not exact. +As we now have a clear idea on gravitational lensing time delays +when the peculiar speeds of the objects are considered while using +a point mass lensing model, let us now investigate the same effect +when a more realistic Singular Isothermal Sphere lensing model is +used for the calculations. +For this also we employ the same simulation with 1000 scenarios +where random peculiar speeds are in random directions. when using +Eq. (15) average velocity dispersion 𝜎𝑣 will be taken as 150𝑘𝑚𝑠−1 +Koopmans (1998). With this average velocity dispersion value and +using Singular Isothermal Sphere model we have a very interesting +result for the non peculiar motion lensing time delay, which is 51.45 +days. this value is almost identical to the observed lensing time delay +value of 51 ± 2 Days. +MNRAS 000, 1–5 (2018) + +1000TimeDelaysinDays +NopeculiarmotionDelay=73.9days +OnlyLensishavingPeculiarSpeeds,ObserverandSourcenotmoving +200 +150 +ber +unN +100 +72 +73 +74 +75 +78 +77 +81 +TimeDelayinDaysUpperCurveforApproachingLens +LowerCurveforRecedingLens +MiddleCurveforLensMovinginTransverseDirection +75.5 +75.0 +Days +ApproachingLense +74.5 +TransverseLense +74:0 +Dela +Receding Lense +73.5 +73.0 +72.5 +0.000 +0.001 +0.002 +0.003 +0:004 +0.005 +β=IIRelativisticLens +UpperCurveforApproachingLens +LowerCurveforRecedingLens +MiddleCurveforLensMovinginTransverseDirection +120 +110 +Days +100 +ApproachingLense +90 +Transverse Lense +80 +Receding Lense +Del +Time +70 +60 +50F +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +1%1=sGravitational Lensing Time Delay with Peculiar Motions +5 +Figure 8. Singular Isothermal Sphere lens model. The Lens is having non +relativistic peculiar speeds in the range of 0 to 0.01c in any random direction +The simulation for the non relativistic peculiar speeds is given in +the Figure 8. In that the non relativistic peculiar speeds are from 0 +to 0.01𝑐. further it can be noted in this simulation the time delays +are ranging from 50.5 - 52.5 days while having a maximum delay +difference of 1 day from the no peculiar motion instance. therefore +even with non relativistic peculiar speeds it is clear that we can have +measurable and significant time delay difference from the no peculiar +motion instance when peculiar speeds of the lens is considered. +In the next simulation given in the Figure 9. we consider a rela- +tivistic peculiar speed distribution from 0 to 0.05𝑐. it can be noted +in this figure when there is a relativistic peculiar speed distribution +for the lens, the lensing time delays can range from 46-56 days with +a maximum delay difference of 5 days from the no peculiar motion +instance. therefore it is apparent from this simulation when there is a +relativistic peculiar speed for the lens there can be a very significant +gravitational lensing time difference from the non peculiar speed +instance while using a more realistic Singular Isothermal Sphere to +model the lens. +4 CONCLUSIONS +From the above simulations we have found out that in fact there is a +significant measurable time delay difference arising from the peculiar +speeds of the lens using both non realistic point mass lens and more +realistic Singular Isothermal Sphere as the lensing model. +The important observation is that an approaching lens results in +an increase of the time delay while a receding lens gives rise to a +decrease in the delay. +We find that the time delay is not significantly affected by the +source or observer peculiar motions. +We see from Figure 7. and Figure 9. that a relativistically moving +lens in any direction can significantly affect the lensing time delays. +Figure 9. Singular Isothermal Sphere lens model. The Lens is having rela- +tivistic peculiar speeds in the range of 0 to 0.05c in any random direction +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Bradt H., 2008, Astrophysics Processes, Cambridge University Press, UK, +437, 482 +Chen G.H., Kochanek C.S. and Hewitt J.N., 1995, Astrophys. J. 447, 62 +Hobson M. P. , Efstathiou G. P. and Lasenby A. N. , 2006, General Relativity +An Introduction for Physicists, Cambridge University Press, UK, 355, +427 +Kogut A., Lineweaver C., Smoot G.F., Bennett C. L., Banday A., et al, 1993, +Astrophysical Journal 419, 1 (1993) +Koopmans L.V.E, de Bruyn A.G, Jackson N., et al, 1998, MNRAS, vol. 295, +534 (1998) +Perlmutter S. et al, 1999, Astrophys. J. 517, 565 +Schneider P. , Ehlers J. and Falco E.E , 1992 Gravitational Lenses, Springer- +Verlag +Walsh D., Carswell R.F. and Weyman R.J, 1979, Natwe 279, 381 +Weinberg S. 1972, , Gravitation & Cosmology, Wiley, New York, 407, 633, +Weinberg S. , 2008, Cosmology, Oxford +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–5 (2018) + +1000TimeDelaysinDays +NopeculiarmotionDelay=51.45days +sigma =150 +lensmoving +150 +100 +Number +50 +50.5 +51.0 +51.5 +52.0 +52.5 +TimeDelayinDays1000TimeDelaysinDays +NopeculiarmotionDelay=51.45days +sigma=150 +lensmoving +120 +100 +Number +80 +60 +40 +20 +0上 +46 +48 +50 +52 +54 +56 +TimeDelay inDays \ No newline at end of file diff --git a/49FAT4oBgHgl3EQfmR1P/content/tmp_files/load_file.txt b/49FAT4oBgHgl3EQfmR1P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c37d073472d5c60794a0c8923a985f2e5d6f0a06 --- /dev/null +++ b/49FAT4oBgHgl3EQfmR1P/content/tmp_files/load_file.txt @@ -0,0 +1,273 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf,len=272 +page_content='MNRAS 000, 1–5 (2018) Preprint 23 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 The Effect of the Peculiar Motions of the Lens, Source and the Observer on the Gravitational Lensing Time Delay Gihan Weerasekara,1★ Thulsi Wickramasinghe,2 Chandana Jayaratne1 1Department of Physics, University of Colombo, Sri Lanka 2Department of Physics, The College of New Jersey, Ewing, NJ 08628, USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' in original form ZZZ ABSTRACT An intervening galaxy acts as a gravitational lens and produces multiple images of a single source such as a remote galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Galaxies have peculiar speeds in addition to the bulk motion arising due to the expansion of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' There is a difference in light arrival times between lensed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' We calculate more realistic time delays between lensed images when galaxy peculiar motions, that is the motion of the Lens, the Source and the Observer are taken into consideration neglecting the gravitomagnetic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Key words: gravitational lensing: strong – galaxies: peculiar 1 INTRODUCTION A remote galaxy S at redshift 𝑧𝑠 (Shown in Figure 1) is lensed by an intervening galaxy L at redshift 𝑧𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' A light ray from S bends by an angle 𝛼 before arriving at the observer O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The image I of S forms at an angle 𝜃 while S is at 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The distances 𝐷𝑑, 𝐷𝑠 and 𝐷𝑑𝑠 shown are the angular diameter distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Walsh (1979), Chen (1995) From the theory of lensing, we can derive the angular positions 𝜃1 and 𝜃2 of the two lensed images formed due to a single point lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' There is a delay Δ𝜏 of light arrival times from these two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' This delay is arising due to both geometrical path difference and the fact that two light rays are traveling in two different potential wells on either side of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The total time delay is given by, Schneider (1992), Bradt (2008) Δ𝜏 = 𝐷 𝑓 𝑐 (1 + 𝑧𝑑) � 1 2 (𝜃2 1 − 𝜃2 2) + |𝜃1𝜃2| ln ���� 𝜃1 𝜃2 ���� � (1) where, 𝐷 𝑓 = 𝐷𝑑𝐷𝑠 𝐷𝑑𝑠 (2) We calculate analytically a more realistic time delay between the two images when the peculiar speeds of the lens, the source and the observer are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' These peculiar speeds are random speeds With respect to the cosmic microwave background radiation - Hubble flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' But as we already know a point mass lens is a highly idealized and less practical lensing model for a real lensing system, in the next part of the paper we will be considering a more practical Singular Isothermal Sphere (SIS) lensing model to calculate the time delay difference when the peculiar speeds of the objects are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' ★ E-mail: contactgihan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='com Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Gravitational Lensing Diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The peculiar speed 𝑣 of the lens L is measured with respect to a freely falling observer with the Hubble flow at the location of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The angle 𝜖 is measured from the optic axis OL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' © 2018 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='08622v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='CO] 20 Jan 2023 10 S Dds So α Ds β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' PO ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='02 Weerasekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 2 THEORY The angular diameter distance D of a source having no peculiar motion at a red shift 𝑧 is given by, Weinberg (1972), Hobson (2006) 𝐷(𝑧, ΩΛ,0) = 𝑐 𝐻0 1 1 + 𝑧 1 ∫ 1 1+𝑧 𝑑𝑥 √︃ 𝑥4 ΩΛ,0 + 𝑥 Ωm,0 + Ωr,0 (3) where Ωi,0 is the density parameter of the substance 𝑖 of the cosmic fluid measured at the present time 𝑡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' We assume a flat universe (𝑘 = 0) for which Perlmutter (1999),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 + Ωr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 + ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 = 1 (4) The red shift 𝑧𝑑𝑠 of S as measured by L is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 1 + 𝑧𝑠 = (1 + 𝑧𝑑)(1 + 𝑧𝑑𝑠) (5) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' from the equations (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' (4) and (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' neglecting Ωr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 and elimi- nating Ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 and expressing everything with the dark energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' we can derive the value of 𝐷𝑑𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' the angular diameter distance of the source as measured by an observer on the lens as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 𝐷𝑑𝑠 �𝑧𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 𝑧𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 � = 𝑐 𝐻0 1 √︁ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 1 + 𝑧𝑑 1 + 𝑧𝑠 1 ∫ 1+𝑧𝑑 1+𝑧𝑠 𝑑𝑥 √︂ 𝑥4 + 𝑥 � 1 ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 − 1 � (1 + 𝑧𝑑)3 (6) By evaluating the integral analytically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' the value of 𝐷𝑑𝑠 can be written as 𝐷𝑑𝑠 �𝑧𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 𝑧𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 � = 𝑐 𝐻0 1 1 + 𝑧𝑠 � Ψ �𝑧𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 � − Ψ �𝑧𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 �� (7) where in terms of hypergeometric function 2𝐹1 Ψ �𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 � = 1 + 𝑧 √︁ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 2𝐹1 � 1 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 4 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' � 1 − 1 ΩΛ,0 � (1 + 𝑧)3 � (8) In the theory of lensing, the source S, lens L, and the observer O in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 1 are all freely falling with the smooth expansion of the universe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' that is, experiencing no peculiar motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The angular diameter dis- tances 𝐷𝑠, 𝐷𝑑 and 𝐷𝑑𝑠 are then measured between these objects which are freely falling with the Hubble flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Thus, the redshifts entering Eq (8) should be associated with the freely falling objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' However, all galaxies are subjected to peculiar or random motions, for an example in the scenario given here the Source S, the Lens L and the Observer O are having peculiar motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Thus, the redshift of the lens we measure includes this peculiar motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Therefore, the redshifts entering Eq (7), which should be the redshifts of freely falling objects, must be corrected for random peculiar motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' For this, consider initially the random motion of L neglecting the random motions of S and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' This is similar to OS axis being fixed and L having a peculiar motion with respect to this axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' An observer freely falling with the Hubble flow at the location of L will see a Doppler shift of L arising due to the random (peculiar) speed 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In addition to this shift, we have the cosmological redshift of that freely falling observer arising due to the bulk expanding motion of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Thus, the redshift z of the freely falling observer, from special theory of relativity, becomes (see Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 1) 1 + 𝑧 = √︁ 1 − 𝛽2 1 − 𝛽 cos 𝜖 (1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑) (9) where 𝑣 = 𝛽𝑐 is the peculiar speed of the object as seen by the freely falling observer and 𝜖 is the angle between the peculiar velocity vector and the line-of-sight to L (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' It is this redshift 𝑧 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 9) that should enter in (7) for the angular diameter distance calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' If 𝜖 = 0, L is approaching a freely falling observer and if 𝜖 = 𝜋 it is receding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Inserting (9) in (8) and expanding to first order in 𝛽 we get, Ψ �𝑧, ΩΛ,0 � ∼ 1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑 √︁ΩΛ,0 × 2𝐹1 � 1 + � 1 + 3 8 � 1 − 1 ΩΛ,0 � � 1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑�3� 𝛽 cos 𝜖 � (10) where the hypergeometric function is the one appearing in (8) with 𝑧 = 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Now that we have an expression to account for the peculiar motion of L, we can employ the same in our code to calculate the time delay taking all the peculiar motions into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' That is including the peculiar motions of S, L and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' while doing so, we find that the other higher order terms are very small and the time delay is linear to first order in 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Then the form of the observed time delay becomes, Δ𝜏 ≈ Δ𝜏0 (1 + 𝜅 𝛽 cos 𝜖) (11) where Δ𝜏0 is when the peculiar motions are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' As we now have an equation for the gravitational time delay differ- ence when the peculiar speeds are considered for a point mass lens model, let us now proceed to the Singular Isothermal Sphere lensing model and derive the time delay difference equation for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' According to the theory of lensing the time delay difference for a SIS model is given by the equation, Schneider (1992) 𝑐Δ𝜏 = � 4𝜋 � 𝜎𝑣 𝑐 �2�2 𝐷𝑑𝐷𝑑𝑠 𝐷𝑠 (1 + 𝑧𝑑)2𝑦 (12) further by making use of the following equations, 𝑦 = 𝜂 𝜂0 (13) 𝜉0 = 4𝜋 � 𝜎𝑣 𝑐 �2 𝐷𝑑𝐷𝑑𝑠 𝐷𝑠 (14) we can arrive at the following equation that gives us the required time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Δ𝜏 = 4𝜋 𝑐 � 𝜎𝑣 𝑐 �2 𝐷𝑑(1 + 𝑧𝑑)2𝛽 (15) we do a realistic assumption for 𝛽 by making use of the point mass lens model as, 𝛽 = 𝜃1 + 𝜃2 (16) In this equation when we consider the peculiar speeds of the ob- jects, we have to use 𝑧 = 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑 in accordance with (9) similar to the calculation we have carried out with the point mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' MNRAS 000, 1–5 (2018) Gravitational Lensing Time Delay with Peculiar Motions 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Lensing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The Optical and Radio delay for this system has been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Koopmans (1998) 3 RESULTS AND DISCUSSION The example we have used is the lensing system illustrated in the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Koopmans (1998) This lens is referred to as B1600+434 and it has the following characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Optical time delay = 51 ± 2 Days 𝑧𝑠 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='59 𝜃1 = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='14" 𝑧𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='42 𝜃2 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='25" According to the given set of angular distances and angles assum- ing the non-realistic assumption that the lens is a point mass, we can calculate a theoretical lensing delay time of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='92 days for the WMAP cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' When we compare the theoretical time delay and the observed time delays it is clear that they are not matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' We believe that the discrepancy is arising due to the lens point-mass assumption and that we have not taken peculiar speeds into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' However we would like to illustrate the effect of the peculiar motions on the time delay assuming initially a point-mass lens here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' We simulated 1000 scenarios with the above given particular set of lensing parameters (𝑧𝑠 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='59, 𝑧𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='42, 𝜃1 = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='14" and 𝜃2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='25" ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' For each scenario the lens and the observer have random peculiar speeds in random directions with respect to the back ground radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In the simulations of Figure 3/4/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' the peciliar speeds are non relativistic and they range from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='01𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' for this lensing system Eq (11) can be written as, Δ𝜏 ≈ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='92 (1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='69 𝛽 cos 𝜖) (17) The observer, that is the Milky Way has an estimated peculiar speed of 600𝑘𝑚𝑠−1 Kogut (1993) with respect to the back ground radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The directions of the peculiar motions are taken to be random in relation to the OL axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' We have taken ΩΛ,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The simulated time delays as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' are showing a time delay range of 8 days with the contribution of the peculiar motions Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The Source, the Lens, and the Observer all are having peculiar speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='01c in any random direction Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The Source and the Observer are having peculiar speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='01c in any random direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The Lens is stationary while no peculiar motion time delay being 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='9 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Therefore the maximum time delay when all three objects are moving is nearly 4 days and it is a significant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Therefore the peculiar motions will give rise to a measurable and significant difference in the gravitational lensing time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In the second simulation given in Figure 4 we have excluded only the peculiar motion of the Lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In this case it is seen that the maxi- mum time delay difference is about 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' From this result it is clear that the peculiar motions of the source and the Observer alone when the lens is not moving is not creating a significant gravitational lens- ing time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' To further enhance this fact we have taken another simulation with only the Lens having peculiar motions and the ob- server and the source are stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' That result is given in the Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' MNRAS 000, 1–5 (2018) 1000TimeDelaysinDays NopeculiarmotionDelay=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='9days TheSource,theLensandtheObserverhavePeculiarSpeeds 140 120 100 ber Num 80 40F 20 70 72 74 78 78 TimeDelayinDays1000TimeDelavsinDays Nopeculiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='motionDelay=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='9days OnlySourceandObserverhavePeculiarSpeeds,Lensisnotmoving 140 120 100 ber unN 80 60 40 20 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 TimeDelayinDays4 Weerasekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The Lens is having peculiar speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='01c in any random direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The source and the observer are stationary The result we have obtained in Figure 5 is almost identical to the result we have obtained in the Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' From these results it is clear that the gravitational lensing time delay is highly sensitive to the peculiar speeds of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' An- other interesting result of the simulation is the peculiar speeds of the observer and the source is not having a significant effect on the gravitational lensing time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' As we have figured out by now, the gravitational lensing time delay is mostly affected by the peculiar motions of the Lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Thus we can neglect the peculiar motions of the Observer and the Source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In the next simulation given in Figure 6, we have taken a lensing system with only the lens moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In that we have taken the speed and the direction of the lens separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The lens in the simulation is having speeds from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='005𝑐 and the direction is 0 (The lens is approaching the observer) to 𝜋 (The lens is receding from the observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' If the 𝜖 is 𝜋/2 then the Lens is moving in a transverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' From Figure 6, we can identify that when the lens is moving towards the observer the gravitational lensing time delay is increasing and it is attaining larger values directly in proportion with the peculiar speed of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' That is, when the lens is having larger approaching peculiar speeds the gravitational lensing time delay is also larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In contrast to that when the lens is receding from the observer the gravitation lensing time delay is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' It can be also seen that when the receding peculiar speed is becoming larger the gravitational lensing time delay is becoming smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' If the lens is moving in a transverse direction then there is no measurable effect in the gravitational lensing time delay as the effect is in second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The lenses we have considered so far are having small velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' But if we consider lenses having relativistic speeds then the effect become more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' That is the measurable gravitational lensing time delay becomes much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Results are illustrated in the Figure 7, where the peculiar speeds of the lens are relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In the example we have taken, the Lens B1400+434 is having an measured optical time delay of 51 days and a theoretical time delay of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='92 days, assuming a point-mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' From our results we can account for the difference of this time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' That is we can have this particular observed optical time delay difference if the lens is Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The lens is having different peculiar speeds in different directions Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The Lens is having relativistic peculiar speeds having a relativistic peculiar speed in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='05𝑐 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='06𝑐 in a receding direction from us provided that we model the lens as a point mass, which is not exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' As we now have a clear idea on gravitational lensing time delays when the peculiar speeds of the objects are considered while using a point mass lensing model, let us now investigate the same effect when a more realistic Singular Isothermal Sphere lensing model is used for the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' For this also we employ the same simulation with 1000 scenarios where random peculiar speeds are in random directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' when using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' (15) average velocity dispersion 𝜎𝑣 will be taken as 150𝑘𝑚𝑠−1 Koopmans (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' With this average velocity dispersion value and using Singular Isothermal Sphere model we have a very interesting result for the non peculiar motion lensing time delay, which is 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='45 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' this value is almost identical to the observed lensing time delay value of 51 ± 2 Days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' MNRAS 000, 1–5 (2018) 1000TimeDelaysinDays NopeculiarmotionDelay=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='9days OnlyLensishavingPeculiarSpeeds,ObserverandSourcenotmoving 200 150 ber unN 100 72 73 74 75 78 77 81 TimeDelayinDaysUpperCurveforApproachingLens LowerCurveforRecedingLens MiddleCurveforLensMovinginTransverseDirection 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 Days ApproachingLense 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 TransverseLense 74:0 Dela Receding Lense 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='003 0:004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='005 β=IIRelativisticLens UpperCurveforApproachingLens LowerCurveforRecedingLens MiddleCurveforLensMovinginTransverseDirection 120 110 Days 100 ApproachingLense 90 Transverse Lense 80 Receding Lense Del Time 70 60 50F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='10 1%1=sGravitational Lensing Time Delay with Peculiar Motions 5 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Singular Isothermal Sphere lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The Lens is having non relativistic peculiar speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='01c in any random direction The simulation for the non relativistic peculiar speeds is given in the Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In that the non relativistic peculiar speeds are from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='01𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' further it can be noted in this simulation the time delays are ranging from 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 - 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 days while having a maximum delay difference of 1 day from the no peculiar motion instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' therefore even with non relativistic peculiar speeds it is clear that we can have measurable and significant time delay difference from the no peculiar motion instance when peculiar speeds of the lens is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' In the next simulation given in the Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' we consider a rela- tivistic peculiar speed distribution from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='05𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' it can be noted in this figure when there is a relativistic peculiar speed distribution for the lens, the lensing time delays can range from 46-56 days with a maximum delay difference of 5 days from the no peculiar motion instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' therefore it is apparent from this simulation when there is a relativistic peculiar speed for the lens there can be a very significant gravitational lensing time difference from the non peculiar speed instance while using a more realistic Singular Isothermal Sphere to model the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' 4 CONCLUSIONS From the above simulations we have found out that in fact there is a significant measurable time delay difference arising from the peculiar speeds of the lens using both non realistic point mass lens and more realistic Singular Isothermal Sphere as the lensing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The important observation is that an approaching lens results in an increase of the time delay while a receding lens gives rise to a decrease in the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' We find that the time delay is not significantly affected by the source or observer peculiar motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' We see from Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' that a relativistically moving lens in any direction can significantly affect the lensing time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' Singular Isothermal Sphere lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content=' The Lens is having rela- tivistic peculiar speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='05c in any random direction DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='5 TimeDelayinDays1000TimeDelaysinDays NopeculiarmotionDelay=51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'} +page_content='45days sigma=150 lensmoving 120 100 Number 80 60 40 20 0上 46 48 50 52 54 56 TimeDelay inDays' metadata={'source': 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Philip S. Yu‡ +†Walmart Global Tech, Sunnyvale, CA, USA +xiaohan.li@walmart.com +‡University of Illinois at Chicago, Chicago, IL, USA +{yliu363, zliu212, psyu}@uic.edu +Abstract—Session-based Recommendation (SBR) is to predict +users’ next interested items based on their previous browsing +sessions. Existing methods model sessions as graphs or sequences +to estimate user interests based on their interacted items to +make recommendations. In recent years, graph-based methods +have achieved outstanding performance on SBR. However, none +of these methods consider temporal information, which is a +crucial feature in SBR as it indicates timeliness or currency. +Besides, the session graphs exhibit a hierarchical structure and +are demonstrated to be suitable in hyperbolic geometry. But few +papers design the models in hyperbolic spaces and this direction +is still under exploration. +In this paper, we propose Time-aware Hyperbolic Graph +Attention Network (TA-HGAT) — a novel hyperbolic graph +neural network framework to build a session-based recommenda- +tion model considering temporal information. More specifically, +there are three components in TA-HGAT. First, a hyperbolic +projection module transforms the item features into hyperbolic +space. Second, the time-aware graph attention module models +time intervals between items and the users’ current interests. +Third, an evolutionary loss at the end of the model provides +an accurate prediction of the recommended item based on the +given timestamp. TA-HGAT is built in a hyperbolic space to learn +the hierarchical structure of session graphs. Experimental results +show that the proposed TA-HGAT has the best performance +compared to ten baseline models on two real-world datasets. +Index Terms—recommender system, graph neural network, +hyperbolic embedding +I. INTRODUCTION +Recommender systems have been an effective solution to +help users overcome the information overload on the Internet. +Many applications are developed based on this rationale, +including online retail [1], music streaming [2], and con- +tent sharing [3]. To better understand users, modeling their +browsing sessions is a useful solution as sessions indicate +their current interests. Session-based recommendation (SBR) +predicts the users’ next interested items by modeling users’ +sessions. Deep learning models, including Recurrent Neural +Networks (RNNs) [4], [5], Memory Networks [6], and Graph +Neural Networks (GNNs) [7], [8] are applied to this problem +and have achieved state-of-the-art performance. +Recently, the most influential works on dealing with SBR +are GNN-based methods. The GNN-based methods [7]–[12] +∗Both authors contributed equally to this research. +take each session as a graph to learn the items’ internal rela- +tionship and their complex transitions. The most representative +model is SR-GNN [7], which is the first work to apply GNN +on session-based recommendation and achieve state-of-the- +art performance. Based on SR-GNN, [10], [12] improve SR- +GNN with attention layers. [8], [11] consider the item order +in the session graph to build the models. [9], [13], [14] take +additional information such as global item relationship, item +categories, and user representations into account to devise +more extensive models. HCGR [15] models session graphs +into a hyperbolic space to extract hierarchical information. +Although the existing GNN-based methods have achieved +satisfactory performance, they still suffer from two limitations. +First, unlike sequence-based models, graph structure cannot +explicitly show the temporal information between items. Time +interval is a crucial feature and can significantly improve +the recommendation performance [16], [17], but it is ignored +in the existing graph-based SBR models. Moreover, though +modeling sessions into graphs has the advantage of learning +items complex transitions [7], the sequential relation between +items is unclear in the session graph because the beginning +and end of a session are ambiguous under the graph structure. +Second, according to [18], [19], graph data exhibits an un- +derlying non-Euclidean structure, and therefore, learning such +geometry in Euclidean spaces is not a proper choice. As a +result, some recent studies [15], [20], [21] reveal that the +real-world datasets of recommender systems usually exhibit +tree-like hierarchical structures, and hyperbolic spaces can +effectively capture such hierarchical information. Therefore, +it is worth trying to learn session graphs in hyperbolic spaces. +Hyperbolic spaces have the ability to model hierarchical +structure data because they expand faster than Euclidean +spaces. They can expand exponentially, but Euclidean spaces +only expand polynomially. Existing work [15] demonstrates +the hierarchical structure of session graphs. However, model- +ing session graphs in hyperbolic spaces is still under explo- +ration. First, time intervals indicate the correlation between +two items. Since hyperbolic embedding is a better match to +session graphs, it is necessary to define a new framework +to identify the time intervals in the edges of session graphs. +Second, learning users’ current interests in the graph is crucial, +but it is difficult to realize in hyperbolic spaces. Previous works +arXiv:2301.03780v1 [cs.IR] 10 Jan 2023 + +[6], [7] devise models in Euclidean space based on the last +item in the session. The last item plays an important role +in predicting the next item because it represents the users’ +current interest. However, it is more challenging to model this +feature in a hyperbolic space as the operations in hyperbolic +spaces are more complicated than the Euclidean space. Third, +when taking the time information into consideration, we can +not only make next-item recommendations, but also provide +recommendations based on a specific timestamp. +To tackle the above challenges, we propose Time-aware Hy- +perbolic Graph Attention Network (TA-HGAT), a hyperbolic +GNN considering the comprehensive time-relevant features. +Specifically, we project the item’s original features into a +Poincar´e ball space via a hyperbolic projection layer. Then, +we design a time-aware hyperbolic attention mechanism to +learn the time intervals and users’ current interests together +in a hyperbolic space. It includes two modules: hyperbolic +self-attention with time intervals and hyperbolic soft-attention +with users’ current interests. Finally, the model is trained via +an evolutionary loss to predict which item the user may be +interested in at a specific timestamp. All these three compo- +nents are based on a fully hyperbolic graph neural network +framework. +Here, we summarize our contributions as follows: +• To the best of our knowledge, this is the first paper that +models temporal information in a hyperbolic space to +improve the performance of the recommender system. We +go beyond the conventional Euclidean machine learning +models to model users’ time-relevant features in a more +delicate manner. +• We propose TA-HGAT, a hyperbolic GNN-based frame- +work with three main components: hyperbolic projection, +time-aware hyperbolic attention, and evolutionary loss. +These three components work together in an end-to-end +GNN to model items’ time intervals and users’ current +interests. In the end, our model provides a time-specific +recommendation. +• We conduct experiments on two real-world datasets and +compare our model with ten baseline models. The ex- +periment results demonstrate the effectiveness of the TA- +HGAT in MRR and Precision. +II. PRELIMINARY +A. Graph neural network +GNNs [22], [23] are designed to handle the structural graph +data. In GNNs, aggregation is the core operation to extract +structural knowledge. By aggregating neighboring informa- +tion, the central node can gain knowledge from its neighbors +passed through edges and learn the node embedding. GNNs +have been demonstrated to be powerful in learning node +embeddings, so they are widely used on many node-related +tasks such as node classification [23], graph classification [24], +and link prediction [22]. +Based on the aggregation operation, the forward propagation +of a GNN on graph G = (V, E) is to learn the embedding +of node vi ∈ V via aggregating its neighboring nodes. We +suppose that the initial node embedding of each node i is h(0) +i , +which generally is the feature of the node. In each hidden layer +of a GNN, the embedding of the central node h(l) +i +is learned +from the aggregated embedding of the neighboring nodes in +the previous hidden layer h(l−1) +i +. The process is described in +math as follows: +h(l) +i += σ +� +W(l)(AGG +j∈Ni (h(l−1) +j +) +� +, +(1) +where Ni represents the set of all neighbors of node i in the +graph, including the node i itself. The aggregation function +AGG(·) integrates the neighboring information together. A +non-linear activation function σ, e.g., sigmoid or LeakyReLU, +is applied to generate the embedding of node i in the layer l. +Based on the vanilla GNN we mentioned above, GAT [25] +is proposed to improve GNNs with self-attention mechanism +[26]. Specifically, for all the neighbors of node i, we need to +learn the attention coefficients for all its neighbors to calculate +the importance of each neighbor node in the aggregation. +Suppose the attention coefficient of the node pair (i, j) is αij, +the process of learning αij is +αij = softmax(dij) = +exp(dij) +� +k∈Ni exp(dik), +(2) +where dij is the correlation between node i and j. dij here can +be the joint embeddings of node i and j, e.g., concatenation +of node embeddings or similarity of the node pair. +B. Hyperbolic spaces +In definition, hyperbolic space is a homogeneous space +with negative curvature. It is a smooth Riemannian manifold, +which can be modeled in several hyperbolic geometric models, +including Poincar´e ball model [27], Klein model [28], Lorentz +model [29], etc. In this paper, we choose the Poincar´e ball +model because the distance between two points grows expo- +nentially, which fits well with the hierarchical structure of +the session graph. Formally, the space of the d-dimensional +Poincar´e ball Pd +c is defined as +Pd +c = {x ∈ Rd, c∥x∥<1}, +(3) +where c is the radius of the ball and x is any point in manifold +P. If c = 0, then Pd +c = Rd and the ball is equal to the +Euclidean surface. In this paper, we set c = 1. The tangent +space TxP is a d-dimensional vector space approximating P +around x, which is isomorphic to the Euclidean space. With +the exponential map, a vector in the Euclidean space can be +mapped to the hyperbolic space. The logarithmic map is the +inverse of the exponential map, which projects the vector back +to the Euclidean space. +In hyperbolic spaces, the fundamental mathematical oper- +ations of neural networks (e.g., addition and multiplication) +are different from those in Euclidean space. In this paper, we +choose M¨obius transformation as an algebraic operation for +studying hyperbolic geometry. For a pair of random vectors +(a, b), we list the operations that will be used in our model +as follows: + +• M¨obius addition ⊕ [30] is to perform addition operation +of a and b. +a ⊕ b = (1 + 2⟨a, b⟩ + ∥b∥2)a + (1 − ∥a∥2)b +1 + 2⟨a, b⟩ + ∥a∥2∥b∥2 +. +(4) +• M¨obius matrix-vector multiplication ⊗ [31] is employed +to transform a with matrix W. +W ⊗ a = tanh(∥Wa∥ +∥a∥ +tanh−1(∥a∥)), +(5) +• M¨obius scalar multiplication ⊗ is the multiplication of a +scalar α with a vector b. +α ⊗ b = tanh(α tanh−1(∥b∥)) b +∥b∥ +(6) +• Exponential map transforms a from the Euclidean space +to a chosen point x in a hyperbolic space. +expx(a) = x ⊕ (tanh(λx∥a∥ +2 +) a +∥a∥), +(7) +• Logarithmic map projects the vector a back to the Eu- +clidean space. +logx(a) = 2 +λx +arctanh(∥ − x ⊕ a∥) +−x ⊕ a +∥ − x ⊕ a∥ +(8) +• λx is the conformal factor. +λx = +2 +1 − ∥x∥2 . +(9) +III. MODEL +In this section, we present the framework of our pro- +posed Time-aware Hyperbolic Graph Attention Network (TA- +HGAT), which is designed to model the temporal information +in the hyperbolic session graph. First, we define the session- +based recommendation task. Then we illustrate the three main +components of the model: hyperbolic projection, time-aware +hyperbolic attention, and hyperbolic evolutionary loss. These +three components train the model with time-relevant features +and provide the recommendation results given a specific +timestamp. The overall structure of TA-HGAT is shown in +Figure1. +A. Problem definition +Session-based recommendation (SBR) is to predict the item +a user will click next based on the user-item interaction +sessions. Generally, it models the user’s short-term browsing +session data to learn the user’s current interest. Here we +formulate the SBR problem mathematically as below. +In the SBR problem, a session is denoted as S = {v1, v2, ·· +·, vn} ordered by timestamps. Each v in S is an item, and +the item set is Vs, which consists of all unique items in this +session. To model the session into a directed graph, we take +all items as nodes and the item-item sequential dependency as +the edges to construct the session graph. The graph is denoted +as Gs = (Vs, Es), where Vs, Es are the node and edge sets, +respectively. Each edge connects two consecutive items, which +is formulated as e = (vt−1, vt). Our target is to learn the +embeddings of items and the session and generate the ranking +of the items that the user may be interested in at the next +timestamp. +B. Hyperbolic projection +In GNN, each node needs input as the initial embedding. +Accommodated to SBR, the input of a GNN is the feature of +items such as category or description. The initial embedding +of item i is h0 +i . However, most feature embedding methods +are based on the Euclidean space. To make the item features +available in the hyperbolic space, we use the exponential map +defined in Eq. 7 to project the initial item embeddings to +the hyperbolic space. Specifically, the projection process is +formulated as +mi = expx(h0 +i ), +(10) +where mi is the mapped embedding in the hyperbolic space +and x is the chosen point in the tangent space. +To achieve a high-level latent representation of the node +features, we also add a linear transformation parameterized by +a weight matrix W1 ∈ Rd′×d, where d′ is the dimension of mi +and d is the dimension of the node’s final embedding. Please +note that W1 is a shared weight matrix for all nodes. M¨obius +matrix-vector multiplication defined in Eq. 4 is employed to +transform mi and the process is +h1 +i = W1 ⊗ mi, +(11) +where h1 +i is the transformed embedding, which is also used +as the initial node embedding in the following steps. +C. Time-aware hyperbolic attention +According to [15], [20], [32], [33], embedding users and +items in hyperbolic spaces is a significant improvement of +graph-based recommender systems. However, none of these +works model the time intervals and users’ current interests in +hyperbolic spaces. Our proposed model TA-HGAT is the first +attempt to solve the problem, in which time-aware hyperbolic +attention is the core component. It is composed of two +attention layers: 1) Hyperbolic self-attention in the aggregation +process, which considers time intervals between items; 2) +Hyperbolic soft-attention in the session embedding learning, +which models the user’s current interest. +1) Hyperbolic self-attention with time intervals: According +to Section II-A, a key step in graph attention is to learn the +attention coefficient αij for each node pair (i, j). αij means +the importance of the neighbors to the central node. To learn +the αij, unlike the traditional attention networks which apply +linear transformation [25] or inner product [26], here we use +the distance of the node embeddings in the hyperbolic space. +Specifically, we denote the distance of node pair (i, j) as +(hi, hj), which is calculated as +d(hl +i, hl +j) = arcosh(1 + 2 +∥hl +i − hl +j∥2 +(1 − ∥hl +i∥2)(1 − ∥hl +j∥2)). +(12) + +... +v5 +v1 +v6 +v7 +t' +t' +t' +v2 +v1 +v3 +v7 +v4 +v5 +v6 +v1 +v2 +v3 +v5 +v6 +v4 +v7 +Hyperbolic Projection +... +Time-aware Hyperbolic Attention +v2 +v1 +v3 +t' +t' +v1 +v2 +v5 +v4 +t' +t' +t' +Hyperbolic Self-attention +with Time Intervals +Hyperbolic Soft-attention +with Users' Current Interests +Hyper bolic Attention +Networ k +s +vn +vn-1 +Hyperbolic +Evolutionary Loss +v1 +v2 +v3 +v5 +v6 +v4 +v7 +v7 +s +Fig. 1. Illustration of TA-HGAT. First, it builds directed session graphs based on the session sequences, and then projects the embeddings from the Euclidean +space to the hyperbolic space. Next, hyperbolic self-attention is adopted to aggregate neighboring information and time intervals t′. After that, each session +graph is represented as a session embedding using a hyperbolic soft-attention mechanism. Finally, TA-HGAT predicts top-k items that are most likely to be +clicked at the next timestamp for each session. +Then with the node distances, we further learn the attention +coefficient αij of node i with all its neighbors (including itself) +Ni as +αij = softmax(dij) = +exp(dij) +� +k∈Ni exp(dik), +(13) +The reason that we use distance in the hyperbolic space to +calculate attention coefficients is because of two advantages. +First, attention coefficients in Euclidean spaces are usually +calculated by linear transformation [25] or inner product [26], +which fail to meet the triangle inequality. In hyperbolic space, +the learned attention coefficients are able to meet this criterion +and preserve the transitivity among nodes. Second, the atten- +tion coefficient of the node i with itself is αii = d(hi, hi) = 0, +so the effect of the central node itself will not affect the +calculation of attention coefficients. +After we achieve attention coefficients, the next step is +to aggregate the node embeddings to learn the central node +embedding of the next layer. Here the learned attention co- +efficients serve as the weights applied to the embeddings of +neighbor nodes. The process is formulated as +hl+1 +i += σ( +⊕ +� +j∈Ni +αij ⊗ hl +j), +(14) +where �⊕ is the M¨obius addition of the weighted neighbor +node embeddings and σ is a nonlinear function such as +sigmoid and LeakyReLU. Different from Eq. 11, the ⊗ in +Eq. 14 is M¨obius scalar multiplication defined in Eq. 6. +To integrate the temporal information into the attention +layer, the core idea is to incorporate the time intervals into +the aggregation process. Specifically, we transform the time +intervals to the vectors in the hyperbolic space and combine +the time vectors with the neighbor node embeddings for ag- +gregation. As time intervals are continuous values, we project +the time interval values into vectors with a mapping function. +The mapping process is +ht′ = wt ⊗ (t+ − t), +(15) +where t′ = t+ − t is the time interval, ⊗ here is M¨obius +matrix-vector multiplication, and wt is the transition vector +to project the time interval to a vector. In this paper, if two +items have multiple time intervals between them, we choose +the closest one. This process is done in the data preprocessing +part before modeling. +Motivated by TransE [34], time-aware hyperbolic attention +translates the neighbor node embedding to the central node +embedding via temporal information, so the joint embedding +of nodes embedding and time embedding is generated by +M¨obius addition, which is represented as hl +j ⊕ ht′. +In Eq. 14, all neighbors of the central node i are aggregated +by M¨obius addition. As the M¨obius addition is complicated +and consumes more computation resources than the addition +in the Euclidean space, here we simplify the calculation in +Eq. 14 using the logarithmic map to project the embeddings +into a tangent space (Euclidean space) to conduct aggregation +operation. Then the embeddings are projected back to the +hyperbolic manifold with the exponential map. Therefore, we +can re-write the aggregation process in Eq. 14 as +hl+1 +i += exp +� +σ +� � +j∈Ni +log(αij ⊗ (hl +j ⊕ ht′)) +�� +. +(16) +2) Hyperbolic soft-attention with users’ current interests: +In the process above, we update the embedding of node i with +its neighbors and time intervals. To make recommendations +based on the learned node embeddings, we also need to know +the global embedding of the session graph by aggregating +all node embeddings. Instead of simply adding all node +embeddings together, we also provide another solution to learn +the graph embedding while considering users’ current interests +based on the most recent interacted items. + +Understanding users’ current interests are one of the main +tasks in SBR. In the previous studies [6], [7], [12], the last +item in the session is the most related feature in this task. +To learn from the correlation of the last item p with each +of the other items in the session, we adopt a soft-attention +mechanism to generate attention coefficients for item p with all +other items, which represent the importance of items w.r.t. the +current timestamp. The learning process of the global session +embedding hs is +βpq = x⊺ ⊗ σ +� +W2 ⊗ hp) ⊕ (W3 ⊗ hq) ⊕ c +� +, +(17) +hs = exp +� +σ +� � +q∈Vs +log(βpq ⊗ hq) +�� +, +(18) +where βpq is the attention coefficient of item p to another +item q in the session S. x ∈ Rd and W2, W3 ∈ Rd×d are +weight matrices. hs is the session embedding that contains +the session graph structure, temporal information, and user’s +current intent, so we can use hs to infer the user’s next +interaction in our next step. +D. Hyperbolic evolutionary loss +Here we introduce how to leverage evolutionary loss to +provide recommendations given a specific timestamp. Unlike +other works [3], [35], our evolutionary loss is also fully +hyperbolic. +1) Evolution formulas: The core idea of evolutionary loss +is to predict the future session and next-item embeddings +given a future timestamp and then make recommendations. +The prediction results of evolutionary loss do not rely on the +sequences like RNN-based models [4], [5] but are based on +the final embeddings learned by the TA-HGAT. +As hs is the predicted session embedding in the future, we +also need an estimated future session embedding to measure +whether the predicted embedding is accurate. Assume that the +growth of the session embedding is smooth. The embedding +vector of the session evolves in a contiguous space. Therefore, +we devise a projection function to infer the future session +embedding based on the element-wise product of the previous +embedding and the time interval. The embedding projection of +session S after current time t to the future time t+ is defined +as follows: +�ht+ +s += σ +� +ht +s ⊙ (1 ⊕ ht′) +� +, +(19) +where 1 ∈ Rd is a vector with all elements 1 and ⊙ is M¨obius +element-wise product. ht′ is the time interval vector, which +is learned in the same way as Eq. 15. The 1 vector is to +provide the minimum difference between the last and next +session embeddings. With this projection function, the future +session embedding grows in a smooth trajectory w.r.t. the time +interval. +After learning the projected embedding �ht+ +s +of the session +S, the next step is to apply another projection function to gen- +erate the future embedding of the next item v, which is denoted +as �ht+ +v . The projected future item embedding is composed of +three components: the projected session embedding, the last +item embedding, and the time interval, which are learned in +the previous steps. Here, we define the projection formula of +next item v as +�ht+ +v = σv +� +(W4 ⊗ �ht+ +s ) ⊕ (W5 ⊗ hvn) ⊕ ht′ +� +, +(20) +where W4 and W5 denote the weight matrix. +2) Loss function: With the above projection functions, we +can achieve the estimated future embeddings of the session and +the next item. They are utilized as ground truth embeddings +in our loss function. To train the model, the loss function is +designed to minimize the distances between model-generated +embeddings ht +s, hvn and estimated ground truth embeddings +�ht+ +s , �ht+ +v +at each interaction time t. Also, another constraint +for the item embeddings is necessary to avoid overfitting. We +constrain the distance between the embeddings of the most re- +cent two items vn−1 and vn to ensure the last item embeddings +are consistent with the previous one. This constraint assumes +that the last and next items reflect similar user intent, and the +session embedding tends to be stable in a short time. Finally, +the loss function is as follows: +L = +� +(s,v,t)∈{Si}I +i=0 +d(�ht+ +v , hvn) ⊕ +� +λs ⊗ d(�ht+ +s , ht +s) +� +⊕ +� +λv ⊗ d(hvn, hvn−1) +� +, +(21) +where {St}I +i=0 denotes all sessions in the datasets, and λs +and λv are smooth coefficients, which are used to prevent +the embeddings of the session and items from deviating too +much during the update process. d(·) is the hyperbolic distance +function which is described in Eq. 12. +To make recommendations for a user, we calculate the +hyperbolic distances between the predicted item embedding +obtained from the loss function and all other item embeddings. +Then the nearest top-k items are what we predict for the user. +Compared with traditional BPR loss [36], the evolutionary +loss is more suitable for time-aware recommendations because +it takes time intervals into account. As a result, the changing +trajectories are modeled by this loss [3], and it can make more +precise recommendations for the next item given a specific +timestamp. +IV. EXPERIMENTS +In this section, we describe the experimental results on two +public datasets and compare our proposed TA-HGAT with ten +state-of-the-art baseline models. Our experiments are designed +to solve the following research questions: +• RQ1: How does TA-HGAT compare with other state-of- +the-art session-based recommendation models? +• RQ2: How do the two modules of time-aware hyperbolic +attention, i.e., hyperbolic self-attention with time intervals +and hyperbolic soft-attention with users’ current interests, +affect the performance of TA-HGAT? +• RQ3: How does the hyperbolic evolutionary loss compare +with other loss functions? +• RQ4: How is the influence of different hyper-parameters, +i.e. embedding dimensions? + +TABLE I +THE NUMBER OF ITEMS, TRAINING SESSIONS, TESTING SESSIONS, THE +AVERAGE LENGTH, AND CLICKS FOR EACH DATASET. +Datasets +Items +train sessions +test sessions +Avg. len +clicks +Diginetica +43,097 +719,470 +60,858 +5.12 +982,961 +Yoochoose1/64 +16,766 +369,859 +55,898 +6.16 +557,248 +Yoochoose1/4 +29,618 +5,917,746 +55,898 +5.71 +8,326,407 +A. Experiment settings +1) Datasets: We conduct our experiments on two widely +used public datasets: Yoochoose and Diginetica. The statistics +of these datasets are listed in Table I. +• Yoochoose1 is a public dataset released by the RecSys +Challenge 2015, which contains click streams from yoo- +choose.com within 6 months. +• Diginetica2 is obtained from the CIKM Cup 2016. We +use the item categories to initialize the item embeddings. +2) Evaluation Metrics: We evaluate the performance of our +model with Mean Reciprocal Rank (MRR@K) and Precision +(P@K) in the comparison experiments. +MRR@K considers the position of the target item in the +list of recommended items. It is set to 0 if the target item is +not in the top-k of the ranking list, or otherwise is calculated +as follows: +MRR@K = 1 +N +N +� +i=1 +1 +Rank(vt), +(22) +where vt is the target item and N is the number of test +sequences in the dataset. +P@K measures whether the target item is included in the +top-k list of recommended items, which is calculated as +P@K = nhit +N +(23) +3) Implementation: Our model 3 is implemented with Py- +Torch 1.12.1 [37] and CUDA 10.2. In the testing phase, we +take the interval between the session’s last timestamp and +the testing item’s timestamp as a part of the input to obtain +the recommendation list. This setting is different from other +baseline models as they cannot deal with temporal information. +In fact, this setting meets the actual situation in the industry +because our model can provide recommendations as soon as +the user logs into the website, and we can easily obtain the +real-time time interval. +B. Performance comparison (RQ1) +To demonstrate the effectiveness of TA-HGAT, we conduct +experiments on two public datasets and compare the model +with ten state-of-the-art baseline models. +1https://www.kaggle.com/datasets/chadgostopp/recsys-challenge-2015 +2https://competitions.codalab.org/competitions/11161 +3The datasets and codes will be available after accepted +TABLE II +EXPERIMENTS ON DIGINETICA AND YOOCHOOSE DATASETS COMPARE +TA-HGAT WITH TEN BASELINE MODELS BASED ON THE TOP-20 OF THE +RANKING LIST IN MEAN RECIPROCAL RANK (MRR@20) AND PRECISION +(P@20). THE BOLD AND UNDERLINED NUMBERS ON EACH DATASET AND +METRIC REPRESENT THE BEST AND SECOND-BEST RESULTS, +RESPECTIVELY. ”IMPROV.” REFERS TO THE MINIMUM IMPROVEMENT +AMONG ALL BASELINES. +Models +Diginetica +Yoochoose 1/64 +Yoochoose 1/4 +MRR@20 +P@20 +MRR@20 +P@20 +MRR@20 +P@20 +S-POP +13.68 +21.06 +18.35 +30.44 +17.75 +27.08 +FPMC +8.92 +31.55 +15.01 +45.62 +- +- +GRU4REC +8.33 +29.45 +22.89 +60.64 +22.60 +59.53 +NARM +16.17 +49.70 +28.63 +68.32 +29.23 +69.73 +STAMP +14.32 +45.64 +29.67 +68.74 +30.00 +70.44 +SR-GNN +17.59 +50.73 +30.94 +70.57 +31.89 +71.36 +TAGNN +18.03 +51.31 +31.12 +71.02 +32.03 +71.51 +HCGR +18.51 +52.47 +31.46 +71.13 +32.39 +71.66 +NISER+ +18.72 +53.39 +31.61 +71.27 +31.80 +71.80 +SGNN-HN +19.45 +55.67 +32.61 +72.06 +32.55 +72.85 +TA-HGAT +19.73 +56.28 +32.90 +72.75 +32.94 +73.56 +Improv. +1.44% +1.10 % +0.89% +0.96% +1.20% +0.97% +1) Baseline models: +• S-POP takes the most popular items of each session as +the recommended list. +• FPMC [38] is a Markov chain-method for sequential +recommendation, which only takes the item sequences +in session-based recommendation since user features are +unavailable. +• GRU4REC [39] is the first work that applies RNN to +the session-based recommendation to learn the sequential +dependency of items. +• NARM [5] utilizes an attention mechanism to model the +sequential behaviors and the user’s primary purpose with +global and local encoders. +• STAMP [6] employs an attention and memory mecha- +nism to learn the user’s preference and takes the last item +as recent intent in the session to make recommendations. +• SR-GNN [7] is the first work that model a session into +a graph. It resorts to the gated graph neural networks to +learn the complex item transitions in the sessions. +• TAGNN [12] improves SR-GNN by learning the interest +representation vector with different target items to im- +prove the performance of the model. +• NISER+ [40] handles the long-tail problem in SBR with +L2 normalization and dropout to alleviate the overfitting +problem. +• SGNN-HN [41] applies a star graph neural network to +consider the items without direct connections. +• HCGR [15] models the session graphs in hyperbolic +space and makes use of multi-behavior information to +improve performance. In our experiments, we don’t use +the behavior information as the datasets didn’t provide it +and we are modeling a more general scenario. +2) Result analysis: The complete experimental results of +the comparison study are shown in Table II. From the results, +we have the following observations: +• Our proposed TA-HGAT outperforms all baseline models + +on all datasets and metrics, which demonstrates the +effectiveness of the model. Besides, HCGR, which is +another hyperbolic graph-based SBR model, has achieved +better performance than the graph-based SBR model SR- +GNN but worse than our model. Compared to HCGR, our +model improves 4.3% and 4.1% on average over three +datasets on metrics MRR@20 and P@20, respectively. +HCGR is better than SR-GNN, indicating that hyperbolic +embeddings match session graphs. And the improvement +of TA-HGAT over HCGR shows the importance of tem- +poral information in the SBR task. +• In Table II, we also observe that our model has a better +performance on dataset Diginetica than Yoochoose. On +average, the performance of TA-HGAT on Diginetica +outperforms Yoochoose for 37.8% and 14.0% on metrics +MRR@20 and P@20, respectively. This phenomenon +may result from the initial features of items. In Diginetica, +each item has its category label, and we transform this +feature into a one-hot vector as the initial embedding of +the item. In HCGR, we model the initial feature to a +feature vector in the hyperbolic space, which is shown +in Eq. 10 and 11. Differences in performance between +Diginetica and Yoochoose indicate that the hyperbolic +embeddings have a better expression ability on the item +features. +C. Ablation study (RQ2) +In the TA-HGAT, we have two main modules in time-aware +hyperbolic attention: hyperbolic self-attention with time inter- +vals and hyperbolic soft-attention with users’ current interests. +In this section, we evaluate their effectiveness separately to +show the improvement compared with the ablation models +without these two modules. +We set up four separate ablation models to compare the +effectiveness of each attention layer. The first ablation model +is no-att, in which we remove both the attention layers and +only conduct the aggregation operations directly. The second +and third ones are self-att and soft-att, and these two ablation +models only include the self-attention and soft-attention layers, +respectively. The fourth one is TA-HGAT, which is the com- +plete model. The comparison results of the ablation models on +datasets Diginetica and Yoochoose are illustrated in Figure 2. +From Figure 2, we observe the following results: +• On both Diginetica and Yoochoose datasets, the non-att +performs worst, and TA-HGAT performs best. The results +show the effectiveness of the attention layers. This is +because the TA-HGAT makes full use of the temporal +information. Compared to the GNNs without temporal +information, our model builds the relations between items +with time intervals and also considers users’ current +interests. Hence, the rich information helps the model to +achieve better results. +• Self-att performs better than soft-att, which means time +intervals are relatively more meaningful than users’ cur- +rent interests. This phenomenon may be due to the fact +that users’ current interests are more complicated, so the +TABLE III +COMPARISON OF PERFORMANCE FOR DIFFERENT LOSS FUNCTIONS. +Loss +Diginetica +Yoochoose 1/64 +Yoochoose 1/4 +MRR@20 +P@20 +MRR@20 +P@20 +MRR@20 +P@20 +Softmax +19.38 +55.81 +32.67 +72.10 +32.72 +72.95 +BPR +19.43 +55.97 +32.53 +72.28 +32.66 +72.84 +TA-HGAT +19.73 +56.28 +32.90 +72.75 +32.94 +73.56 +last item cannot fully represent them. In contrast, the time +interval is a more straightforward feature, so our proposed +hyperbolic self-attention layer can handle this information +effectively. +D. Comparison of loss functions (RQ3) +In this section, we compare our proposed hyperbolic evolu- +tionary loss to conventional loss functions, i.e., BPR [36] and +softmax loss [7]. Because the learned session embeddings in +the output of our model are in the hyperbolic space, we need +to use the logarithmic map to project the embeddings back to +the Euclidean space before applying BPR and softmax loss. +The comparison results are shown in Table III. The hy- +perbolic evolutionary loss is denoted as TA-HGAT in the +table. From this table, we can find that the performance of +BPR and softmax loss is similar, but our proposed hyperbolic +evolutionary loss has a clear improvement compared to the +other losses. This observation demonstrates that considering +the specific timestamp is effective for the SBR task models +designed in hyperbolic space. +E. Hyperparameter analysis (RQ4) +The embedding dimension is the hyperparameter in our pro- +posed model, so we test the influence of different embedding +dimensions in this section. The embedding dimensions range +from 20 to 100. The results of the hyperparameter analysis are +illustrated in Figure 3. +It is observed that a proper embedding dimension is essen- +tial for learning the item and session representations. From +Figure 3, we can see that the Diginetica and Yoochoose +1/64 all achieve the best performance when the embedding +dimension is 60, and the best result of Yoochoose 1/4 is 80. +Because Yoochoose 1/4 is much larger than the other two +datasets, it indicates that larger datasets need larger embedding +space. +V. RELATED WORKS +A. Hyperbolic spaces +Recent research has shown that many types of complex +data exhibit a highly non-Euclidean structure [19]. In many +domains, e.g., natural language [42], computer vision [43], +and healthcare [44], data usually has a tree-like structure +or can be represented hierarchically. Since this type of data +contains an underlying hierarchical structure, capturing such +representations in Euclidean space is difficult. To solve this +problem, current studies are increasingly attracted by the idea +of building neural networks in Riemannian space, such as + +P@20 +MRR@20 +Metric +0 +10 +20 +30 +40 +50 +Value +non-att +Self-att +Soft-att +TA-HGAT +(a) Diginetica +P@20 +MRR@20 +Metric +0 +10 +20 +30 +40 +50 +60 +70 +Value +non-att +Self-att +Soft-att +TA-HGAT +(b) Yoochoose 1/64 +P@20 +MRR@20 +Metric +0 +10 +20 +30 +40 +50 +60 +70 +Value +non-att +Self-att +Soft-att +TA-HGAT +(c) Yoochoose 1/4 +Fig. 2. +The ablation study of TA-HGAT. ’non-att’ is our model without attention layers. ’Self-att’ and ’Soft-att’ are composed of only self-attention and +soft-attention layers, respectively. TA-HGAT is the complete model. +(a) Diginetica MRR +(b) Yoochoose MRR +(c) Diginetica Precision +(d) Yoochoose Precision +Fig. 3. The hyperparameter analysis of the embedding dimensions. +the hyperbolic space, which is a homogeneous space with +constant negative curvature [27]. Compared with Euclidean +space, hyperbolic space in which the volume of a ball grows +exponentially with radius instead of growing polynomially. +Because of its powerful representation ability, hyperbolic +space has been applied in many areas. For instance, [45] +learns word and sentence embeddings in hyperbolic space in +an unsupervised manner from text corpora. [46] demonstrates +that hyperbolic embeddings are beneficial for visual data. [47] +proposes Hyperbolic Graph Convolutional Neural Networks, +which combines the expressiveness of GCNs and hyperbolic +geometry to learn graph representations. These works show +the potential and advantages of hyperbolic space in learning +hierarchical structures of complex data. +Based on the performance of hyperbolic space in these +fields, it is natural for researchers to think of applying hyper- +bolic learning to recommender systems. [48] justifies the use +of hyperbolic representations for neural recommender systems. +[49] proposes HyperML to bridge the gap between Euclidean +and hyperbolic geometry in recommender systems through a +metric learning approach. [21] proposes a hyperbolic GCN +model for collaborative filtering. [50] presents HyperSoRec, a +novel graph neural network (GNN) framework with multiple- +aspect learning for social recommendation. +B. Session-based Recommendation +Session-based Recommendation (SBR) has increasingly en- +gaged attention in both industry and academia due to its +effectiveness in modeling users’ current interests. In the recent +SBR studies, there are mainly three types of methods that +apply deep learning to SBR and have achieved state-of-the-art +performance, which are sequence-based [4], [51], attention- +based [5], [6] and graph-based models [7]. GRU4REC [4] +is the most representative work in the sequence-based SBR +models. It employs GRU, a variant of RNN, to model the +item sequences and make the next-item prediction. Follow- +ing GRU4REC, some other papers [39], [51], [52] improve +it with data augmentation, hierarchical structure, and top-k +gains. Attention-based models aim to learn the importance of +different items in the session and make the model focus on the +important ones. NARM [5] utilizes an attention mechanism to +model both local and global features of the session to learn +users’ interests. STAMP [6] combines the attention model and +memory network to learn the short-term priority of sessions. +Graph-based models connect the items in a graph to learn their +complex transitions. SR-GNN [7] is the first work that models +the sessions into graphs. It leverages the Gated Graph Neural +Network (GGNN) to model the session graphs and achieve +state-of-the-art performance. Based on SR-GNN, [10], [12] +improve SR-GNN with attention layers. [8], [11] consider the +item order in the session graph in the model. [9], [13], [14] +take additional information such as global item relationship, +item categories, and user representations into account to design +more extensive models. However, all these methods fail to +consider the hierarchical geometry of the session graphs and +the temporal information. + +19.73 +Dataset +Diginetica +19.72 +19.71 +19.70 +Value +19.69 +19.68 +19.67 +19.66 +19.65 +20 +40 +60 +80 +100 +DimensionDataset +32.94 +Yoochoose1/4 +Yoochoose1/64 +32.92 +32.90 +32.88 +Value +32.86 +32.84 +32.82 +32.80 +20 +40 +60 +80 +100 +Dimension56.28 +Dataset +Diginetica +56.26 +Value +56.24 +56.22 +56.20 +20 +40 +60 +80 +100 +Dimension73.6 +73.4 +73.2 +Dataset +alue +Yoochoose1/4 +Yoochoose1/64 +73.0 +72.8 +72.6 +20 +40 +60 +80 +100 +DimensionC. GNN-based Recommendation Models +GNNs have proven to be useful in different research fields +[53]–[58]. There also exist many works considering the graph +structures in data modeling of recommender systems. Gener- +ally, there are two main ways regarding the graph structure +and embedding space. One way is to model the user-item +interaction graph in Euclidean spaces. Among them, [59]–[61] +perform graph convolution on the user-item graph to explore +their interactions. [62]–[64] utilize layer-to-layer neighbor- +hood aggregation in GNNs to capture the high-order connec- +tions. [65] pre-trains user-user and item-item graphs separately +to learn the initial embeddings of the user-item interaction +graph. [35] models the changes in user-item interactions with a +dynamic graph and evolutionary loss. These works apply GNN +to learn from high-dimensional graph data and generate low- +dimensional node embeddings without feature engineering, but +the learned embeddings are all in Euclidean spaces, while +some graph data may be more suitable to other geometries +in the representation learning. +The other way is to model the recommendation graph in hy- +perbolic space to learn the hierarchical geometry. HGCF [21] +applies hyperbolic GCN to learn the node embeddings using +a user-item graph. Wang et al. [20] propose a fully hyperbolic +GCN where all operations are conducted in hyperbolic space. +Xu et al. [32] model the product graph in a knowledge graph +and learn the node embeddings in hyperbolic space. HCGR +[15] is a novel hyperbolic contrastive graph representation +learning method to make session-based recommendations. +None of these models utilize the time-relevant information in +the session graphs to improve the recommendation accuracy. +In this paper, we propose a novel framework incorporating +a time-aware graph attention mechanism in hyperbolic space, +which is specifically devised for the session-based recommen- +dation. +VI. CONCLUSION +Session-based Recommendation (SBR) is to predict users’ +next interested items based on their previous sessions. Existing +works model the graph structure in the sessions and have +achieved state-of-the-art performance. However, they fail to +consider the hierarchical geometry and temporal information +in the sessions. In this paper, we propose TA-HGAT, a +hyperbolic GNN-based model that considers the time interval +between items and users’ current interests. Experiment results +demonstrate that TA-HGAT outperforms other SBR models on +two real-world datasets. +For future work, we will extend our model to more general +recommender systems. The time intervals are not only in the +SBR problem, but also in almost all recommender systems. +As a result, we want to test how our model performs on other +recommendation problems, e.g., next-basket recommendation +and point-of-interest recommendation, where temporal infor- +mation plays a crucial role in providing recommendations. +VII. ACKNOWLEDGEMENT +This work is supported in part by NSF under grants III- +1763325, III-1909323, III-2106758, and SaTC-1930941. +REFERENCES +[1] G. Zhou, X. Zhu, C. Song, Y. Fan, H. Zhu, X. Ma, Y. Yan, J. Jin, H. Li, +and K. Gai, “Deep interest network for click-through rate prediction,” +in SIGKDD. +ACM, 2018, pp. 1059–1068. +[2] A. Van den Oord, S. Dieleman, and B. 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Achan, “Pre- +training recommender systems via reinforced attentive multi-relational +graph neural network,” in 2021 IEEE International Conference on Big +Data (Big Data). +IEEE, 2021, pp. 457–468. + diff --git a/A9E2T4oBgHgl3EQfRQfr/content/tmp_files/load_file.txt b/A9E2T4oBgHgl3EQfRQfr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..06ea7970512e1109d309b7ff82ffd44437832842 --- /dev/null +++ b/A9E2T4oBgHgl3EQfRQfr/content/tmp_files/load_file.txt @@ -0,0 +1,1049 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf,len=1048 +page_content='Time-aware Hyperbolic Graph Attention Network for Session-based Recommendation Xiaohan Li∗†, Yuqing Liu∗‡, Zheng Liu‡, Philip S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Yu‡ †Walmart Global Tech, Sunnyvale, CA, USA xiaohan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='li@walmart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='com ‡University of Illinois at Chicago, Chicago, IL, USA {yliu363, zliu212, psyu}@uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='edu Abstract—Session-based Recommendation (SBR) is to predict users’ next interested items based on their previous browsing sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Existing methods model sessions as graphs or sequences to estimate user interests based on their interacted items to make recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In recent years, graph-based methods have achieved outstanding performance on SBR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' However, none of these methods consider temporal information, which is a crucial feature in SBR as it indicates timeliness or currency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Besides, the session graphs exhibit a hierarchical structure and are demonstrated to be suitable in hyperbolic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' But few papers design the models in hyperbolic spaces and this direction is still under exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In this paper, we propose Time-aware Hyperbolic Graph Attention Network (TA-HGAT) — a novel hyperbolic graph neural network framework to build a session-based recommenda- tion model considering temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' More specifically, there are three components in TA-HGAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' First, a hyperbolic projection module transforms the item features into hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Second, the time-aware graph attention module models time intervals between items and the users’ current interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Third, an evolutionary loss at the end of the model provides an accurate prediction of the recommended item based on the given timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' TA-HGAT is built in a hyperbolic space to learn the hierarchical structure of session graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Experimental results show that the proposed TA-HGAT has the best performance compared to ten baseline models on two real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Index Terms—recommender system, graph neural network, hyperbolic embedding I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' INTRODUCTION Recommender systems have been an effective solution to help users overcome the information overload on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Many applications are developed based on this rationale, including online retail [1], music streaming [2], and con- tent sharing [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To better understand users, modeling their browsing sessions is a useful solution as sessions indicate their current interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Session-based recommendation (SBR) predicts the users’ next interested items by modeling users’ sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Deep learning models, including Recurrent Neural Networks (RNNs) [4], [5], Memory Networks [6], and Graph Neural Networks (GNNs) [7], [8] are applied to this problem and have achieved state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Recently, the most influential works on dealing with SBR are GNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The GNN-based methods [7]–[12] ∗Both authors contributed equally to this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' take each session as a graph to learn the items’ internal rela- tionship and their complex transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The most representative model is SR-GNN [7], which is the first work to apply GNN on session-based recommendation and achieve state-of-the- art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Based on SR-GNN, [10], [12] improve SR- GNN with attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [8], [11] consider the item order in the session graph to build the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [9], [13], [14] take additional information such as global item relationship, item categories, and user representations into account to devise more extensive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' HCGR [15] models session graphs into a hyperbolic space to extract hierarchical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Although the existing GNN-based methods have achieved satisfactory performance, they still suffer from two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' First, unlike sequence-based models, graph structure cannot explicitly show the temporal information between items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Time interval is a crucial feature and can significantly improve the recommendation performance [16], [17], but it is ignored in the existing graph-based SBR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Moreover, though modeling sessions into graphs has the advantage of learning items complex transitions [7], the sequential relation between items is unclear in the session graph because the beginning and end of a session are ambiguous under the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Second, according to [18], [19], graph data exhibits an un- derlying non-Euclidean structure, and therefore, learning such geometry in Euclidean spaces is not a proper choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' As a result, some recent studies [15], [20], [21] reveal that the real-world datasets of recommender systems usually exhibit tree-like hierarchical structures, and hyperbolic spaces can effectively capture such hierarchical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Therefore, it is worth trying to learn session graphs in hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Hyperbolic spaces have the ability to model hierarchical structure data because they expand faster than Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' They can expand exponentially, but Euclidean spaces only expand polynomially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Existing work [15] demonstrates the hierarchical structure of session graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' However, model- ing session graphs in hyperbolic spaces is still under explo- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' First, time intervals indicate the correlation between two items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Since hyperbolic embedding is a better match to session graphs, it is necessary to define a new framework to identify the time intervals in the edges of session graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Second, learning users’ current interests in the graph is crucial, but it is difficult to realize in hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Previous works arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='03780v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='IR] 10 Jan 2023 [6], [7] devise models in Euclidean space based on the last item in the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The last item plays an important role in predicting the next item because it represents the users’ current interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' However, it is more challenging to model this feature in a hyperbolic space as the operations in hyperbolic spaces are more complicated than the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Third, when taking the time information into consideration, we can not only make next-item recommendations, but also provide recommendations based on a specific timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To tackle the above challenges, we propose Time-aware Hy- perbolic Graph Attention Network (TA-HGAT), a hyperbolic GNN considering the comprehensive time-relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Specifically, we project the item’s original features into a Poincar´e ball space via a hyperbolic projection layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Then, we design a time-aware hyperbolic attention mechanism to learn the time intervals and users’ current interests together in a hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' It includes two modules: hyperbolic self-attention with time intervals and hyperbolic soft-attention with users’ current interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Finally, the model is trained via an evolutionary loss to predict which item the user may be interested in at a specific timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' All these three compo- nents are based on a fully hyperbolic graph neural network framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Here, we summarize our contributions as follows: To the best of our knowledge, this is the first paper that models temporal information in a hyperbolic space to improve the performance of the recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' We go beyond the conventional Euclidean machine learning models to model users’ time-relevant features in a more delicate manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' We propose TA-HGAT, a hyperbolic GNN-based frame- work with three main components: hyperbolic projection, time-aware hyperbolic attention, and evolutionary loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' These three components work together in an end-to-end GNN to model items’ time intervals and users’ current interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In the end, our model provides a time-specific recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' We conduct experiments on two real-world datasets and compare our model with ten baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The ex- periment results demonstrate the effectiveness of the TA- HGAT in MRR and Precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' PRELIMINARY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Graph neural network GNNs [22], [23] are designed to handle the structural graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In GNNs, aggregation is the core operation to extract structural knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' By aggregating neighboring informa- tion, the central node can gain knowledge from its neighbors passed through edges and learn the node embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' GNNs have been demonstrated to be powerful in learning node embeddings, so they are widely used on many node-related tasks such as node classification [23], graph classification [24], and link prediction [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Based on the aggregation operation, the forward propagation of a GNN on graph G = (V, E) is to learn the embedding of node vi ∈ V via aggregating its neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' We suppose that the initial node embedding of each node i is h(0) i , which generally is the feature of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In each hidden layer of a GNN, the embedding of the central node h(l) i is learned from the aggregated embedding of the neighboring nodes in the previous hidden layer h(l−1) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The process is described in math as follows: h(l) i = σ � W(l)(AGG j∈Ni (h(l−1) j ) � , (1) where Ni represents the set of all neighbors of node i in the graph, including the node i itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The aggregation function AGG(·) integrates the neighboring information together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' A non-linear activation function σ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=', sigmoid or LeakyReLU, is applied to generate the embedding of node i in the layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Based on the vanilla GNN we mentioned above, GAT [25] is proposed to improve GNNs with self-attention mechanism [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Specifically, for all the neighbors of node i, we need to learn the attention coefficients for all its neighbors to calculate the importance of each neighbor node in the aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Suppose the attention coefficient of the node pair (i, j) is αij, the process of learning αij is αij = softmax(dij) = exp(dij) � k∈Ni exp(dik), (2) where dij is the correlation between node i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' dij here can be the joint embeddings of node i and j, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=', concatenation of node embeddings or similarity of the node pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Hyperbolic spaces In definition, hyperbolic space is a homogeneous space with negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' It is a smooth Riemannian manifold, which can be modeled in several hyperbolic geometric models, including Poincar´e ball model [27], Klein model [28], Lorentz model [29], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In this paper, we choose the Poincar´e ball model because the distance between two points grows expo- nentially, which fits well with the hierarchical structure of the session graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Formally, the space of the d-dimensional Poincar´e ball Pd c is defined as Pd c = {x ∈ Rd, c∥x∥<1}, (3) where c is the radius of the ball and x is any point in manifold P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' If c = 0, then Pd c = Rd and the ball is equal to the Euclidean surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In this paper, we set c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The tangent space TxP is a d-dimensional vector space approximating P around x, which is isomorphic to the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' With the exponential map, a vector in the Euclidean space can be mapped to the hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The logarithmic map is the inverse of the exponential map, which projects the vector back to the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In hyperbolic spaces, the fundamental mathematical oper- ations of neural networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=', addition and multiplication) are different from those in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In this paper, we choose M¨obius transformation as an algebraic operation for studying hyperbolic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' For a pair of random vectors (a, b), we list the operations that will be used in our model as follows: M¨obius addition ⊕ [30] is to perform addition operation of a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' a ⊕ b = (1 + 2⟨a, b⟩ + ∥b∥2)a + (1 − ∥a∥2)b 1 + 2⟨a, b⟩ + ∥a∥2∥b∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' (4) M¨obius matrix-vector multiplication ⊗ [31] is employed to transform a with matrix W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' W ⊗ a = tanh(∥Wa∥ ∥a∥ tanh−1(∥a∥)), (5) M¨obius scalar multiplication ⊗ is the multiplication of a scalar α with a vector b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' α ⊗ b = tanh(α tanh−1(∥b∥)) b ∥b∥ (6) Exponential map transforms a from the Euclidean space to a chosen point x in a hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' expx(a) = x ⊕ (tanh(λx∥a∥ 2 ) a ∥a∥), (7) Logarithmic map projects the vector a back to the Eu- clidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' logx(a) = 2 λx arctanh(∥ − x ⊕ a∥) −x ⊕ a ∥ − x ⊕ a∥ (8) λx is the conformal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' λx = 2 1 − ∥x∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' (9) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' MODEL In this section, we present the framework of our pro- posed Time-aware Hyperbolic Graph Attention Network (TA- HGAT), which is designed to model the temporal information in the hyperbolic session graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' First, we define the session- based recommendation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Then we illustrate the three main components of the model: hyperbolic projection, time-aware hyperbolic attention, and hyperbolic evolutionary loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' These three components train the model with time-relevant features and provide the recommendation results given a specific timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The overall structure of TA-HGAT is shown in Figure1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Problem definition Session-based recommendation (SBR) is to predict the item a user will click next based on the user-item interaction sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Generally, it models the user’s short-term browsing session data to learn the user’s current interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Here we formulate the SBR problem mathematically as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In the SBR problem, a session is denoted as S = {v1, v2, ·· , vn} ordered by timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Each v in S is an item, and the item set is Vs, which consists of all unique items in this session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To model the session into a directed graph, we take all items as nodes and the item-item sequential dependency as the edges to construct the session graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The graph is denoted as Gs = (Vs, Es), where Vs, Es are the node and edge sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Each edge connects two consecutive items, which is formulated as e = (vt−1, vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Our target is to learn the embeddings of items and the session and generate the ranking of the items that the user may be interested in at the next timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Hyperbolic projection In GNN, each node needs input as the initial embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Accommodated to SBR, the input of a GNN is the feature of items such as category or description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The initial embedding of item i is h0 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' However, most feature embedding methods are based on the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To make the item features available in the hyperbolic space, we use the exponential map defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 7 to project the initial item embeddings to the hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Specifically, the projection process is formulated as mi = expx(h0 i ), (10) where mi is the mapped embedding in the hyperbolic space and x is the chosen point in the tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To achieve a high-level latent representation of the node features, we also add a linear transformation parameterized by a weight matrix W1 ∈ Rd′×d, where d′ is the dimension of mi and d is the dimension of the node’s final embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Please note that W1 is a shared weight matrix for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' M¨obius matrix-vector multiplication defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 4 is employed to transform mi and the process is h1 i = W1 ⊗ mi, (11) where h1 i is the transformed embedding, which is also used as the initial node embedding in the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Time-aware hyperbolic attention According to [15], [20], [32], [33], embedding users and items in hyperbolic spaces is a significant improvement of graph-based recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' However, none of these works model the time intervals and users’ current interests in hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Our proposed model TA-HGAT is the first attempt to solve the problem, in which time-aware hyperbolic attention is the core component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' It is composed of two attention layers: 1) Hyperbolic self-attention in the aggregation process, which considers time intervals between items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 2) Hyperbolic soft-attention in the session embedding learning, which models the user’s current interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 1) Hyperbolic self-attention with time intervals: According to Section II-A, a key step in graph attention is to learn the attention coefficient αij for each node pair (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' αij means the importance of the neighbors to the central node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To learn the αij, unlike the traditional attention networks which apply linear transformation [25] or inner product [26], here we use the distance of the node embeddings in the hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Specifically, we denote the distance of node pair (i, j) as (hi, hj), which is calculated as d(hl i, hl j) = arcosh(1 + 2 ∥hl i − hl j∥2 (1 − ∥hl i∥2)(1 − ∥hl j∥2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' (12) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=" v5 v1 v6 v7 t' t' t' v2 v1 v3 v7 v4 v5 v6 v1 v2 v3 v5 v6 v4 v7 Hyperbolic Projection ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=" Time-aware Hyperbolic Attention v2 v1 v3 t' t' v1 v2 v5 v4 t' t' t' Hyperbolic Self-attention with Time Intervals Hyperbolic Soft-attention with Users' Current Interests Hyper bolic Attention Networ k s vn vn-1 Hyperbolic Evolutionary Loss v1 v2 v3 v5 v6 v4 v7 v7 s Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Illustration of TA-HGAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' First, it builds directed session graphs based on the session sequences, and then projects the embeddings from the Euclidean space to the hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Next, hyperbolic self-attention is adopted to aggregate neighboring information and time intervals t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' After that, each session graph is represented as a session embedding using a hyperbolic soft-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Finally, TA-HGAT predicts top-k items that are most likely to be clicked at the next timestamp for each session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Then with the node distances, we further learn the attention coefficient αij of node i with all its neighbors (including itself) Ni as αij = softmax(dij) = exp(dij) � k∈Ni exp(dik), (13) The reason that we use distance in the hyperbolic space to calculate attention coefficients is because of two advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' First, attention coefficients in Euclidean spaces are usually calculated by linear transformation [25] or inner product [26], which fail to meet the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In hyperbolic space, the learned attention coefficients are able to meet this criterion and preserve the transitivity among nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Second, the atten- tion coefficient of the node i with itself is αii = d(hi, hi) = 0, so the effect of the central node itself will not affect the calculation of attention coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' After we achieve attention coefficients, the next step is to aggregate the node embeddings to learn the central node embedding of the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Here the learned attention co- efficients serve as the weights applied to the embeddings of neighbor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The process is formulated as hl+1 i = σ( ⊕ � j∈Ni αij ⊗ hl j), (14) where �⊕ is the M¨obius addition of the weighted neighbor node embeddings and σ is a nonlinear function such as sigmoid and LeakyReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Different from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 11, the ⊗ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 14 is M¨obius scalar multiplication defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To integrate the temporal information into the attention layer, the core idea is to incorporate the time intervals into the aggregation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Specifically, we transform the time intervals to the vectors in the hyperbolic space and combine the time vectors with the neighbor node embeddings for ag- gregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' As time intervals are continuous values, we project the time interval values into vectors with a mapping function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The mapping process is ht′ = wt ⊗ (t+ − t), (15) where t′ = t+ − t is the time interval, ⊗ here is M¨obius matrix-vector multiplication, and wt is the transition vector to project the time interval to a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In this paper, if two items have multiple time intervals between them, we choose the closest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' This process is done in the data preprocessing part before modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Motivated by TransE [34], time-aware hyperbolic attention translates the neighbor node embedding to the central node embedding via temporal information, so the joint embedding of nodes embedding and time embedding is generated by M¨obius addition, which is represented as hl j ⊕ ht′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 14, all neighbors of the central node i are aggregated by M¨obius addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' As the M¨obius addition is complicated and consumes more computation resources than the addition in the Euclidean space, here we simplify the calculation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 14 using the logarithmic map to project the embeddings into a tangent space (Euclidean space) to conduct aggregation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Then the embeddings are projected back to the hyperbolic manifold with the exponential map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Therefore, we can re-write the aggregation process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 14 as hl+1 i = exp � σ � � j∈Ni log(αij ⊗ (hl j ⊕ ht′)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' (16) 2) Hyperbolic soft-attention with users’ current interests: In the process above, we update the embedding of node i with its neighbors and time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To make recommendations based on the learned node embeddings, we also need to know the global embedding of the session graph by aggregating all node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Instead of simply adding all node embeddings together, we also provide another solution to learn the graph embedding while considering users’ current interests based on the most recent interacted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Understanding users’ current interests are one of the main tasks in SBR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In the previous studies [6], [7], [12], the last item in the session is the most related feature in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To learn from the correlation of the last item p with each of the other items in the session, we adopt a soft-attention mechanism to generate attention coefficients for item p with all other items, which represent the importance of items w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' the current timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The learning process of the global session embedding hs is βpq = x⊺ ⊗ σ � W2 ⊗ hp) ⊕ (W3 ⊗ hq) ⊕ c � , (17) hs = exp � σ � � q∈Vs log(βpq ⊗ hq) �� , (18) where βpq is the attention coefficient of item p to another item q in the session S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' x ∈ Rd and W2, W3 ∈ Rd×d are weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' hs is the session embedding that contains the session graph structure, temporal information, and user’s current intent, so we can use hs to infer the user’s next interaction in our next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Hyperbolic evolutionary loss Here we introduce how to leverage evolutionary loss to provide recommendations given a specific timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Unlike other works [3], [35], our evolutionary loss is also fully hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 1) Evolution formulas: The core idea of evolutionary loss is to predict the future session and next-item embeddings given a future timestamp and then make recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The prediction results of evolutionary loss do not rely on the sequences like RNN-based models [4], [5] but are based on the final embeddings learned by the TA-HGAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' As hs is the predicted session embedding in the future, we also need an estimated future session embedding to measure whether the predicted embedding is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Assume that the growth of the session embedding is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The embedding vector of the session evolves in a contiguous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Therefore, we devise a projection function to infer the future session embedding based on the element-wise product of the previous embedding and the time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The embedding projection of session S after current time t to the future time t+ is defined as follows: �ht+ s = σ � ht s ⊙ (1 ⊕ ht′) � , (19) where 1 ∈ Rd is a vector with all elements 1 and ⊙ is M¨obius element-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' ht′ is the time interval vector, which is learned in the same way as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The 1 vector is to provide the minimum difference between the last and next session embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' With this projection function, the future session embedding grows in a smooth trajectory w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' the time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' After learning the projected embedding �ht+ s of the session S, the next step is to apply another projection function to gen- erate the future embedding of the next item v, which is denoted as �ht+ v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The projected future item embedding is composed of three components: the projected session embedding, the last item embedding, and the time interval, which are learned in the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Here, we define the projection formula of next item v as �ht+ v = σv � (W4 ⊗ �ht+ s ) ⊕ (W5 ⊗ hvn) ⊕ ht′ � , (20) where W4 and W5 denote the weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 2) Loss function: With the above projection functions, we can achieve the estimated future embeddings of the session and the next item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' They are utilized as ground truth embeddings in our loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To train the model, the loss function is designed to minimize the distances between model-generated embeddings ht s, hvn and estimated ground truth embeddings �ht+ s , �ht+ v at each interaction time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Also, another constraint for the item embeddings is necessary to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' We constrain the distance between the embeddings of the most re- cent two items vn−1 and vn to ensure the last item embeddings are consistent with the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' This constraint assumes that the last and next items reflect similar user intent, and the session embedding tends to be stable in a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Finally, the loss function is as follows: L = � (s,v,t)∈{Si}I i=0 d(�ht+ v , hvn) ⊕ � λs ⊗ d(�ht+ s , ht s) � ⊕ � λv ⊗ d(hvn, hvn−1) � , (21) where {St}I i=0 denotes all sessions in the datasets, and λs and λv are smooth coefficients, which are used to prevent the embeddings of the session and items from deviating too much during the update process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' d(·) is the hyperbolic distance function which is described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To make recommendations for a user, we calculate the hyperbolic distances between the predicted item embedding obtained from the loss function and all other item embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Then the nearest top-k items are what we predict for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Compared with traditional BPR loss [36], the evolutionary loss is more suitable for time-aware recommendations because it takes time intervals into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' As a result, the changing trajectories are modeled by this loss [3], and it can make more precise recommendations for the next item given a specific timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' EXPERIMENTS In this section, we describe the experimental results on two public datasets and compare our proposed TA-HGAT with ten state-of-the-art baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Our experiments are designed to solve the following research questions: RQ1: How does TA-HGAT compare with other state-of- the-art session-based recommendation models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' RQ2: How do the two modules of time-aware hyperbolic attention, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=', hyperbolic self-attention with time intervals and hyperbolic soft-attention with users’ current interests, affect the performance of TA-HGAT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' RQ3: How does the hyperbolic evolutionary loss compare with other loss functions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' RQ4: How is the influence of different hyper-parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' embedding dimensions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' TABLE I THE NUMBER OF ITEMS, TRAINING SESSIONS, TESTING SESSIONS, THE AVERAGE LENGTH, AND CLICKS FOR EACH DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Datasets Items train sessions test sessions Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' len clicks Diginetica 43,097 719,470 60,858 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='12 982,961 Yoochoose1/64 16,766 369,859 55,898 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='16 557,248 Yoochoose1/4 29,618 5,917,746 55,898 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='71 8,326,407 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Experiment settings 1) Datasets: We conduct our experiments on two widely used public datasets: Yoochoose and Diginetica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The statistics of these datasets are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Yoochoose1 is a public dataset released by the RecSys Challenge 2015, which contains click streams from yoo- choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='com within 6 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Diginetica2 is obtained from the CIKM Cup 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' We use the item categories to initialize the item embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 2) Evaluation Metrics: We evaluate the performance of our model with Mean Reciprocal Rank (MRR@K) and Precision (P@K) in the comparison experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' MRR@K considers the position of the target item in the list of recommended items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' It is set to 0 if the target item is not in the top-k of the ranking list, or otherwise is calculated as follows: MRR@K = 1 N N � i=1 1 Rank(vt), (22) where vt is the target item and N is the number of test sequences in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' P@K measures whether the target item is included in the top-k list of recommended items, which is calculated as P@K = nhit N (23) 3) Implementation: Our model 3 is implemented with Py- Torch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='1 [37] and CUDA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In the testing phase, we take the interval between the session’s last timestamp and the testing item’s timestamp as a part of the input to obtain the recommendation list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' This setting is different from other baseline models as they cannot deal with temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In fact, this setting meets the actual situation in the industry because our model can provide recommendations as soon as the user logs into the website, and we can easily obtain the real-time time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Performance comparison (RQ1) To demonstrate the effectiveness of TA-HGAT, we conduct experiments on two public datasets and compare the model with ten state-of-the-art baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='com/datasets/chadgostopp/recsys-challenge-2015 2https://competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='codalab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='org/competitions/11161 3The datasets and codes will be available after accepted TABLE II EXPERIMENTS ON DIGINETICA AND YOOCHOOSE DATASETS COMPARE TA-HGAT WITH TEN BASELINE MODELS BASED ON THE TOP-20 OF THE RANKING LIST IN MEAN RECIPROCAL RANK (MRR@20) AND PRECISION (P@20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' THE BOLD AND UNDERLINED NUMBERS ON EACH DATASET AND METRIC REPRESENT THE BEST AND SECOND-BEST RESULTS, RESPECTIVELY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' ”IMPROV.” REFERS TO THE MINIMUM IMPROVEMENT AMONG ALL BASELINES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Models Diginetica Yoochoose 1/64 Yoochoose 1/4 MRR@20 P@20 MRR@20 P@20 MRR@20 P@20 S-POP 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='68 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='06 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='35 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='44 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='75 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='08 FPMC 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='44 SR-GNN 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='59 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='73 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='94 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='57 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='89 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='36 TAGNN 18.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='47 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='46 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='13 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='39 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='66 NISER+ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='72 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='39 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='61 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='27 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='80 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='80 SGNN-HN 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='45 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='67 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='61 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='06 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='55 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='85 TA-HGAT 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='73 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='28 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='90 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='75 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='94 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='56 Improv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='44% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='10 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='89% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='96% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='20% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='97% 1) Baseline models: S-POP takes the most popular items of each session as the recommended list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' FPMC [38] is a Markov chain-method for sequential recommendation, which only takes the item sequences in session-based recommendation since user features are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' GRU4REC [39] is the first work that applies RNN to the session-based recommendation to learn the sequential dependency of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' NARM [5] utilizes an attention mechanism to model the sequential behaviors and the user’s primary purpose with global and local encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' STAMP [6] employs an attention and memory mecha- nism to learn the user’s preference and takes the last item as recent intent in the session to make recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' SR-GNN [7] is the first work that model a session into a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' It resorts to the gated graph neural networks to learn the complex item transitions in the sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' TAGNN [12] improves SR-GNN by learning the interest representation vector with different target items to im- prove the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' NISER+ [40] handles the long-tail problem in SBR with L2 normalization and dropout to alleviate the overfitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' SGNN-HN [41] applies a star graph neural network to consider the items without direct connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' HCGR [15] models the session graphs in hyperbolic space and makes use of multi-behavior information to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In our experiments, we don’t use the behavior information as the datasets didn’t provide it and we are modeling a more general scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 2) Result analysis: The complete experimental results of the comparison study are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' From the results, we have the following observations: Our proposed TA-HGAT outperforms all baseline models on all datasets and metrics, which demonstrates the effectiveness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Besides, HCGR, which is another hyperbolic graph-based SBR model, has achieved better performance than the graph-based SBR model SR- GNN but worse than our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Compared to HCGR, our model improves 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='3% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='1% on average over three datasets on metrics MRR@20 and P@20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' HCGR is better than SR-GNN, indicating that hyperbolic embeddings match session graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' And the improvement of TA-HGAT over HCGR shows the importance of tem- poral information in the SBR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In Table II, we also observe that our model has a better performance on dataset Diginetica than Yoochoose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' On average, the performance of TA-HGAT on Diginetica outperforms Yoochoose for 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='8% and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='0% on metrics MRR@20 and P@20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' This phenomenon may result from the initial features of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In Diginetica, each item has its category label, and we transform this feature into a one-hot vector as the initial embedding of the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In HCGR, we model the initial feature to a feature vector in the hyperbolic space, which is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Differences in performance between Diginetica and Yoochoose indicate that the hyperbolic embeddings have a better expression ability on the item features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Ablation study (RQ2) In the TA-HGAT, we have two main modules in time-aware hyperbolic attention: hyperbolic self-attention with time inter- vals and hyperbolic soft-attention with users’ current interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In this section, we evaluate their effectiveness separately to show the improvement compared with the ablation models without these two modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' We set up four separate ablation models to compare the effectiveness of each attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The first ablation model is no-att, in which we remove both the attention layers and only conduct the aggregation operations directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The second and third ones are self-att and soft-att, and these two ablation models only include the self-attention and soft-attention layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The fourth one is TA-HGAT, which is the com- plete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The comparison results of the ablation models on datasets Diginetica and Yoochoose are illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' From Figure 2, we observe the following results: On both Diginetica and Yoochoose datasets, the non-att performs worst, and TA-HGAT performs best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The results show the effectiveness of the attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' This is because the TA-HGAT makes full use of the temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Compared to the GNNs without temporal information, our model builds the relations between items with time intervals and also considers users’ current interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Hence, the rich information helps the model to achieve better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Self-att performs better than soft-att, which means time intervals are relatively more meaningful than users’ cur- rent interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' This phenomenon may be due to the fact that users’ current interests are more complicated, so the TABLE III COMPARISON OF PERFORMANCE FOR DIFFERENT LOSS FUNCTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Loss Diginetica Yoochoose 1/64 Yoochoose 1/4 MRR@20 P@20 MRR@20 P@20 MRR@20 P@20 Softmax 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='38 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='81 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='67 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='10 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='72 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='95 BPR 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='43 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='97 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='53 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='28 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='66 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='84 TA-HGAT 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='73 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='28 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='90 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='75 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='94 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='56 last item cannot fully represent them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In contrast, the time interval is a more straightforward feature, so our proposed hyperbolic self-attention layer can handle this information effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Comparison of loss functions (RQ3) In this section, we compare our proposed hyperbolic evolu- tionary loss to conventional loss functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=', BPR [36] and softmax loss [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Because the learned session embeddings in the output of our model are in the hyperbolic space, we need to use the logarithmic map to project the embeddings back to the Euclidean space before applying BPR and softmax loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The comparison results are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The hy- perbolic evolutionary loss is denoted as TA-HGAT in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' From this table, we can find that the performance of BPR and softmax loss is similar, but our proposed hyperbolic evolutionary loss has a clear improvement compared to the other losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' This observation demonstrates that considering the specific timestamp is effective for the SBR task models designed in hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Hyperparameter analysis (RQ4) The embedding dimension is the hyperparameter in our pro- posed model, so we test the influence of different embedding dimensions in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The embedding dimensions range from 20 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The results of the hyperparameter analysis are illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' It is observed that a proper embedding dimension is essen- tial for learning the item and session representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' From Figure 3, we can see that the Diginetica and Yoochoose 1/64 all achieve the best performance when the embedding dimension is 60, and the best result of Yoochoose 1/4 is 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Because Yoochoose 1/4 is much larger than the other two datasets, it indicates that larger datasets need larger embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' RELATED WORKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Hyperbolic spaces Recent research has shown that many types of complex data exhibit a highly non-Euclidean structure [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In many domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=', natural language [42], computer vision [43], and healthcare [44], data usually has a tree-like structure or can be represented hierarchically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Since this type of data contains an underlying hierarchical structure, capturing such representations in Euclidean space is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' To solve this problem, current studies are increasingly attracted by the idea of building neural networks in Riemannian space, such as P@20 MRR@20 Metric 0 10 20 30 40 50 Value non-att Self-att Soft-att TA-HGAT (a) Diginetica P@20 MRR@20 Metric 0 10 20 30 40 50 60 70 Value non-att Self-att Soft-att TA-HGAT (b) Yoochoose 1/64 P@20 MRR@20 Metric 0 10 20 30 40 50 60 70 Value non-att Self-att Soft-att TA-HGAT (c) Yoochoose 1/4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The ablation study of TA-HGAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' ’non-att’ is our model without attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' ’Self-att’ and ’Soft-att’ are composed of only self-attention and soft-attention layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' TA-HGAT is the complete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' (a) Diginetica MRR (b) Yoochoose MRR (c) Diginetica Precision (d) Yoochoose Precision Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The hyperparameter analysis of the embedding dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' the hyperbolic space, which is a homogeneous space with constant negative curvature [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Compared with Euclidean space, hyperbolic space in which the volume of a ball grows exponentially with radius instead of growing polynomially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Because of its powerful representation ability, hyperbolic space has been applied in many areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' For instance, [45] learns word and sentence embeddings in hyperbolic space in an unsupervised manner from text corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [46] demonstrates that hyperbolic embeddings are beneficial for visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [47] proposes Hyperbolic Graph Convolutional Neural Networks, which combines the expressiveness of GCNs and hyperbolic geometry to learn graph representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' These works show the potential and advantages of hyperbolic space in learning hierarchical structures of complex data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Based on the performance of hyperbolic space in these fields, it is natural for researchers to think of applying hyper- bolic learning to recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [48] justifies the use of hyperbolic representations for neural recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [49] proposes HyperML to bridge the gap between Euclidean and hyperbolic geometry in recommender systems through a metric learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [21] proposes a hyperbolic GCN model for collaborative filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [50] presents HyperSoRec, a novel graph neural network (GNN) framework with multiple- aspect learning for social recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Session-based Recommendation Session-based Recommendation (SBR) has increasingly en- gaged attention in both industry and academia due to its effectiveness in modeling users’ current interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In the recent SBR studies, there are mainly three types of methods that apply deep learning to SBR and have achieved state-of-the-art performance, which are sequence-based [4], [51], attention- based [5], [6] and graph-based models [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' GRU4REC [4] is the most representative work in the sequence-based SBR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' It employs GRU, a variant of RNN, to model the item sequences and make the next-item prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Follow- ing GRU4REC, some other papers [39], [51], [52] improve it with data augmentation, hierarchical structure, and top-k gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Attention-based models aim to learn the importance of different items in the session and make the model focus on the important ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' NARM [5] utilizes an attention mechanism to model both local and global features of the session to learn users’ interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' STAMP [6] combines the attention model and memory network to learn the short-term priority of sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Graph-based models connect the items in a graph to learn their complex transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' SR-GNN [7] is the first work that models the sessions into graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' It leverages the Gated Graph Neural Network (GGNN) to model the session graphs and achieve state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Based on SR-GNN, [10], [12] improve SR-GNN with attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [8], [11] consider the item order in the session graph in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [9], [13], [14] take additional information such as global item relationship, item categories, and user representations into account to design more extensive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' However, all these methods fail to consider the hierarchical geometry of the session graphs and the temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='73 Dataset Diginetica 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='72 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='71 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='70 Value 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='69 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='68 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='67 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='66 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='65 20 40 60 80 100 DimensionDataset 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='94 Yoochoose1/4 Yoochoose1/64 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='92 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='90 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='88 Value 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='86 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='84 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='82 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='80 20 40 60 80 100 Dimension56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='28 Dataset Diginetica 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='26 Value 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='24 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='22 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='20 20 40 60 80 100 Dimension73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='2 Dataset alue Yoochoose1/4 Yoochoose1/64 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='6 20 40 60 80 100 DimensionC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' GNN-based Recommendation Models GNNs have proven to be useful in different research fields [53]–[58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' There also exist many works considering the graph structures in data modeling of recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Gener- ally, there are two main ways regarding the graph structure and embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' One way is to model the user-item interaction graph in Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Among them, [59]–[61] perform graph convolution on the user-item graph to explore their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [62]–[64] utilize layer-to-layer neighbor- hood aggregation in GNNs to capture the high-order connec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [65] pre-trains user-user and item-item graphs separately to learn the initial embeddings of the user-item interaction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [35] models the changes in user-item interactions with a dynamic graph and evolutionary loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' These works apply GNN to learn from high-dimensional graph data and generate low- dimensional node embeddings without feature engineering, but the learned embeddings are all in Euclidean spaces, while some graph data may be more suitable to other geometries in the representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The other way is to model the recommendation graph in hy- perbolic space to learn the hierarchical geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' HGCF [21] applies hyperbolic GCN to learn the node embeddings using a user-item graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [20] propose a fully hyperbolic GCN where all operations are conducted in hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' [32] model the product graph in a knowledge graph and learn the node embeddings in hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' HCGR [15] is a novel hyperbolic contrastive graph representation learning method to make session-based recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' None of these models utilize the time-relevant information in the session graphs to improve the recommendation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In this paper, we propose a novel framework incorporating a time-aware graph attention mechanism in hyperbolic space, which is specifically devised for the session-based recommen- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' CONCLUSION Session-based Recommendation (SBR) is to predict users’ next interested items based on their previous sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Existing works model the graph structure in the sessions and have achieved state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' However, they fail to consider the hierarchical geometry and temporal information in the sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' In this paper, we propose TA-HGAT, a hyperbolic GNN-based model that considers the time interval between items and users’ current interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' Experiment results demonstrate that TA-HGAT outperforms other SBR models on two real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' For future work, we will extend our model to more general recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' The time intervals are not only in the SBR problem, but also in almost all recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' As a result, we want to test how our model performs on other recommendation problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=', next-basket recommendation and point-of-interest recommendation, where temporal infor- mation plays a crucial role in providing recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfRQfr/content/2301.03780v1.pdf'} +page_content=' VII.' 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and Engineering, +University of WisconsinMadison, Madison, Wisconsin 53706, USA +(Dated: January 16, 2023) +Strontium titanate (STO), apart from being a ubiquitous substrate for complex-oxide heterostruc- +tures, possesses a multitude of strongly-coupled electronic and mechanical properties. Surface acous- +tic wave (SAW) generation and detection offers insight into electromechanical couplings that are +sensitive to quantum paraelectricity and other structural phase transitions. +Propagating SAWs +can interact with STO-based electronic nanostructures, in particular LaAlO3/SrTiO3 (LAO/STO). +Here we report generation and detection of SAW within LAO/STO heterointerfaces at cryogenic +temperatures (T ≥ 2 K) using superconducting interdigitated transducers (IDTs). The temper- +ature dependence shows an increase in the SAWs quality factor that saturates at T ≈ 8 K. The +effect of backgate tuning on the SAW resonance frequency shows the possible acoustic coupling +with the ferroelastic domain wall evolution. This method of generating SAWs provides a pathway +towards dynamic tuning of ferroelastic domain structures, which are expected to influence electronic +properties of complex-oxide nanostructures. Devices which incorporate SAWs may in turn help to +elucidate the role of ferroelastic domain structures in mediating electronic behavior. +I. +INTRODUCTION +Strontium titanate holds a unique place among the +growing family of complex-oxide heterostructures and +nanostructures [1]. +Apart from possessing a wealth of +physical phenomena–ferroelectricity [2, 3], ferroelasticity +[4, 5], superconductivity [6–8], high spin-to-charge inter- +conversion [9], large third-order optical susceptibility [10] +– STO also exhibits fascinating transport properties at +interfaces and within conductive nanostructures [11, 12]. +These latter properties arise when STO is capped with +a thin layer, often but not exclusively LaAlO3, which +results in electron doping near the STO interface [13]. +Conductive nanostructures of many types have been cre- +ated by “sketching” with a conductive atomic force mi- +croscope (c-AFM) tip [14] or ultra-low-voltage focused +electron beam [15]. The properties of these devices are +profoundly affected by the intrinsic behavior of STO and +are in many aspects not well understood. +One of the least well-understood property interrela- +tionships concerns the coupling between electronic, ferro- +electric, and ferroelastic degrees of freedom. STO is cen- +trosymmetric at room temperature with ABO3 cubic per- +ovskite structure. Upon cooling below ∼105 K [16, 17], +STO undergoes a cubic-to-tetragonal antiferrodistortive +(AFD) phase transition [18]. Further cooling to ∼35 K +∗ jlevy@pitt.edu +[19] gives rise to an incipient ferroelectric or “quantum +paraelectric” phase transition (QPE) in which the dielec- +tric constant ε saturates at ∼10 K [20]. +At cryogenic +temperature (T < 10 K), STO shows giant piezoelec- +tricity even larger than the best well-known piezoelectric +material such as quartz [21]. At microscopic scales, the +coupling between polar phases in STO and ferroelastic +domains [22] is quite strong and can be directly observed +using scanning single electron transistor microscopy [5]. +Piezoelectric distortions were found to be the result of +reorienting tetragonal domains, whose in-plane and out- +of-plane lattice constants differ by ∼ 10−3. +Surface acoustic waves (SAW), also known as Rayleigh +waves [23], arise from linear piezoelectric coupling, and +propagate parallel to the sample surface with its depth +comparable to the SAW wavelength. SAW propagation is +sensitive to both mechanical and electrical changes at the +sample surface, making it surface-sensitive and useful for +radio-frequency (RF) signal processing. A common tech- +nique to generate and detect SAW is to apply RF signals +to a pair of metallic inter-digitated transducers (IDT). +However, the complexity and subtlety of the STO struc- +ture with multiple phases make SAW generation and de- +tection difficult to achieve. With STO, DC fields have +been used to break cubic symmetry and generate polar- +ization above 150 K [24, 25] via electrostrictive effect. +To generate SAW, an extra piezoelectric layer, PZT, was +deposited on top of LAO [26], and SAW was observed +down to T =110 K. Below this temperature, the signal +disappeared and SAW generation and detection has not +arXiv:2301.05324v1 [cond-mat.str-el] 12 Jan 2023 + +2 +to our knowledge been reported in STO or LAO/STO or +at temperatures lower than T =105 K. +In this paper, we demonstrate direct SAW generation +and detection on LAO/STO surface at cryogenic temper- +atures using superconducting IDTs. The linearly-coupled +SAW shows an ultra-low phase velocity, indicating soft- +ening of STO crystal at low temperature and consistent +with earlier reports of large piezoelectric and electrostric- +tion coefficients [21]. The temperature at which the qual- +ity factor of the SAW resonator saturates coincides with +the quantum-paraelectric (QPE) transition temperature +(TQPE), showing that the quality factor Q is coupled to +the dielectric constant and can be used to identify the +onset of the quantum paraelectric phase. The resonance +frequency can be tuned with a backgate voltage. +The +tunability with applying the backgate at negative side +but not at the positive backgate side coincides with the +tuning effect of ferroelastic domain with the backgate +showing the coupling between ferroelastic domains and +surface phonon. The applied DC bias confirms the elec- +trostrictive effect from STO by showing the quadratic +tuning behavior. +II. +EXPERIMENT +LaAlO3 epitaxial films were grown on TiO2-terminated +STO (001) substrates by pulsed laser deposition [13]. The +thickness of LAO is fixed to 3.4 u.c., close to the critical +thickness of metal-insulator transition [27]. To form a +uniform-type single electrode IDT, an 80 nm thick film +of NbTiN is deposited on top of the LAO/STO, with +IDT fingers oriented along the (010) direction. Supercon- +ducting NbTiN is chosen as the IDT material for three +principal reasons: to help with impedance matching; to +maximize the transmission; and to minimize ohmic losses +and heating. A metallization ratio (m), defined as the fin- +ger width divided by the finger spacing, m ≡ w/(w + d), +is fixed such that m = 0.5 in all devices. SAW-related +experiments are carried out in a physical property mea- +surement system (PPMS) at temperatures T ≥ 2 K. Each +IDT is grounded on one side, and the other side is con- +nected to an input port of a vector network analyzer +(VNA) to enable two-port scattering parameter measure- +ments (Fig. 1(a)). Between the IDT and the VNA, a bias +tee is inserted on each side to allow a DC bias to be (Vbias) +applied between the IDT fingers. SAWs are generated by +an IDT, transmitted along the (100) direction, and de- +tected by the second IDT pair. To reduce contributions +from bulk acoustic waves, the LAO/STO sample bottom +surface is roughened and coated with silver epoxy as a +“soft conductor” [28]. The bottom conducting electrode +is also used to apply a voltage Vbg from the back of the +STO substrate. +Using a P = −10 dBm signal applied to the IDT, a +clear resonant feature can be seen at 127.5 MHz in the +reflection spectrum S11 (Fig. 1(c)), defined as the center +frequency fc. By contrast, in a control device in which +one side of the two comb structures in the IDT is miss- +ing there is no resonance (Fig. 1 (d)), demonstrating that +the resonance feature is a result of the paired comb struc- +tures patterned with NbTiN, and not due to bulk acous- +tic wave transmission or an electrical resonance from the +cable or other parts of the instrument. Meanwhile, the +SAW phase velocity is obtained from the measured fc by +v = fcλ. The wavelength λ is determined by the distance +between a pair of nearest IDT fingers with the same po- +larity. Here we have λ = 8 µm, giving a SAW velocity +on LAO/STO of v = 1, 020 m/s. The IDT comb struc- +ture generates SAW by converting the electrical energy +to elastic energy, causing a resonance dip in the reflection +signal. +The total quality factor Q is defined as +Q ≡ fc/B, +(1) +where fc is the center resonance frequency and B is the +half-power (-3 dB) bandwidth. The resonance spectrum +shows a quality factor Q = 17.5, which is consistent with +previous reports [25] on STO-based acoustic resonators +without resonance-enhanced structures (e.g., Bragg mir- +rors). Theoretically the bandwidth B can be determined +from the IDT geometry according to Ref. [29], +B ∼ 0.9fc/Np, +(2) +where Np = 16 is the number of comb pairs in the IDT. +Here the calculated bandwidth is 7.2 MHz is close to the +expected value of 7.3 MHz. +The < 2% difference can +come from the imperfect edge of IDT geometry related +to the mask-less photolithography precision. +The transmission spectrum S21 (Fig. 2 (a)) shows a +resonance peak at a frequency fc that coincides with the +reflection dip in S11, supporting the scenario that energy +is transmitted efficiently via SAW from the transmitting +IDT to the receiving IDT. When we sweep the temper- +ature, the resonance peak disappears sharply at temper- +atures larger than 13.7 K, corresponding to the temper- +ature Tc above which the NbTiN is no longer super- +conducting (see Supplementary material). The NbTiN +normal resistance 1.57 kΩ gives an impedance mismatch +which causes most of the power to be reflected and dis- +sipated both internally in the IDT and externally into +the transmission line leading to a sharp cut-off on the +transmission signal. +Notably, +the resonance frequency is temperature- +dependent. A quadratic scaling is observed between the +center frequency and temperature, with a lower fc at a +lower temperature. To understand this scaling, we may +consider a Helmholtz free energy of phonons, F, descrip- +tion of its equilibrium state, +F(t, ψ) = a(t) + b(t)ψ2 + c(t)ψ4 + · · · , +(3) +where ψ is the order parameter, and t = (T − Tc)/Tc +is the reduced temperature. When we only consider the +equilibrium states, we obtain +b(t)ψ + 2c(t)ψ3 = 0 + +3 +|ψ| ≈ (b1/2c0)|t|1/2. +The asymptotic expression for F becomes: +F(t, ψ) ≈ a0 − b2 +1 +4c0 +t2 + · · · +(4) +Therefore, the free energy is expected to scale quadrat- +ically with temperature. +The resonance frequency fc, +depending linearly on F, scales quadratically with tem- +perature (Fig. 2 (b)). +With a constant IDT geometry, such that the wave- +length λ is kept fixed, a smaller fc corresponds to a lower +SAW phase velocity v. We observe that lowering the tem- +perature reduces the SAW velocity. This trend contrasts +with results on most piezoelectric materials such as PZT, +in which SAW have been reported to increase as tem- +perature is lowered, because of the decreasing thermal +fluctuations and increasing stiffness of the sample [30]. +Both the piezoelectric coefficient (d311) and the elec- +trostriction coefficient (R311) of STO increase signifi- +cantly with decreasing temperature, especially for T < +10 K [21]. This finding implies both a softer crystal and +a more efficient electro-mechanical energy conversion at +low temperature. The quality factor, plotted versus tem- +perature in Figure 2 (c) first increases as temperature +is decreased, and then saturates at T ≈ 8 K. The sat- +uration temperature coincides with the STO quantum +paraelectric phase transition (TQPE) where the dielectric +permittivity ε saturates, described by Barrett’s formula +[20]. This correspondence indicates that SAW is sensitive +to the quantum paraelectric phase transition and its Q +factor is related to the ε variance. When T > Tc, Q drops +quickly to near zero due to the increasing resistance R +for the IDT. +To verify the linear dispersion of the SAW (v = fλ), +two different pairs of IDTs with different electrode spac- +ing are compared, keeping the metallization ratio fixed +such that m = 0.5. The IDT geometry is as shown in +Fig. 3 (a). The IDT finger widths w = d = 2 µm and +w = d = 3 µm correspond to wavelengths λ = 8 µm and +λ = 12 µm, respectively. The measured resonance fre- +quency fc, labeled with black arrows in Fig. 3 (b,c) shows +the expected inverse linear scaling with wavelength, pro- +viding further confirmation of the SAW origin of the res- +onance feature. +The transmission resonances in both devices show an +appreciable hardening as a function of back gate volt- +age Vbg (Fig. 3 (b,c)). The rise in acoustic velocity is +associated with induced ferroelectric displacement which +breaking the inversion symmetry of the crystal structure +and couples to the strain field S. This phenomenon can +be modeled using a Landau-Ginsburg-Devonshire (LGD) +free-energy expression, expanding in powers of the dis- +placement D up to the second order (Eq. 5) [25, 31–33]. +For STO when it is paraelectric, the dielectric electro- +mechanical response is described within LGD theory [34]. +F − F0 = −pSD + 1 +2χ−1D2 + 1 +2GSD2 + · · · +(5) +In Eq. 5, p is the piezoelectric tensor, χ is the dielectric +permittivity tensor and G is the electrostrictive tensor. +Surprisingly, the dependence of fc on Vbg is much +stronger when Vbg < 0 compared to Vbg > 0 (Fig. 4). +This dependence contrasts with pure electrical tuning of +the dielectric constant through Vbg, in which the tun- +ing effect is symmetric across zero bias [35]. The LAO +thickness is below the critical thickness for spontaneous +formation of a conductive LAO/STO interface [27]. The +interface remains insulating during the experiment, even +for the maximum backgate voltage that has been applied, +and thus carrier screening or other effects associated with +a gate-induced insulator-to-metal transition can be ruled +out. One is left with explanations that are tied to the +formation and evolution of ferroelastic domains. The mo- +tion of such domains under back gate bias is consistent +with prior imaging from Honig et al. [5] which showed +that under large negative backgate voltage, tetragonal +ferroelastic domains are observed, leading to the anoma- +lously large piezoelectricity at low temperature. The for- +mation and drifting of the ferroelastic domain under neg- +ative backgate voltages play an important role coupling +with the SAW. +To investigate how the magnitude of the SAW coupling +depends on the static bias across the IDT, we incorporate +a bias tee between the VNA port and IDT connection to +apply a dc bias Vbias between IDT neighboring fingers +with opposite polarity, and measure the change in the +resonance amplitude as a function of Vbias. The result +(Fig. 5 (a)) shows a quadratic relationship for S11 ampli- +tude. The quadratic dependence can be understood as +an electrostriction effect in which electric field couples to +strain up to the second order (Eq. 5). The Vbias induced +first-order electric field breaks the inversion symmetry of +STO, yielding a linear coupling. The scaling indicates +a built-in STO polarization that can be modulated with +an applied bias across the IDT. For comparison, port 2 is +not subject to a dc bias, and as a result no tuning of the +S22 amplitude (Fig. 5 (a)) is observed. When the applied +Vbias cancels the built-in polarization, the resonance am- +plitude is minimized, which happens Vbias ∼ −1 V. Sim- +ilar tuning is observed for the transmission signals S12 +and S21, as expected. +III. +DISCUSSION AND CONCLUSION +With an acoustic speed five orders lower than the speed +of light, a relatively short acoustic wavelength, and high +degree of surface sensitivity, SAW generation, propaga- +tion, and detection can be regarded as useful building +blocks for manipulating electronic and lattice degrees of +freedom in complex-oxide heterostructures and nanos- +tructures. +Specifically, SAW has the potential to con- +tribute to quantum information processing architectures +[29] both in superconducting qubits [36–38] and elec- +tron spin-based quantum computing architectures [39– +41]. Coupling the superconducting qubits with SAW can + +4 +help control and measure quantum states [42]. Further- +more, SAW generates moving potential wells with meso- +scopic scale which transport electron charges with spin +information propagating at speed of sound in a confined +one-dimensional channel, helping to meet architectural +challenges of long-range transport of spin information +[43–45]. At the same time, SAW manipulation of elec- +tronic properties may help provide insight into the nature +of correlated electronic phases such as superconductivity. +In conclusion, we demonstrate the direct generation +and detection of SAW on LAO/STO at cryogenic tem- +perature using superconducting IDTs. Spurious contri- +butions arising from possible bulk acoustic wave compo- +nents and electronic resonances from the instrument are +carefully ruled out. The SAW shows an ultra-low phase +velocity which reveals the coupling to the high permit- +tivity of the STO at low temperatures. The SAW quality +factor saturates at quantum paraelectric phase transition +temperature corroborating the related Q-factor with di- +electric permittivity. This method can thus be used to +probe behavior near the quantum phase transition. The +behavior is consistent with a linear electro-mechanical +coupling that is tightly coupled with the ferroelastic do- +main evolution. +Supporting Information +Supporting Information is available as Supplementary +information.pdf. +Acknowledgements +JL acknowledges support from the Vannevar Bush Fac- +ulty Fellowship program sponsored by the Basic Research +Office of the Assistant Secretary of Defense for Research +and Engineering and funded by the Office of Naval Re- +search through grant N00014-15-1-2847. +The work at +University of Wisconsin-Madison was supported by the +National Science Foundation under DMREF Grant No. +DMR-1629270, AFOSR FA9550-15-1-0334 and AOARD +FA2386-15-1-4046. This research is funded by the Gor- +don and Betty Moore Foundations EPiQS Initiative, +grant GBMF9065 to C.B.E., Vannevar Bush Faculty Fel- +lowship (N00014-20-1-2844 (C.B.E.). +Transport mea- +surement at the University of WisconsinMadison was +supported by the US Department of Energy (DOE), Of- +fice of Science, Office of Basic Energy Sciences (BES), +the Materials Sciences and Engineering (MSE) Division +under award number DE-FG02-06ER46327. +Conflict of Interest +The authors declare no conflict of interest. +Data Availability Statement +Data generated or analysed during this study are in- +cluded in this published article (and its supplementary +information). +[1] Y.-Y. Pai, A. Tylan-Tyler, P. Irvin, and J. Levy, Physics +of SrTiO3-based heterostructures and nanostructures: a +review, Rep. Prog. Phys. 81, 036503 (2018). +[2] W. J. Burke and R. J. Pressley, Stress induced ferroelec- +tricity in SrTiO3, Solid State Commun. 9, 191 (1971). +[3] O. Tikhomirov, H. Jiang, and J. 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Anderson, G. A. C. Jones, I. Farrer, and D. A. +Ritchie, On-demand single-electron transfer between dis- + +6 +tant quantum dots, Nature 477, 439 (2011). + +7 +VNA +DC +DC +SAW +(a) +(b) +(c) +(d) +FIG. 1. Surface Acoustic Wave (SAW) generated and detected by Interdigitated Transducers (IDTs). (a) Schematic diagram of +the experiment setup. The orange parts are IDTs. Blue circuits denote the bias tee inserted between vector network analyzer +(VNA) port and IDT. (b) Optical image of an IDT patterned with NbTiN. The black scale bar is 20 µm. (c) S11 from an +experiment device with normal IDT comb structure in pair. The resonances is observable. (d) Reflection signal S11 from a +control device without one side of IDT comb structure. There is no resonance observed from this control device. +T = 2K +(a) +(b) +(c) +TQPE +FIG. 2. Temperature dependence of resonance. (a) The upper intensity plot shows transmission signals with respect to the +temperature between 2 K and 16 K. The lower figure is a transmission signal line cut through 2 K temperature showing +the resonance peak. +(b) Temperature dependence of resonance center frequency and calculated SAW phase velocity (blue +diamonds). The red dashed line is a quadratic fit. (c) Quality factor Q = fc/B plotted with respect to the temperature. The +red arrow shows the STO quantum paraelectric saturation temperature. + +SSZSS20 +41.0 +100 +200 +f (MHz)13.8 ++ +- ++ +- +w +d +l +LAO +STO +(a) +(b) +(c) +FIG. 3. Resonance frequency shift due to different wavelengths and negative backgate voltages. (a) Schematic diagram of +NbTiN IDT geometry, where w is the finger width and d is the gap distance. +Wavelength λ is determined by the center +distance between two nearest same polarity fingers. (b) Transmission spectrum of Device “A” (λ = 8 µm) as a function of +Vbg. Black arrow denotes the resonance frequency. (c) Transmission spectrum of Device “B” (λ = 12 µm) as a function of Vbg. +Black arrow denotes the resonance frequency. All data taken at T = 2 K +(a) +(b) +FIG. 4. Positive and negative Vbg dependence of reflection spectrum and resonance frequency. (a) Reflection spectrum as a +function of Vbg. There is a resonance dip and a second harmonic dip observed in the spectrum. (b) Resonance center frequency +fc as a function of Vbg. The green region highlights where the resonance frequency can be tuned with negative Vbg. + +入=12μm^=8μm9 +(a) +(b) +FIG. 5. Reflection scattering parameters (S11, S22) measured as a function of Vbias applied across the IDT fingers in port +1, with zero bias at port 2. (a) Reflection amplitude for ports 1 (S11) and 2 (S22), referenced against the zero-bias value +Sii (Vbias = 0). (b)Transmission amplitudes (S12, S21), measured as a function of Vbias. + diff --git a/CNE4T4oBgHgl3EQf5g54/content/tmp_files/load_file.txt b/CNE4T4oBgHgl3EQf5g54/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e2c54953f2f7aafd3c97e19502047e4c75ef5cb --- /dev/null +++ b/CNE4T4oBgHgl3EQf5g54/content/tmp_files/load_file.txt @@ -0,0 +1,752 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf,len=751 +page_content='Surface acoustic wave generation and detection in quantum paraelectric regime of SrTiO3-based heterostructure Dengyu Yang1,2, Muqing Yu1,2, Yun-Yi Pai1,2, Patrick Irvin1,2, Hyungwoo Lee3, Kitae Eom3, Chang-Beom Eom3, and Jeremy Levy1,2∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Pittsburgh Quantum Institute, Pittsburgh, Pennsylvania 15260, USA and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Department of Materials Science and Engineering, University of WisconsinMadison, Madison, Wisconsin 53706, USA (Dated: January 16, 2023) Strontium titanate (STO), apart from being a ubiquitous substrate for complex-oxide heterostruc- tures, possesses a multitude of strongly-coupled electronic and mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Surface acous- tic wave (SAW) generation and detection offers insight into electromechanical couplings that are sensitive to quantum paraelectricity and other structural phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Propagating SAWs can interact with STO-based electronic nanostructures, in particular LaAlO3/SrTiO3 (LAO/STO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Here we report generation and detection of SAW within LAO/STO heterointerfaces at cryogenic temperatures (T ≥ 2 K) using superconducting interdigitated transducers (IDTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The temper- ature dependence shows an increase in the SAWs quality factor that saturates at T ≈ 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The effect of backgate tuning on the SAW resonance frequency shows the possible acoustic coupling with the ferroelastic domain wall evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' This method of generating SAWs provides a pathway towards dynamic tuning of ferroelastic domain structures, which are expected to influence electronic properties of complex-oxide nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Devices which incorporate SAWs may in turn help to elucidate the role of ferroelastic domain structures in mediating electronic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' INTRODUCTION Strontium titanate holds a unique place among the growing family of complex-oxide heterostructures and nanostructures [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Apart from possessing a wealth of physical phenomena–ferroelectricity [2, 3], ferroelasticity [4, 5], superconductivity [6–8], high spin-to-charge inter- conversion [9], large third-order optical susceptibility [10] – STO also exhibits fascinating transport properties at interfaces and within conductive nanostructures [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' These latter properties arise when STO is capped with a thin layer, often but not exclusively LaAlO3, which results in electron doping near the STO interface [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Conductive nanostructures of many types have been cre- ated by “sketching” with a conductive atomic force mi- croscope (c-AFM) tip [14] or ultra-low-voltage focused electron beam [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The properties of these devices are profoundly affected by the intrinsic behavior of STO and are in many aspects not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' One of the least well-understood property interrela- tionships concerns the coupling between electronic, ferro- electric, and ferroelastic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' STO is cen- trosymmetric at room temperature with ABO3 cubic per- ovskite structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Upon cooling below ∼105 K [16, 17], STO undergoes a cubic-to-tetragonal antiferrodistortive (AFD) phase transition [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Further cooling to ∼35 K ∗ jlevy@pitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='edu [19] gives rise to an incipient ferroelectric or “quantum paraelectric” phase transition (QPE) in which the dielec- tric constant ε saturates at ∼10 K [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' At cryogenic temperature (T < 10 K), STO shows giant piezoelec- tricity even larger than the best well-known piezoelectric material such as quartz [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' At microscopic scales, the coupling between polar phases in STO and ferroelastic domains [22] is quite strong and can be directly observed using scanning single electron transistor microscopy [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Piezoelectric distortions were found to be the result of reorienting tetragonal domains, whose in-plane and out- of-plane lattice constants differ by ∼ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Surface acoustic waves (SAW), also known as Rayleigh waves [23], arise from linear piezoelectric coupling, and propagate parallel to the sample surface with its depth comparable to the SAW wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' SAW propagation is sensitive to both mechanical and electrical changes at the sample surface, making it surface-sensitive and useful for radio-frequency (RF) signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' A common tech- nique to generate and detect SAW is to apply RF signals to a pair of metallic inter-digitated transducers (IDT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' However, the complexity and subtlety of the STO struc- ture with multiple phases make SAW generation and de- tection difficult to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' With STO, DC fields have been used to break cubic symmetry and generate polar- ization above 150 K [24, 25] via electrostrictive effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' To generate SAW, an extra piezoelectric layer, PZT, was deposited on top of LAO [26], and SAW was observed down to T =110 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Below this temperature, the signal disappeared and SAW generation and detection has not arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='05324v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='str-el] 12 Jan 2023 2 to our knowledge been reported in STO or LAO/STO or at temperatures lower than T =105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' In this paper, we demonstrate direct SAW generation and detection on LAO/STO surface at cryogenic temper- atures using superconducting IDTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The linearly-coupled SAW shows an ultra-low phase velocity, indicating soft- ening of STO crystal at low temperature and consistent with earlier reports of large piezoelectric and electrostric- tion coefficients [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The temperature at which the qual- ity factor of the SAW resonator saturates coincides with the quantum-paraelectric (QPE) transition temperature (TQPE), showing that the quality factor Q is coupled to the dielectric constant and can be used to identify the onset of the quantum paraelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The resonance frequency can be tuned with a backgate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The tunability with applying the backgate at negative side but not at the positive backgate side coincides with the tuning effect of ferroelastic domain with the backgate showing the coupling between ferroelastic domains and surface phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The applied DC bias confirms the elec- trostrictive effect from STO by showing the quadratic tuning behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' EXPERIMENT LaAlO3 epitaxial films were grown on TiO2-terminated STO (001) substrates by pulsed laser deposition [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The thickness of LAO is fixed to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='4 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=', close to the critical thickness of metal-insulator transition [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' To form a uniform-type single electrode IDT, an 80 nm thick film of NbTiN is deposited on top of the LAO/STO, with IDT fingers oriented along the (010) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Supercon- ducting NbTiN is chosen as the IDT material for three principal reasons: to help with impedance matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' to maximize the transmission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' and to minimize ohmic losses and heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' A metallization ratio (m), defined as the fin- ger width divided by the finger spacing, m ≡ w/(w + d), is fixed such that m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='5 in all devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' SAW-related experiments are carried out in a physical property mea- surement system (PPMS) at temperatures T ≥ 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Each IDT is grounded on one side, and the other side is con- nected to an input port of a vector network analyzer (VNA) to enable two-port scattering parameter measure- ments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Between the IDT and the VNA, a bias tee is inserted on each side to allow a DC bias to be (Vbias) applied between the IDT fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' SAWs are generated by an IDT, transmitted along the (100) direction, and de- tected by the second IDT pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' To reduce contributions from bulk acoustic waves, the LAO/STO sample bottom surface is roughened and coated with silver epoxy as a “soft conductor” [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The bottom conducting electrode is also used to apply a voltage Vbg from the back of the STO substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Using a P = −10 dBm signal applied to the IDT, a clear resonant feature can be seen at 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='5 MHz in the reflection spectrum S11 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 1(c)), defined as the center frequency fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' By contrast, in a control device in which one side of the two comb structures in the IDT is miss- ing there is no resonance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 1 (d)), demonstrating that the resonance feature is a result of the paired comb struc- tures patterned with NbTiN, and not due to bulk acous- tic wave transmission or an electrical resonance from the cable or other parts of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Meanwhile, the SAW phase velocity is obtained from the measured fc by v = fcλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The wavelength λ is determined by the distance between a pair of nearest IDT fingers with the same po- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Here we have λ = 8 µm, giving a SAW velocity on LAO/STO of v = 1, 020 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The IDT comb struc- ture generates SAW by converting the electrical energy to elastic energy, causing a resonance dip in the reflection signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The total quality factor Q is defined as Q ≡ fc/B, (1) where fc is the center resonance frequency and B is the half-power (-3 dB) bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The resonance spectrum shows a quality factor Q = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='5, which is consistent with previous reports [25] on STO-based acoustic resonators without resonance-enhanced structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=', Bragg mir- rors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Theoretically the bandwidth B can be determined from the IDT geometry according to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' [29], B ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='9fc/Np, (2) where Np = 16 is the number of comb pairs in the IDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Here the calculated bandwidth is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='2 MHz is close to the expected value of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='3 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The < 2% difference can come from the imperfect edge of IDT geometry related to the mask-less photolithography precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The transmission spectrum S21 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 2 (a)) shows a resonance peak at a frequency fc that coincides with the reflection dip in S11, supporting the scenario that energy is transmitted efficiently via SAW from the transmitting IDT to the receiving IDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' When we sweep the temper- ature, the resonance peak disappears sharply at temper- atures larger than 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='7 K, corresponding to the temper- ature Tc above which the NbTiN is no longer super- conducting (see Supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The NbTiN normal resistance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='57 kΩ gives an impedance mismatch which causes most of the power to be reflected and dis- sipated both internally in the IDT and externally into the transmission line leading to a sharp cut-off on the transmission signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Notably, the resonance frequency is temperature- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' A quadratic scaling is observed between the center frequency and temperature, with a lower fc at a lower temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' To understand this scaling, we may consider a Helmholtz free energy of phonons, F, descrip- tion of its equilibrium state, F(t, ψ) = a(t) + b(t)ψ2 + c(t)ψ4 + · · · , (3) where ψ is the order parameter, and t = (T − Tc)/Tc is the reduced temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' When we only consider the equilibrium states, we obtain b(t)ψ + 2c(t)ψ3 = 0 3 |ψ| ≈ (b1/2c0)|t|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The asymptotic expression for F becomes: F(t, ψ) ≈ a0 − b2 1 4c0 t2 + · · · (4) Therefore, the free energy is expected to scale quadrat- ically with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The resonance frequency fc, depending linearly on F, scales quadratically with tem- perature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 2 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' With a constant IDT geometry, such that the wave- length λ is kept fixed, a smaller fc corresponds to a lower SAW phase velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' We observe that lowering the tem- perature reduces the SAW velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' This trend contrasts with results on most piezoelectric materials such as PZT, in which SAW have been reported to increase as tem- perature is lowered, because of the decreasing thermal fluctuations and increasing stiffness of the sample [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Both the piezoelectric coefficient (d311) and the elec- trostriction coefficient (R311) of STO increase signifi- cantly with decreasing temperature, especially for T < 10 K [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' This finding implies both a softer crystal and a more efficient electro-mechanical energy conversion at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The quality factor, plotted versus tem- perature in Figure 2 (c) first increases as temperature is decreased, and then saturates at T ≈ 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The sat- uration temperature coincides with the STO quantum paraelectric phase transition (TQPE) where the dielectric permittivity ε saturates, described by Barrett’s formula [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' This correspondence indicates that SAW is sensitive to the quantum paraelectric phase transition and its Q factor is related to the ε variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' When T > Tc, Q drops quickly to near zero due to the increasing resistance R for the IDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' To verify the linear dispersion of the SAW (v = fλ), two different pairs of IDTs with different electrode spac- ing are compared, keeping the metallization ratio fixed such that m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The IDT geometry is as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The IDT finger widths w = d = 2 µm and w = d = 3 µm correspond to wavelengths λ = 8 µm and λ = 12 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The measured resonance fre- quency fc, labeled with black arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 3 (b,c) shows the expected inverse linear scaling with wavelength, pro- viding further confirmation of the SAW origin of the res- onance feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The transmission resonances in both devices show an appreciable hardening as a function of back gate volt- age Vbg (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 3 (b,c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The rise in acoustic velocity is associated with induced ferroelectric displacement which breaking the inversion symmetry of the crystal structure and couples to the strain field S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' This phenomenon can be modeled using a Landau-Ginsburg-Devonshire (LGD) free-energy expression, expanding in powers of the dis- placement D up to the second order (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 5) [25, 31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' For STO when it is paraelectric, the dielectric electro- mechanical response is described within LGD theory [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' F − F0 = −pSD + 1 2χ−1D2 + 1 2GSD2 + · · · (5) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 5, p is the piezoelectric tensor, χ is the dielectric permittivity tensor and G is the electrostrictive tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Surprisingly, the dependence of fc on Vbg is much stronger when Vbg < 0 compared to Vbg > 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' This dependence contrasts with pure electrical tuning of the dielectric constant through Vbg, in which the tun- ing effect is symmetric across zero bias [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The LAO thickness is below the critical thickness for spontaneous formation of a conductive LAO/STO interface [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The interface remains insulating during the experiment, even for the maximum backgate voltage that has been applied, and thus carrier screening or other effects associated with a gate-induced insulator-to-metal transition can be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' One is left with explanations that are tied to the formation and evolution of ferroelastic domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The mo- tion of such domains under back gate bias is consistent with prior imaging from Honig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' [5] which showed that under large negative backgate voltage, tetragonal ferroelastic domains are observed, leading to the anoma- lously large piezoelectricity at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The for- mation and drifting of the ferroelastic domain under neg- ative backgate voltages play an important role coupling with the SAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' To investigate how the magnitude of the SAW coupling depends on the static bias across the IDT, we incorporate a bias tee between the VNA port and IDT connection to apply a dc bias Vbias between IDT neighboring fingers with opposite polarity, and measure the change in the resonance amplitude as a function of Vbias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The result (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 5 (a)) shows a quadratic relationship for S11 ampli- tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The quadratic dependence can be understood as an electrostriction effect in which electric field couples to strain up to the second order (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The Vbias induced first-order electric field breaks the inversion symmetry of STO, yielding a linear coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The scaling indicates a built-in STO polarization that can be modulated with an applied bias across the IDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' For comparison, port 2 is not subject to a dc bias, and as a result no tuning of the S22 amplitude (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 5 (a)) is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' When the applied Vbias cancels the built-in polarization, the resonance am- plitude is minimized, which happens Vbias ∼ −1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Sim- ilar tuning is observed for the transmission signals S12 and S21, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION With an acoustic speed five orders lower than the speed of light, a relatively short acoustic wavelength, and high degree of surface sensitivity, SAW generation, propaga- tion, and detection can be regarded as useful building blocks for manipulating electronic and lattice degrees of freedom in complex-oxide heterostructures and nanos- tructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Specifically, SAW has the potential to con- tribute to quantum information processing architectures [29] both in superconducting qubits [36–38] and elec- tron spin-based quantum computing architectures [39– 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Coupling the superconducting qubits with SAW can 4 help control and measure quantum states [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Further- more, SAW generates moving potential wells with meso- scopic scale which transport electron charges with spin information propagating at speed of sound in a confined one-dimensional channel, helping to meet architectural challenges of long-range transport of spin information [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' At the same time, SAW manipulation of elec- tronic properties may help provide insight into the nature of correlated electronic phases such as superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' In conclusion, we demonstrate the direct generation and detection of SAW on LAO/STO at cryogenic tem- perature using superconducting IDTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Spurious contri- butions arising from possible bulk acoustic wave compo- nents and electronic resonances from the instrument are carefully ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The SAW shows an ultra-low phase velocity which reveals the coupling to the high permit- tivity of the STO at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The SAW quality factor saturates at quantum paraelectric phase transition temperature corroborating the related Q-factor with di- electric permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' This method can thus be used to probe behavior near the quantum phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The behavior is consistent with a linear electro-mechanical coupling that is tightly coupled with the ferroelastic do- main evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Supporting Information Supporting Information is available as Supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Acknowledgements JL acknowledges support from the Vannevar Bush Fac- ulty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Re- search through grant N00014-15-1-2847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The work at University of Wisconsin-Madison was supported by the National Science Foundation under DMREF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' DMR-1629270, AFOSR FA9550-15-1-0334 and AOARD FA2386-15-1-4046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' This research is funded by the Gor- don and Betty Moore Foundations EPiQS Initiative, grant GBMF9065 to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=', Vannevar Bush Faculty Fel- lowship (N00014-20-1-2844 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Transport mea- surement at the University of WisconsinMadison was supported by the US Department of Energy (DOE), Of- fice of Science, Office of Basic Energy Sciences (BES), the Materials Sciences and Engineering (MSE) Division under award number DE-FG02-06ER46327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Conflict of Interest The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Data Availability Statement Data generated or analysed during this study are in- cluded in this published article (and its supplementary information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Surface Acoustic Wave (SAW) generated and detected by Interdigitated Transducers (IDTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (a) Schematic diagram of the experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The orange parts are IDTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Blue circuits denote the bias tee inserted between vector network analyzer (VNA) port and IDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (b) Optical image of an IDT patterned with NbTiN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The black scale bar is 20 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (c) S11 from an experiment device with normal IDT comb structure in pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The resonances is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (d) Reflection signal S11 from a control device without one side of IDT comb structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' There is no resonance observed from this control device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' T = 2K (a) (b) (c) TQPE FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Temperature dependence of resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (a) The upper intensity plot shows transmission signals with respect to the temperature between 2 K and 16 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The lower figure is a transmission signal line cut through 2 K temperature showing the resonance peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (b) Temperature dependence of resonance center frequency and calculated SAW phase velocity (blue diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The red dashed line is a quadratic fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (c) Quality factor Q = fc/B plotted with respect to the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The red arrow shows the STO quantum paraelectric saturation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' SSZSS20 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='0 100 200 f (MHz)13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content='8 + + w d l LAO STO (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Resonance frequency shift due to different wavelengths and negative backgate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (a) Schematic diagram of NbTiN IDT geometry, where w is the finger width and d is the gap distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Wavelength λ is determined by the center distance between two nearest same polarity fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (b) Transmission spectrum of Device “A” (λ = 8 µm) as a function of Vbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Black arrow denotes the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (c) Transmission spectrum of Device “B” (λ = 12 µm) as a function of Vbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Black arrow denotes the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' All data taken at T = 2 K (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Positive and negative Vbg dependence of reflection spectrum and resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (a) Reflection spectrum as a function of Vbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' There is a resonance dip and a second harmonic dip observed in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (b) Resonance center frequency fc as a function of Vbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' The green region highlights where the resonance frequency can be tuned with negative Vbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 入=12μm^=8μm9 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' Reflection scattering parameters (S11, S22) measured as a function of Vbias applied across the IDT fingers in port 1, with zero bias at port 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (a) Reflection amplitude for ports 1 (S11) and 2 (S22), referenced against the zero-bias value Sii (Vbias = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} +page_content=' (b)Transmission amplitudes (S12, S21), measured as a function of Vbias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQf5g54/content/2301.05324v1.pdf'} diff --git a/CtAzT4oBgHgl3EQfiP1R/content/tmp_files/2301.01496v1.pdf.txt b/CtAzT4oBgHgl3EQfiP1R/content/tmp_files/2301.01496v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..68f5ccd405919bfa89c5af7ed5195687b68f0a78 --- /dev/null +++ b/CtAzT4oBgHgl3EQfiP1R/content/tmp_files/2301.01496v1.pdf.txt @@ -0,0 +1,1503 @@ +arXiv:2301.01496v1 [math.DS] 4 Jan 2023 +UNIQUE ERGODICITY OF SIMPLE SYMMETRIC RANDOM +WALKS ON THE CIRCLE +KLAUDIUSZ CZUDEK +Abstract. Fix an irrational number α and a smooth, positive, real function +p on the circle. If current position is x ∈ R/Z then in the next step jump to +x + α with probability p(x) or to x − α with probability 1 − p(x). In 1999 +Sinai has proven that if p is asymmetric (in certain sense) or α is Diophantine +then the Markov process possesses a unique stationary distribution. Next year +Conze and Guivarc’h showed the uniqueness of stationary distribution for an +arbitrary irrational angle α. +In this note we present a new proof of latter +result. +1. Introduction +Fix an irrational number α ∈ R, and consider the family of Markov processes +with the evolution governed by the transition kernel +(1.1) +p(x, ·) = p(x)δx+α + q(x)δx−α, +p : T × B(T) → [0, 1], +where B(S1) stands for the σ-algebra of Borel subsets of S1 and q(x) = 1 − p(x), +x ∈ T. We call the function p symmetric if +� +T +f(x)dx = 0, +where +(1.2) +f(x) = ln p(x) +q(x), +x ∈ T, +and asymmetric otherwise. We call a measure µ invariant for transition kernel (1.1) +if distributing the starting point according to µ makes the Markov process with this +transition kernel stationary (thus µ is called also often a stationary measure). Since +T is compact, the Krylov-Bogoliubov technique yields existence of an invariant +distribution for (1.1) for every choice of continuous p. However, it is far from being +obvious if there exists more than one invariant distribution. +The earliest paper known to the author dealing with similar (but still slightly +different) system was by Sine [Sin79]. +More recently it was proven by Sinai in +[Sin99] that if p ∈ C∞(T) is asymmetric or p ∈ C∞(T) is symmetric and α is +Diophantine then the uniqueness follows. +One year later Conze and Guivarc’h +proved in [CG00] that in the symmetric case +p(x) +q(x+α) ∈ BV implies uniqueness no +matter if α is Diophantine or not. The present paper contains another proof of the +latter statement assuming p ∈ C1 is symmetric. The advantage of the new proof is +that it gives more insight to the problem of mixing and the problem of uniqueness +2020 Mathematics Subject Classification. +Primary 37A50, 60F05. +Key words and phrases. random rotations, Diophantine approximation, random walk, circle. +1 + +2 +KLAUDIUSZ CZUDEK +in higher dimensional analogs (where T is replaced by Td). See Section 5 for more +details. +The strategy is based on Sinai’s. To explain it, fix x ∈ T and consider a Markov +process (Xn) started at x with transition kernel (1.1). It is evident that the process +can achieve only the points of the form x+jα, j ∈ Z. Thus to learn the distribution +of (Xn) on T we consider a Markov chain (ξn) on Z, started at 0, with +P(ξn+1 = k + 1|ξn = k) = p(x + kα) +and +P(ξn+1 = k − 1|ξn = k) = q(x + kα) +for n ≥ 0 and k ∈ Z. Let us now restrict to the symmetric case, which is in our scope +of interest. In that case the system on Z is recurrent. If p ∈ C∞(T) is symmetric +and α is Diophantine then the cohomological equation f(x) = g(x + α) − g(x), +where f is defined in (1.2), possesses a solution. Using the solution g we can easily +check that the measure with density h(z)/q(z) is invariant, where h = exp(g). Now +the whole difficulty in Sinai’s approach was to show the local limit theorem for (ξn) +on Z. More precisely, in the symmetric case Sinai has proven that +P(ξn = k) ∼ h(x + kα) +p(x + kα) +1 +√ +2πσ2n +exp −k2 +2nσ2 , +for some σ > 0 and all x ∈ T, where ∼ means the ratio of both sides tends to one. +With this fact one can show that +Eϕ(Xn) → +� +T +ϕ(z)h(z) +q(z) dz, +which easily implies the unique ergodicity (in fact it’s even a stronger property +called mixing or stability). +Unfortunately we cannot follow exactly the same path when generalizing result +to all irrational α. Recently Dolgopyat, Fayad and Saprykina [DFS21] have proven +that if α is Liouville then the behaviour of (ξn) on Z is erratic for the generic +choice of smooth and symmetric p (see Theorems A-E therein). +In particular, +neither annealed, nor quenched central limit theorem holds (see Corollary D and +G therein). However, we can still modify something in Sinai’s idea to get desired +assertion. The main result of this work is the following. +Theorem 1. If p ∈ C1(T) is symmetric and separated from 0 and 1 (i.e. 0 < +p(x) < 1 for each x ∈ T) then there exists exactly one invariant measure for the +transition kernel (1.1). +As it was mentioned, the proof is some sense is in the spirit of Sinai’s. We still +concentrate on the process (ξn) on Z but instead of proving the local limit theorem +we focus on the limits +lim +n→∞ +P(ξ0 = k) + · · · + P(ξn−1 = k) +P(ξ0 = m) + · · · + P(ξn−1 = m), +where k, m ∈ Z are two states. The problem of existence of such limits for general +(including countable space, null recurrent) Markov chains was raised by Kolmogorov +in 1936 and answered two years later by Doeblin [Doe38] without identification of +the value of the limit. It has been done only later by Chung [Chu50]. It turns out +we can define certain infinite measure on Z, k �−→ ax,k (depending on x ∈ T since + +UNIQUE ERGODICITY OF SIMPLE SYMMETRIC... +3 +(ξn) depends on x ∈ T) such that the limit above tends to ax,k/ax,m for arbitrary +two states k and m. +In Section 2 we identify the measure k �−→ ax,k on Z and reproduce the proof +of Doeblin ratio limit theorem. In Section 3 it is proved that if one takes a large +interval of integers A of length q and projects the measure k �−→ ax,k to the circle +(by identifying k with x + kα) then what we obtain is, after normalization and up +to ε, independent of the choice of the interval A and the point x, provided q is +sufficiently large. Section 4 contains how to complete the proof of Theorem 1 using +the above results. Section 5 contains some final remarks. +2. Acknowledgments and personal remarks +When I proved the main theorem I wasn’t aware of Conze, Guivarc’h result. +After discovering it, I started thinking if my proof can be used to show something +more. +I realized the advantage of mine is it can be modified to obtain mixing +(assuming p is C1 and symmetric, no matter if α is Diophantine or not). Then +I gave several talks about it, e.g. in the conference “Probabilistic techniques in +random and time-varying dynamical systems”, Luminy 3-7.10.2022 or in the KTH +dynamical systems seminar, where I announced “mixing” result. Although I still +think this result is true, I didn’t predicted certain difficulties in the proof and I +need more time and effort to complete it. Meanwhile I’m publishing the proof of +uniqueness. It’s not going to be submitted to any journal. +The research was supported by the Polish National Science Center grant Pre- +ludium UMO-2019/35/N/ST1/02363. +3. Basic facts about symmetric random walks on Z +Fix x ∈ T and define (ξn) to be the Markov process on Z, started at 0, with +P(ξn+1 = k + 1|ξn = k) = p(x + kα) +and +P(ξn+1 = k − 1|ξn = k) = q(x + kα) +for n ≥ 0 and k ∈ Z. In present section we are going to prove recurrence of this +random walk and some related results. We say that (ξn) is recurrent if almost +surely there exists n > 0 with ξn = 0. We say (ξn) is null recurrent if it is recurrent +and the expected time of the first return to 0 is infinite. +Proposition 1. If p is of bounded variation, symmetric and separated from 0 and +1 then the process (ξn) is recurrent. Moreover, for every r > 0 there exists m0 that +can be chosen uniformly in x ∈ T such that the expected number of returns of (ξn) +to zero until m0 is greater than r, i.e. +P(ξ1 = 0) + · · · + P(ξn = 0) = E +� +1{0}(ξ1) + · · · + 1{0}(ξn) +� +> r +for n ≥ m0, whatever x ∈ T. +Proof. To show the recurrence of (ξn), we reproduce the analysis from [DFS21], +Section 3.2. Let us define a function M : Z → R by M(0) = 0, M(1) = 1, +M(n) = 1 + +n−1 +� +k=1 +k +� +j=1 +q(x + jα) +p(x + jα) +for n ≥ 2, + +4 +KLAUDIUSZ CZUDEK +and +M(−n) = − +n +� +k=0 +k +� +j=0 +p(x − jα) +q(x − jα) +for n ≥ 1. +To avoid complicated notation, we do not stress the dependence of M on x. It can +be checked that (M(ξn)) is a martingale. Let a < 0 < b and let us define τ to be +the first moment when (ξn) hits a or b. By Doob’s theorem EM(ξτ) = M(ξ0) = 0. +On the other hand +EM(ξτ) = M(a)P(ξτ = a) + M(b)P(ξτ = b) += M(a)P(ξτ = a) + M(b)(1 − P(ξτ = a)), +which combined with EM(ξτ) = 0 yields +P(ξτ = a) = +M(b) +M(b) − M(a). +If ξτ = a then (ξn) returns to 0 before hitting b. Setting a = −1 above we get +therefore +(3.1) +P +� +(ξn) returns to 0 before hitting b +� +≥ +M(b) +M(b) − M(−1). +Similarly +(3.2) +P +� +(ξn) returns to 0 before hitting a +� +≥ +−M(a) +M(1) − M(a). +This easy implies the random walk (ξn) is recurrent provided M(n) → ∞ as n → ∞ +and M(n) → −∞ as n → −∞. The latter is implied by the following consequence +of the Denjoy-Koksma inequality. +Lemma 1. For every A > 0 there exists n0 > 0 that is independent of x ∈ T such +that M(n) > A for n ≥ n0 and M(n) < −A for n ≤ n0. +Proof. Take n > 0. The function M(n) is a sum of expressions of the form +k +� +j=1 +q(x + jα) +p(x + jα) +for k < n, +therefore to show the assertion it is sufficient to find δ > 0 such that the product +above is greater than δ for infinitely many k’s. Define f(x) = ln q(x) − ln p(x), +x ∈ T, and observe we can write +k +� +j=1 +q(x + jα) +p(x + jα) = exp +� +k +� +j=1 +f(x + jα) +� +. +The function f is of bounded variation and � +T f(t)dt = 0 so the Denjoy-Koksma +inequality (Theorem 3.1 in [Her79], p. 73) yields +���� +q +� +j=1 +f(x + jα) +���� = +���� +q +� +j=1 +f(x + jα) − q +� +T +f(t)dt +���� < var(f) + +UNIQUE ERGODICITY OF SIMPLE SYMMETRIC... +5 +for an arbitrary x ∈ T and an arbitrary closest return time q. But this means for +an arbitrary closest return time q we have +exp +� +k +� +j=1 +f(x + jα) +� +> e−var(f) > 0. +Thus the assertion follows with δ = e−var(f). +□ +To show the remaining part of Proposition, fix r > 0 and take ε > 0 so small +that (1 − ε)2r > 1/2. By (3.1), (3.2) and Lemma 1 there exists a > 0 (suitable for +all x ∈ T) such that +P +� +(ξn) returns to 0 before hitting −a or a +� +≥ 1 − ε +2. +Since p and q are separated from 0, there exists n0 so large (suitable for all x ∈ T) +such that probability that (ξn) stays in (−a, a) for the first n0 steps is less than +ε/2. Combining these two facts yields +P +� +(ξn) returns to 0 before n0 +� +> 1 − ε. +By the strong Markov property +P +� +(ξn) returns 2r-times to 0 before 2rn0 +� +> (1 − ε)2r > 1/2, +by the choice of ε. The assertion follows with m0 = 2rn0 since the expected number +of returns to 0 before m0 is greater than 2r with probability 1/2. +□ +It is advantageous to use the following notation in the remaining part of this +section. Let pn +i,j denote the probability of transition from state i to state j in n +steps. We simply write pi,j instead of p1 +i,j. Let kpn +i,j stands for the probability of +transition from state i to state j in n steps under the restriction that state k is +visited in neither of steps 1, . . . , n − 1. Again, these values depend on chosen point +x ∈ T but we refrain from stressing that in the notation. +Clearly jpn +k,j is the probability of the first visit in j starting at k occurring in +step n and kpn +k,j is the probability of transition to j from k in n steps with the +restriction that the state k is not visited in steps 1, . . . , n. The series �∞ +n=1 kpn +k,j +is interpreted as the expected number of visits in j starting at k before the first +return to k. It is not difficult to show the convergence of this series. +Lemma 2. If p is of bounded variation, symmetric and separated from 0 and 1 then +the series �∞ +n=1 kpn +i,j is convergent. Moreover, for any q ≥ 1 its sum is uniformly +bounded over all k, i, j with |k − i|, |k − j|, |j − i| < q, x ∈ T. For every ε > 0 +and natural q ≥ 1 there exists N with �∞ +n=N kpn +i,j < ε whatever x ∈ T, provided +|k − j| ≤ q. +Proof. Let m ∈ N be such that kpm +j,k > η for some η > 0 and all j, k with the same +parity and |j − k| ≤ q (remember the Markov chain is periodic with period two). +It is clear m and η can be chosen uniformly in x ∈ T since p is separated from 0 +and 1. We have +kpn +i,j · kpm +j,k ≤ kpn+m +i,k + +6 +KLAUDIUSZ CZUDEK +for n ∈ N, hence +∞ +� +n=N +kpn +i,j ≤ +1 +kpm +j,k +∞ +� +n=N +kpn+m +i,k +≤ 1 +η +∞ +� +n=N +kpn+m +i,k +. +The last series represents the probability that the first transition to k starting at i +occurs at earliest at the step N + m. This number is bounded from above by ε if +N is sufficiently large. Moreover, N can be chosen to be suitable for all x ∈ T by +a reasoning similar to the proof of Lemma 1. +□ +It is not difficult also to recover the value of �∞ +n=1 kpn +k,j, which represents the +expected value of appearances in state j of the process started at k before it returns +to k. +Lemma 3. If p is of bounded variation, symmetric and separated from 0 and 1 and +ax,n is defined by1 ax,0 = 1 and +(3.3) +ax,n = +q(x) +q(x + nα) +n−1 +� +j=0 +p(x + jα) +q(x + jα) +and +(3.4) +ax,−n = +p(x) +p(x − nα) +n−1 +� +j=0 +q(x − jα) +p(x − jα) +for n > 0. Then +∞ +� +n=1 +kpn +k,j = ax,j +ax,k +for any two states k, j ∈ Z. +Proof. Fix k. First of all, the aim is to show the assertion for j = k + 1. Notice +if the process started at k visits k − 1 in the first step then it necessarily visits k +before ever reaching k + 1. Thus the probability of exactly one appearance in k + 1 +before returning to k is p(x + kα) · q(x + (k + 1)α) and the probability of exactly r +appearances is p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α) (since after the first +r − 1 visits it “jumps” to the state k + 2 with probability p(x + (k + 1)α) and right +after r-th to k with probability q(x + (k + 1)α)). Hence the expected number of +appearances is +∞ +� +n=1 +kpn +k,j = +∞ +� +r=1 +r · p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α) += p(x + kα)q(x + (k + 1)α) +∞ +� +r=1 +rp(x + (k + 1)α)r−1 += p(x + kα)q(x + (k + 1)α) +(1 − p(x + (k + 1)α))2 += p(x + kα)q(x + (k + 1)α) +q(x + (k + 1)α)2 += +p(x + kα) +q(x + (k + 1)α), +where in the passing from the second line to the third one the formula �∞ +r=1 rzr−1 = +1 +(1−z)2 was used. Since the last equals ax,k+1 +ax,k , this completes the proof for j = k+1. +1In contrast to other symbols here we stress the dependence on x ∈ T. That is because this +symbol appears in the next section where the dependence on x is significant. + +UNIQUE ERGODICITY OF SIMPLE SYMMETRIC... +7 +To end the proof we proceed by induction. Let us assume the assertion holds +for k + 1, k + 2, ..., j for some j > k. Let us consider the process started at k. Take +r > 0. It is easy to conclude the expected number of appearances of this process in +j + 1 under the condition the number of appearances in k + 1 is r equals, by the +induction assumption, to r · ax,j+1 +ax,k+1 . In turn, the probability of exactly r visits in k +before returning to k is, as before, p(x+kα)·p(x+(k +1)α)r−1 ·q(x+(k +1)α). In +the view of foregoing, the expected number of appearances in j + 1 of the process +started at k before returning to k equals +∞ +� +r=1 +r · ax,j+1 +ax,k+1 +p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α) += ax,j+1 +ax,k+1 +· +p(x + kα) +q(x + (k + 1)α) = ax,j+1 +ax,k+1 +· ax,k+1 +ax,k += ax,j+1 +ax,k +. +This completes the proof of Lemma 3 in the case of any two integers with j > k. +The case j < k is symmetric. +□ +The last result of this section is basically the Doeblin ratio limit theorem (cf. +Corollary 2 to Theorem 4 in Section I.9, p. 48, in [Chu60]). However, reproducing +the proof is necessary because we need a kind of uniform convergence result over +all x ∈ T and states j, k that are sufficiently close to each other. +Proposition 2. If p is of bounded variation, symmetric and separated from 0 and +1 then for every ε > 0 and q ≥ 1 there exists N such that +���� +P(ξ1 = j) + · · · + P(ξn = j) +P(ξ1 = k) + · · · + P(ξn = k) − ax,j +ax,k +���� < ε +for every n ≥ N, x ∈ T, provided |k|, |j| ≤ q and |k − j| ≤ q. +Proof. Take ε > 0. By Lemma 2 there exists B > 0 such that �N +n=1 kpn +0,j ≤ B for +every N and states k, j satisfying the assumptions. The number B can be chosen +also such that +max +|j|,|k|≤q max +x∈T +ax,j +ax,k +≤ B. +Apply Lemma 2 and 3 to get N0 so large that +(3.5) +���� +N−ν +� +n=1 +kpn +k,j − ax,j +ax,k +���� < ε +3 +for N ≥ N0. +The number N ′ +0 > N0 should be so large that +(3.6) +2B +�N +n=1 pn +0,k +< ε +3 +for N ≥ N ′ +0 and +(3.7) +BN0 +�N +n=1 pn +0,k +< ε +3 +The easily proven decomposition formula +pn +0,j = kpn +0,j + +n−1 +� +ν=1 +pν +0,k · kpn−ν +k,j + +8 +KLAUDIUSZ CZUDEK +yields +N +� +n=1 +pn +0,j = +N +� +n=1 +kpn +0,j + +N−1 +� +ν=1 +pν +0,k +N−ν +� +n=1 +kpn +k,j. +We have +���� +P(ξ1 = j) + · · · + P(ξN = j) +P(ξ1 = k) + · · · + P(ξN = k) − ax,j +ax,k +���� = +���� +�N +n=1 pn +0,j +�N +n=1 pn +0,k +− ax,j +ax,k +���� += +���� +�N +n=1 kpn +0,j + �N−1 +ν=1 pν +0,k +�N−ν +n=1 kpn +k,j +�N +n=1 pn +0,k +− +�N +n=1 +ax,j +ax,k pn +0,k +�N +n=1 pn +0,k +���� +≤ +���� +�N +n=1 kpn +0,j − ax,j +ax,k pN +0,k + �N−1 +ν=1 pν +0,k +� �N−ν +n=1 kpn +k,j − ax,j +ax,k +� +�N +n=1 pn +0,k +���� +≤ +���� +�N +n=1 kpn +0,j − ax,j +ax,k pN +0,k +�N +n=1 pn +0,k +���� + +���� +�N−1 +ν=N−N0+1 pν +0,k +� �N−ν +n=1 kpn +k,j − ax,j +ax,k +� +�N +n=1 pn +0,k +���� ++ +���� +�N−N0 +ν=1 +pν +0,k +� �N−ν +n=1 kpn +k,j − ax,j +ax,k +� +�N +n=1 pn +0,k +���� +By (3.5) the third term is less than ε +3. By the very definition of B, the numerator +of the first term is less that 2B and the numerator of the second expression is less +than BN0. Thus (3.6) and (3.7) complete the proof. +□ +Remark 1. Let us consider an interval A ⊆ Z of length q. Let (ξn) be as usually +the process started at 0, and let τ be the moment of the first visit of (ξn) in A. +If N is given in Proposition 2. Since N was independent of x ∈ T, a conditional +argument easily implies +���� +P(ξ0 = j|Fτ) + · · · + P(ξn−1 = j|Fτ) +P(ξ0 = k|Fτ) + · · · + P(ξn−1 = k|Fτ) − ax,j +ax,k +���� < ε +almost surely on {τ < n − N} for any two states k, j ∈ A. +Remark 2. Let us now consider certain function ϕ : Z → R with support contained +in an interval A, as above, and ∥ϕ∥∞ ≤ 1. An easy argument using Remark 1 yields +���� +E +� +ϕ(ξ0) + · · · + ϕ(ξn−1) +��Fτ +� +E +� +1A(ξ0) + · · · + 1A(ξn−1) +��Fτ +� − +� +i∈A ϕ(i)ax,i +� +i∈A ax,i +���� < ε +almost surely on {τ < n − N}. It is clear that N can be chosen uniformly over all +intervals A of fixed length q, x ∈ T and function ϕ as far as ∥ϕ∥∞ ≤ 1. +4. Projection of measures +Put +ax,k = exp +� +Φ(x) + · · · + Φ(x + (k − 1)α) +�1 + exp Φ(x + kα) +1 + exp Φ(x) +for k ≥ 1 and ax,0 = 1. Define +µx,n = +1 +Mx,n +n−1 +� +k=0 +ax,kδx+kα + +UNIQUE ERGODICITY OF SIMPLE SYMMETRIC... +9 +for x ∈ T and n ≥ 1, where Mx,n is the normalizing constant, +Mx,n = +n−1 +� +k=0 +ax,k. +Lemma 4. If x ∈ T, k1, k2 ∈ N, then +ax,k1+k2 = ax,k1 · ax+k1α,k2 +and +µx,k1+k2 = +Mx,k1 +Mx,k1+k2 +µx,k1 + ax,k1 +Mx+k1α,k2 +Mx,k1+k2 +µx+k1α,k2. +The proof is straightforward. +Lemma 5. For every ε > 0 there exists N such that if q ≥ N is the closest return +time then (1 − ε)ay,n ≤ ax,n ≤ (1 + ε)ay,n for every natural n ≤ q and x, y ∈ T +with |x − y| < 2 +q . +Proof. Take δ > 0. We can find n0 so large that +(4.1) +1 +n +� +f ′� +x +� ++ · · · + f ′� +x + (n − 1)α +�� +< δ +for n ≥ n0 and every x ∈ T. +Indeed, this is the consequence of the Birkhoff ergodic theorem applied to the +rotation by angle α and the Lebesgue measure (uniform convergence in x follows +from unique ergodicity and continuity of f ′, see e.g. Proposition 4.1.13 in [KH95]). +Let q ≥ n0 be so large that +(4.2) +1 +q +� +f ′� +x +� ++ · · · + f ′� +x + jα +�� +< δ +for j ≤ n0 and every x ∈ T. +Finally, by uniform continuity, let us assume q to be so large that +(4.3) +1 − δ ≤ 1 + exp f(x) +1 + exp f(y) ≤ 1 + δ +for x, y ∈ T, |x − y| ≤ 2/q. +Take x, y ∈ T with |x − y| ≤ 2/q, a natural n ≤ q. By the mean value theorem +there exists z in the shorter arc joining x and y such that +ax,n +ay,n += exp +�� +f ′(z) + · · · + f ′(z + (n − 1)α) +� +|x − y| +� +×1 + exp f(x) +1 + exp f(y) · 1 + exp f(x + (n + 1)α) +1 + exp f(y + (n + 1)α). +If n ≥ n0 then apply (4.1) and the fact that |x − y| ≤ 2/q to get +� +f ′(z)+· · ·+f ′(z +(n−1)α) +� +|x−y| ≤ 1 +n +� +f ′� +x +� ++· · ·+f ′� +x+(n−1)α +�� +· n +q ≤ 2δ, +as n ≤ q. This combined with (4.3) yields +e−2δ(1 − δ)2 ≤ ax,n +ay,n +≤ e2δ(1 + δ)2. +Using (4.2) and (4.3) we can deduce similar statement in the case n < n0. If δ → 0 +then the values on the left and right above tend to 1, thus the assertion follows. +□ + +10 +KLAUDIUSZ CZUDEK +Proposition 3. Let ϕ ∈ C(T). For every ε > 0 there exists N such that if q ≥ N +is a closest return time then���� +� +T +ϕdµx,q − +� +T +ϕdµy,q +���� < ε +for every x, y ∈ T. +Proof. Take η > 0 and ϕ ∈ C(T). Choose δ > 0 small (to be determined), and let +q be the closes return time such that Lemma 5 is satisfied with ε replaced by δ. As +a consequence +(4.4) +1 − δ < az1,n +az2,n +< 1 + δ +and +1 − δ < Mz1,n +Mz2,n +< 1 + δ +for n ≤ q and z1, z2 ∈ T with |z1 − z2| < 2/q. Further, using Lemma ?? we easily +see az,qn → 1 uniformly in z, when (qn) is the sequence of closest return times. +Thus q can be chosen so large that 1 − δ ≤ az,q ≤ 1 + δ for all z ∈ T. Using the +first assertion in Lemma 4 it implies +(4.5) +1 − δ ≤ az,naz+nα,n−q ≤ 1 + δ +for n < q and z ∈ T. +The last thing we want to assume on q it is so large that +(4.6) +sup +z∈T +sup +|h|≤ 2 +q +|ϕ(z + h) − ϕ(z)| < δ. +Let us take x, y ∈ T. Denote xj = x + jα, yj = y + jα, j ∈ [0, q]. Let t be +the smallest natural number with d(xt, y) ≤ 1 +q . Since rotation is an isometry we +immediately see d(xt+j, yj) < 1 +q for j = 0, 1, · · · q − t. In particular d(xq, yq−t) < 1 +q , +hence d(yq−t, x) ≤ d(yq−t, xq) + d(xq, x) < 1/q + 1/q = 2/q and, since the rotation +is isometry, d(yq−t+j, xj) < 2 +q for j = 0, · · · , t. +The measure µx,q is an atomic measure with atoms at the points x, x+α, . . . , x+ +(q −1)α. The idea is to represent µx,q as a convex combination of measures concen- +trated on two disjoint subsets {x, x+α, . . . , x+(t−1)α} and {x+tα, . . . , x+(q−1)α} +and, similarly, represent µy,q and a convex combinations of measures concentrated +on two disjoint subsets {y, y +α, . . ., y +(q −t−1)α} and {y +(q −t)α, . . . , y +qα}. +Namely, it is easy to check using Lemma 4 that +µx,q = Mx,t +Mx,q +µx,t + ax,t +Mxt,q−t +Mx,q +µxt,q−t +and +µy,q = My,q−t +My,q +µy,q−t + ay,q−t +Myq−t,t +My,q +µyq−t,t. +Since d(xt, y) ≤ 1/q, in view of (4.4) we expect the second measure in the decom- +position of µx,q to be close to the first measure in decomposition of µy,q. Similar +reasoning applies to two remaining terms since d(yq−t, x) < 2/q. We have +���� +� +T +ϕdµx,q − +� +T +ϕdµy,q +���� ≤ +���� +Mx,t +Mx,q +� +T +ϕdµx,t − ay,q−t +Myq−t,t +My,q +� +T +ϕdµyq−t,t +���� +(4.7) ++ +����ax,t +Mxt,q−t +Mx,q +� +T +ϕdµxt,q−t − My,q−t +My,q +� +T +ϕdµy,q−t +����. + +UNIQUE ERGODICITY OF SIMPLE SYMMETRIC... +11 +Let us now focus on the second term on the right hand side. The analysis of the +first term proceeds analogously. We have +����ax,t +Mxt,q−t +Mx,q +� +T +ϕdµxt,q−t − My,q−t +My,q +� +T +ϕdµy,q−t +���� +(4.8) +≤ +����ax,t +Mxt,q−t +Mx,q +− My,q−t +My,q +���� +� +T +|ϕ|dµxt,q−t ++My,q−t +My,q +���� +� +T +ϕdµxt,q−t − +� +T +ϕdµy,q−t +����. +We are going to show the first term in (4.8) is bounded by ∥ϕ∥∞η and the second +by δ + ∥ϕ∥∞η. Since exactly the same estimates can be derived for the first term +on the right-hand side of (4.7), it will give +���� +� +T +ϕdµx,q − +� +T +ϕdµy,q +���� ≤ 2δ + 4∥ϕ∥∞η +and will complete the proof. Thus what remains to do is to find the desired bounds +on the right-hand side of (4.8). +A. Analysis of the first term on the right-hand side of (4.8) +We have +(4.9) +����ax,t +Mxt,q−t +Mx,q +− My,q−t +My,q +���� = My,q−t +My,q +����ax,t · My,q +Mx,q +· Mxt,q−t +My,q−t +− 1 +����. +Since d(y, xt) < 1/q ≤ 2/q we can apply (4.4) to get that 1 − δ ≤ Mxt,q−t +My,q−t ≤ 1 + δ. +Further, d(yq−t, x) ≤ 2/q, thus Lemma 4 and (4.4) give +My,q = My,q−t + ay,q−tMyq−t,t ≤ (1 + δ)Mxt,q−t + (1 + δ)2axt,q−tMx,t. +From (4.5) we have axt,q−t ≤ 1+δ +ax,t . Finally +My,q ≤ (1 + δ)Mxt,q−t + (1 + δ)2axt,q−tMx,t ≤ (1 + δ)Mxt,q−t + (1 + δ)3 +ax,t +Mx,t +≤ (1 + δ)3 +� +Mxt,q−t + +1 +ax,t +Mx,t +� += (1 + δ)3 +ax,t +� +ax,tMxt,q−t + Mx,t +� += (1 + δ)3 Mx,q +ax,t +. +So far we used only the bounds from above in (4.4 ) and (4.5 ). Applying the same +reasoning with estimates from below we see that +My,q ≥ (1 − δ)3 Mx,q +ax,t +. +Going back to (4.9) we have +(1 − δ)4 ≤ ax,t · My,q +Mx,q +· Mxt,q−t +My,q−t +≤ (1 + δ)4. +Take η > 0. If δ was chosen sufficiently small then +����ax,t · My,q +Mx,q +· Mxt,q−t +My,q−t +− 1 +���� < η. + +12 +KLAUDIUSZ CZUDEK +Since My,q−t +My,q +≤ 1 it leads to the estimate +����ax,t +Mxt,q−t +Mx,q +− My,q−t +My,q +���� < η. +Thus the first term on the right-hand side of (4.8) is bounded by η∥ϕ∥∞. +B. Analysis of the second term on the right-hand side of (4.8) +To deal with the second expression we have clearly My,q−t +My,q +≤ 1 and +���� +� +T +ϕdµxt,q−t − +� +T +ϕdµy,q−t +���� = +���� +q−t−1 +� +k=0 +axt,k +Mxt,q−t +ϕ(xt +kα)− +q−t−1 +� +k=0 +ay,k +My,q−t +ϕ(y+kα) +���� +≤ +���� +q−t−1 +� +k=0 +axt,k +Mxt,q−t +ϕ(xt + kα) − +q−t−1 +� +k=0 +ay,k +My,q−t +ϕ(xt + kα) +���� ++ +���� +q−t−1 +� +k=0 +ay,k +My,q−t +ϕ(xt + kα) − +q−t−1 +� +k=0 +ay,k +My,q−t +ϕ(y + kα) +���� +≤ +q−t−1 +� +k=0 +axt,k +Mxt,q−t +��ϕ(xt + kα) +�� +����1 − ay,k +axt,k +· Mxt,q−t +My,q−t +���� ++ +q−t−1 +� +k=0 +ay,k +My,q−t +��ϕ(xt + kα) − ϕ(y + kα) +��. +Since d(xt, y) < 1/q, (4.4) yields +(1 − δ)2 ≤ ay,k +axt,k +· Mxt,q−t +My,q−t +≤ (1 + δ)2, +thus +����1 − ay,k +axt,k +· Mxt,q−t +My,q−t +���� < η +if δ is sufficiently small. This leads us to the estimate +q−t−1 +� +k=0 +axt,k +Mxt,q−t +��ϕ(xt + kα) +�� +����1 − ay,k +axt,k +· Mxt,q−t +My,q−t +���� ≤ ∥ϕ∥η. +Clearly, +q−t−1 +� +k=0 +ay,k +My,q−t +��ϕ(xt + kα) − ϕ(y + kα) +�� ≤ δ +by (4.6), which completes the proof. +□ + +UNIQUE ERGODICITY OF SIMPLE SYMMETRIC... +13 +5. Proof of Theorem 1 +We shall use the following criterion for the uniqueness of the stationary distri- +bution. +If for every ε > 0 and nonnegative ϕ ∈ C(T) with 1/2 < ∥ϕ∥∞ < 1 there exist +β ∈ R and N > 0 such that +���� +ϕ(x) + · · · + P n−1ϕ(x) +n +− β +���� < ε +for every x ∈ T and n ≥ N, then there exists exactly one stationary distribution. +Let us take ε > 0 and ϕ ∈ C(T) as stated in the criterion. Let y ∈ T be arbitrary, +and let β = +� +T ϕdµy,q, where q is chosen so large that Proposition 3 holds with ε +replaced by ε/3. +Take x ∈ T. Set Ak = [kq, (k + 1)q), k ∈ Z, and define +ϕk(j) = 1Ak(j) · ϕ(x + jα), +ϕk : Z → R, k ∈ Z. +Observe that +� +i∈Ak ϕk(i)ax,i +� +i∈Ak ax,i += +� +T +ϕdµx+kα,q +for every k, thus Proposition 3 gives +(5.1) +���� +� +i∈Ak ϕk(i)ax,i +� +i∈Ak ax,i +− β +���� < ε +3, +for an arbitrary k ∈ Z. For k ∈ Z denote by τk the moment of the first visit of (ξn) +in Ak. Fix n sufficiently large and set Γ ⊆ Z to be the set of k’s such that Ak is +visited with positive probability till n. Apply Proposition 2 and Remark 2 to get a +number N such that +(5.2) +���� +E +� +ϕk(ξ0) + · · · + ϕk(ξn−1) +��Fτk +� +E +� +1Ak(ξ0) + · · · + 1Ak(ξn−1) +��Fτk +� − β +���� < ε +a.s. on {τk < n − N}. +Let (Xn) be the process with transition kernel (1.1) started at x ∈ T. We have +(5.3) +|E +� +ϕ(X0) + · · · + ϕ(Xn) +� +− βn| += +����E +� � +k∈Γ +ϕk(ξ0) + · · · + ϕk(ξn−1) +� +− βE +� � +k∈Γ +1Ak(ξ0) + · · · + 1Ak(ξn−1) +����� +≤ +� +k∈Γ +E +����E +� +ϕk(ξ0) + · · · + ϕk(ξn−1) +��Fτk +� +− βE +� +1Ak(ξ0) + · · · + 1Ak(ξn−1) +��Fτk +����� += +� +k∈Γ +E +� +E +� +1Ak(ξ0) + · · · + 1Ak(ξn−1) +��Fτk +� +· +���� +E +� +ϕk(ξ0) + · · · + ϕk(ξn−1) +��Fτk +� +E +� +1Ak(ξ0) + · · · + 1Ak(ξn−1) +��Fτk +� − β +���� +� +. +Let us fix k ∈ Γ and split the expectation above as follows. +E +� +E +� +1Ak(ξ0) + · · · + 1Ak(ξn−1) +��Fτk +� +· +���� +E +� +ϕk(ξ0) + · · · + ϕk(ξn−1) +��Fτk +� +E +� +1Ak(ξ0) + · · · + 1Ak(ξn−1) +��Fτk +� − β +���� +� + +14 +KLAUDIUSZ CZUDEK += E1{τk 0, +ǫ is the electrical permittivity and ω is the current frequency. +Given a boundary value f ∈ H1/2(∂Ω) we can determine the respective electrical potential +u ∈ H1(Ω) by uniquely solving +� +∇ · (γ∇u) = 0, in Ω, +u|∂Ω = f. +(1) +This is the so-called conductivity equation which describes the behavior of the electrical +potential, u, in a conductive body when a voltage potential is applied on the boundary, f. +In 1980, A.P. Calder´on, [11] introduced the problem of whether one can recover uniquely +a conductivity σ ∈ L∞(Ω) from the boundary measurements, i.e., from the Dirichlet-to- +Neumann map +Λσ : H1/2(∂Ω) → H−1/2(∂Ω), +(2) +f �→ σ ∂u +∂ν +���� +∂Ω +which connects the voltage and electrical current at the boundary. The normal derivative +exists as an element of H−1/2(∂Ω) by +⟨Λσf, g⟩ = +� +Ω +σ∇u · ∇v dx +(3) +where v ∈ H1(Ω) with v|∂Ω = g and u solves (1). +In the same paper, Calder´on was able to prove that the linearized problem at constant +real conductivities has a unique solution. Thereafter, many authors extended is work into +global uniqueness results. Sylvester and Uhlmann [31] used ideas of scattering theory, namely +the exponential growing solutions of Faddeev [14] to obtain global uniqueness in dimensions +n ≥ 3 for smooth conductivities. +Using this foundations the uniqueness for lesser regular +conductivities was further generalized for dimensions n ≥ 3 in the works of ([1], [7], [8], +[12], [13], [18], [22], [24], [27]). Currently, the best know result is due to Haberman [17] for +conductivities γ ∈ W 1,3(Ω). The reconstruction procedure for n ≥ 3 was obtained in both [22] +and [25] independently. As far as we are aware, there seems to be no literature concerning +reconstruction for conductivities with less than two derivatives. +In two dimensions the problem seems to be of a different nature and tools of complex analy- +sis were used to establish uniqueness. Nachman [23] obtained uniqueness and a reconstruction +method for conductivities with two derivatives. The uniqueness result was soon extend for +once-differentiable conductivities in [9] and a corresponding reconstruction method was ob- +tained in [20]. In 2006, Astala and P¨aiv¨arinta [3] gave a positive answer Calder´on’s problem +1 + +for σ ∈ L∞(Ω), σ ≥ c > 0, by providing the uniqueness proof through the reconstruction +process. +All of this definitions can be extended to the complex-conductivity case with the assump- +tion Re γ ≥ c > 0, in particular, we can define the Dirichlet-to-Neumann as above Λγ. +In this scenario, the first works was done in two-dimensions by Francini [15], by extending +the work of Brown and Uhlmann [9] in two-dimensions proving uniqueness for small frequencies +ω and γ ∈ W 2,∞. Afterwards, Bukgheim influential paper [10] proved the general result in +two-dimensions for complex-conductivities in W 2,∞. +He reduced the (1) to a Schr¨odinger +equation and shows uniqueness through the stationary phase method (based on is work many +extensions followed in two-dimensions [2], [4], [26]). +Recently, by mixing techniques of [9] +and [10], Lakshtanov, Tejero and Vainberg obtained [21] uniqueness for Lipschitz complex- +conductivities in R2. In [28], the author followed up their work to show that it is possible to +reconstruct complex-conductivity with a jump at least in a certain set of points. +In three dimensions, the uniqueness results presented in [31] and [24] hold for twice- +differentiable complex-conductivities in W 2,∞, but there was no reconstruction process pre- +sented. Nachman’s reconstruction method in three dimensions [22] was used in [19] to re- +construct complex conductivities from boundary measurements. Even though the Nachman’s +proof was presented only for real conductivities, the paper [29] structures the proof in order +to show the result holds for complex-conductivities. As far as we aware, the works with lower +regularity require real-conductivities. +In this paper our interest resides in Calder´on’s problem for once-differentiable complex- +conductivities in three-dimensions. The aim is to prove the following theorem: +Theorem 1.1. Let Ω ⊂ R3 a bounded Lipschitz domain, γi ∈ W 1,∞(Ω), i = 1, 2 be two +complex-valued conductivities with Re γi ≥ c > 0. +If Λγ1 = Λγ2, then γ1 = γ2. +Our work basis itself on a transformation of (1) into a Dirac system of equation in three- +dimensions with the help of quaternions. In this scenario, we obtain a potential q that we want +to determine from boundary data. The main ideas follow the work of Brown and Uhlmann [9] +for real conductivities in two-dimensions and Lakshtanov, Vainberg and Tejero [21], as well as +the authors work [28], for complex-conductivities. +In this paper, we provide a novel reconstruction of the bounded potential q from the +boundary data, but we are yet to be able to establish a relation between this boundary +data and the Dirichlet-to-Neumann map. This is essentially to answer Calder´on problem for +Lipschitz complex conductivities, but the lack of a well-suited Poincar´e lemma that fits the +quaternion structure does not allow such a simple work as in 2D. +2 +Minimalistic lesson of Quaternions +We present the basis of the quaternionic framework we will use for our work. Let R(2) be the +real universal Clifford Algebra over R2. By definition, it is generated as an algebra over R by +the elements {e0, e1, e2}, where e1, e2 is a basis of R2 with eiej + ejei = −2δij, for i, j = 1, 2 +and e0 = 1 is the identity and commutes with the basis elements. This algebra has dimension +4 and is identified with the algebra of the quaternions, H. As such it holds e3 = e1e2. In the +following, we refer to this algebra as the quaternions. An element of the quaternions can be +written as: +x = x0 + x1e1 + x2e2 + x3e3, +(4) +where xj, j = 0, ..., 3 are real. We define the quaternionic conjugate ¯x of an element x as +¯x = x0 − x1e1 − x2e2 − x3e3. +(5) +Let x, y ∈ H, we write xy for the resulting quaternionic product. The product ¯xy defines +a Clifford valued inner product on H. Further, we have xy = ¯y¯x and the conjugate of the +conjugate of quaternion is that same quaternion. Let x ∈ H then Sc x = x0 denotes the scalar +of x and Vec x = x − Sc x. The scalar of a Clifford inner product Sc(¯xy) is the usual inner +product in R4 for x, y identified as vectors. +With this inner product H is an Hilbert space and the resulting norm is the usual Euclidean +norm. +In order to introduce some of the concepts we also extend the real quaternions to complex +quaternions C2 = C ⊗ H. Here, we use the same generators (e1, e2) as above, with the same +2 + +multiplication rules, however, the coefficients of the quaternion can be complex-valued. That +is, λ ∈ C ⊗ H may be written as +λ = λ0 + λ1e1 + λ2e2 + λ3e3, +λj ∈ C, j = 0, ..., 3 +(6) +or still as +λ = x + iy, +x, y ∈ H. +(7) +Due to the complexification we can still take another conjugation, to which we define has +Hermitian conjugation and denote it by ·†. Explicitly, for λ ∈ C ⊗ H one has +¯λ† = λc +0 − λc +1e1 − λc +2e2 − λc +3e3, +(8) +where ·c denotes complex conjugation, or +¯λ† = ¯x − i¯y. +(9) +Similarly, one can introduce an associated inner product and norm in C ⊗ H by means of +this conjugation: +⟨λ, µ⟩ = Sc +� +¯λ†µ +� +; +|λ|C2 = +� +Sc +�¯λ†λ +� +. +(10) +For ease of notation, we also define for λ ∈ C2 the complex conjugation as +¯λc = λc +0 + λc +1e1 + λc +2e2 + λc +3e3. +(11) +Now, we can also introduce Quaternion-valued functions f : R3 → C2 written as f = +f0 + f1e1 + f2e2 + f3e3, where fj : R3 → C. +The Banach spaces Lp, W n,p of C2-valued functions are defines by requiring that each +component is in such space. On L2(Ω) we introduce the C2-valued inner product +⟨f, g⟩ = +� +Ω +¯f †(x)g(x)dx. +(12) +Analogously to the Wirtinger derivatives in complex analysis, we have the Cauchy-Riemann +operators under (x0, x1, x2) coordinates of R3 defined as +D = ∂0 + e1∂1 + e2∂2, +(13) +where ∂j is the derivative with respect to the xj, j = 0, 1, 2 variable; and +¯D = ∂0 − e1∂1 − e2∂2. +(14) +The vector part of the Cauchy-Riemann operator is designated as Dirac operator. It holds +that D ¯D = ∆ where ∆ is the Laplacian. +We designate any function f fulfilling Df = 0 as a monogenic function, analogous to the +holomorphic functions in complex analysis. +2.1 +A bit of Operator Theory +Let Ω be a bounded domain and f : Ω → C2. All the results in this subsection were taken out +from the classical book on quaternionic analysis of G¨urlebeck and Spr¨ossig [16] +The Cauchy-Riemann operator has a right-inverse in the form +(T f) (x) = − 1 +ω +� +Ω +y − x +|y − x|3 f(y) dy, for x ∈ Ω, +(15) +where E(x, y) = − 1 +ω +y−x +|y−x|3 is the generalized Cauchy kernel and ω = 4π stands for the surface +area of the unit sphere in R3, that is, DT f = f. This operator acts from W k,p(Ω) to W k+1,p(Ω) +with 1 < p < ∞ and k ∈ N0. +Furthermore, we introduce the boundary integral operator for x /∈ ∂Ω +(F∂Ωf) (x) = 1 +ω +� +∂Ω +y − x +|y − x|3 α(y)f(y) dS(y), +(16) +where α(y) is the outward pointing normal unit vector to ∂Ω at y. We get the well-known +Borel-Pompeiu formula +(F∂Ωf) (x) + (T Df) (x) = f(x) for x ∈ Ω. +Obviously, DF∂Ω = 0 holds through this formula it it holds that F∂Ω acts from W k− 1 +p ,p(∂Ω) +into W k,p(Ω), for k ∈ N and 1 < p < ∞. +One of the other well-known results we will need for our work is the Plemelj-Sokhotzki +formula is obtaining by taking the trace of the boundary integral operator. +First we introduce an operator over the boundary of Ω. +3 + +Proposition 2.1. If f ∈ W k,p(∂Ω), then there exists the integral +(S∂Ωf) = 1 +2π +� +∂Ω +y − x +|y − x|3 α(y)f(y) dS(y) +(17) +for all points x ∈ Ω in the sense of Cauchy principal value. +Furthermore, the operator S∂Ω is continuous in W k,p(∂Ω), for 1 < p < ∞, k ∈ N. +From this the Plemelj-Sokhotzki formula is given as: +Theorem 2.2. Let f ∈ W k,p(∂Ω) where by taking the non-tangential limit we have: +lim +x→x0, +x∈Ω, x0∈∂Ω +(F∂Ωf) (x) = 1 +2 (f(x0) + (S∂Ωf) (x0)) . +One of the corollaries concerns the limit to the boundary acting as a projector. That is, +Corollary 2.3. The operator P∂Ω denoting the projection onto the space of all H−valued +functions which may be monogenicaly extended into the domain Ω. +Then, this projection may be represented as +P∂Ω = 1 +2 (I + S∂Ω) . +The proofs of this results and others to follow in our proofs may be found in [16]. +Now we are ready to start constructing our work on the inverse conductivity problem. +3 +Inverse Dirac scattering problem +Transforming our conductivity equation into another type of equation also changes the in- +verse problem we are concerned. We transform it into a system of equations based on the +Cauchy-Riemann operator D (also called Dirac operator in some contexts) and thus we need +to solve the inverse Dirac scattering problem first and only afterwards we care about the +inverse conductivity problem. +Let u be a solution to (1) for some boundary function. We define +φ = γ1/2 � ¯Du, Du +�T , +remark that γ1/2 is well-defined since it is contained in C+. Then, φ solves the system +� +Dφ1 += φ2q1, +¯Dφ2 += φ1q2, +in R3. +(18) +where q1 = − 1 +2 +¯ +Dγ +γ +and q2 = − 1 +2 +Dγ +γ . +This transformation arises as follows: +Dφ1 = D +� +γ1/2 ¯Du +� += Dγ1/2 ¯Du + γ1/2∆u += Dγ1/2 ¯Du − γ−1/2∇γ · ∇u += Dγ1/2 ¯Du − 1 +2γ−1/2 � +Dγ ¯Du + Du ¯Dγ +� += −1 +2 +� +γ1/2Du +� ¯Dγ +γ += φ2q1 +Carefully, we can extend our potential to the outside by setting γ ≡ 1 outside of Ω, which +lead us to treat the study the equation in R3. +3.1 +Exponentially Growing Solutions +We devise new exponentially growing solutions from the classical ones used in three dimensions. +In most literature works, the exponential behavior is defined through the function ex·ζ, with +ζ ∈ C3 fulfilling ζ · ζ = 0. However, in our scenario this function does not fulfill Deix·ζ = 0, +which brings the simplicity in all of the literature works. +Since we know that it is harmonic we can generate a monogenic function through it. Let +ζ ∈ C3 such that ζ · ζ := ζ2 +0 + ζ2 +1 + ζ2 +2 = 0, then it holds +∆ex·ζ = 0 ⇔ D +� +¯Dex·ζ� += 0 ≡ D +� +ex·ζ ¯ζ +� +4 + +where now ζ is also defined as a quaternion through ζ = ζ0 + e1ζ1 + e2ζ2 ∈ C2. Thus +the function E(x, ζ) = ex·ζ ¯ζ is monogenic. This also arises from the choice of ζ, since ζ ¯ζ = +ζ2 +0 + ζ2 +1 + ζ2 +2 = 0. +We make a clear statement of when ζ is a complex-quaternion or complex-a vector, but +in most cases it is clear from context: it is a vector if it is in the exponent and a quaternion +otherwise. +We assume the following asymptotic behaviour for φ: +φ1 = ex·ζ ¯ζµ1, +(19) +φ2 = ex·¯ζc ¯ζcµ2 +(20) +Setting ˜µ1 = ¯ζµ1 and ˜µ2 = ¯ζcµ2 we have the equations: +� +D˜µ1 += e−x·(ζ− ¯ζc)˜µ2q1 +¯D˜µ2 += ex·(ζ− ¯ζc)˜µ1q2 +(21) +Further, we assume ˜µ → +� +1 +0 +� +as |x| → ∞. These system of equations will lead us to an +integral equation from which we can extract interesting behaviour for ζ → ∞. +The main point of this subsection is to demonstrate how we can obtain the system of +integral equations related with (21). Here, the approach is similar to [21], but we need to be +careful due to the non-commutative nature of quaternions. +Recall, that DT = ¯D ¯T = I (in appropriate spaces). Hence, applying this to (21) it holds: + + + + + + + +˜µ1 = 1 + T +� +e−x·(ζ−¯ζc)˜µ2q1 +� +˜µ2 = T +� +ex·(ζ−¯ζc)˜µ1q2 +� +Thus, we can obtain two integral equations with respect to their function: + + + + + + + +˜µ1 = 1 + T +� +e−x·(ζ−¯ζc) ¯T +� +ex·(ζ−¯ζc)˜µ1q2 +� +q1 +� +˜µ2 = ¯T +� +ex·(ζ−¯ζc)q2 +� ++ ¯T +� +ex·(ζ−¯ζc)T +� +e−x·(ζ−¯ζc)˜µ2q1 +� +q2 +� + + + +˜µ1 = 1 + M 1˜µ1 +˜µ2 = T +� +e +x· +� +ζ−ζC� +q2 +� ++ M 2˜µ2 +⇔ +� +[I − M 1](˜µ1 − 1) = M 11 +[I − M 2](˜µ2) = ¯T +� +ex·(ζ−¯ζC)q2 +� +(22) +Our objective now is to study the uniqueness and existence of this equations, we approach +this task by proving that M j, j = 1, 2 are contractions. +Instead of working with all possible ζ ∈ C(2) fulfilling ζ ¯ζ = 0, we choose them for k ∈ R3 +as +ζ = k⊥ + ik +2 , +k⊥ · k = 0 +and k⊥ can be algorithmically found. +We now describe our space of functions in terms of the space variable and k ∈ R3 as +S = L∞ +x (Lp +k(|k| > R)) +(23) +where R > 0 is a constant. In this space we prove that the operators M 1, M 2 are indeed +contractions: +Lemma 3.1. Let p > 2. Then +lim +R→∞ ∥M j∥S = 0. +To further study the system (22), we also need to show that the right-hand side is in S for +an R large enough: +Lemma 3.2. Let p > 2. Then there exists R > 0 such that +M 11 ∈ S, +(24) +¯T +� +ex·(ζ−¯ζC)q2 +� +∈ S +(25) +The above Lemmas imply the existence and uniqueness of (˜µ1, ˜µ2) solving the system (22) +with respect to the potential q. This is essential for the reconstruction procedure we show up +next. +5 + +3.2 +Reconstruction from scattering data +In this section, we are mixing ideas from [21] and [22] with quaternionic theory to obtain the +potential from the scattering data. +Starting from Clifford-Green theorem +� +Ω +� +g(x) +� ¯Df(x) +� ++ +� +g(x) ¯D +� +f(x) +� +dx = +� +∂Ω +g(x)η(x)f(x) dSx +and using g(x; iξ + ζ) = (iξ + ζ)e−x·(iξ+ζ) for ξ ∈ R3 such that (iξ + ζ) · (iξ + ζ) = 0. This +implies that g ¯D = 0. Thus we define the scattering data as: +h(ξ, ζ) = (iξ + ζ) +� +∂Ω +e−x·(iξ+ζ)η(x)φ2(x, ζ) dx +(26) +Applying now Clifford-Green theorem we obtain another form for the scattering data: +h(ξ, ζ) = (iξ + ζ) +� +Ω +e−x·(iξ+ζ) ¯Dφ2(x, ζ) dx += (iξ + ζ) +� +Ω +e−ix·ξ � +e−x·ζφ1(x, ζ) +� +q2(x) dx, +by Dφ2 = φ1q2 += (iξ + ζ) +� +Ω +e−ix·ξ (ζµ1(x, ζ)) q2(x) dx += iξ +� +Ω +e−ix·ξ ˜µ1(x, ζ)q2(x) dx, +since ¯ζζ = 0 += iξˆq2(ξ) + iξ +� +Ω +e−ix·ξ [˜µ1(x, ζ) − 1] q2(x) dx. +Thus, we have: +ˆq2(ξ) = h(ξ, ζ) +iξ +− +� +Ω +e−ix·ξ [˜µ1(x, ζ) − 1] q2(x) dx +(27) +This is yet not enough to reconstruct the potential, since the integral acts as a residual in +the reconstruction and requires data that we technically do not have. Therefore, we integrate +everything over an annulus in k +� +R<|k|<2R +ˆq2(ξ) +|k|3 dk = 1 +iξ +� +R<|k|<2R +h(ξ, ζ(k)) +|k|3 +dk +− +� +R<|k|<2R +1 +|k|3 +� +Ω +e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx, +(28) +since the potential does not depend on k it can be taken out of the integral and taking the limit +as R → ∞ leads the second integral on the right to decay to zero, obtaining a reconstruction +formula. +Theorem 3.3. Let Ω ⊂ R3 a bounded Lipschitz domain, q ∈ L∞(Ω) be a complex-valued po- +tential obtained through a conductivity γ ∈ W 1,∞(Ω), Re γ ≥ c > 0. Then, we can reconstruct +the potential from +ˆq2(ξ) = lim +R→∞ +C +iξ +� +R<|k|<2R +h(ξ, ζ(k)) +|k|3 +dk, +(29) +where C = +1 +4π ln(2). +Proof. The scattering data is defined from the solutions of the Dirac system (22) and therefore +it holds that ˜µ1 − 1 ∈ S. Starting from (28) we obtain by integrating the right-hand side for +any ξ ∈ R3: +4π ln 2 ˆq2(ξ) = 1 +iξ +� +R<|k|<2R +h(ξ, ζ(k)) +|k|3 +dk +− +� +R<|k|<2R +1 +|k|3 +� +Ω +e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx +6 + +Let p > 2 and 1/p + 1/q = 1. We estimate the last integral: +����� +� +R<|k|<2R +1 +|k|3 +� +Ω +e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx +����� ≤ +≤ +� +R<|k|<2R +1 +|k|3 +� +Ω +���e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) +��� dx +≤ CΩ∥q∥∞ +� +R<|k|<2R +1 +|k|3 sup +x +|˜µ1(x, ζ(k)) − 1| dk +≤ CΩ∥q∥∞ +�� +R<|k|<2R +1 +|k|3q dk +�1/q �� +R<|k|<2R +sup +x |˜µ1(x, ζ(k)) − 1|p dk +�1/p +≤ CΩ∥q∥∞∥˜µ1 − 1∥S +�� +R<|k|<2R +1 +|k|3q dk +�1/q +Taking the limit as R → 0 the integral that is left goes to zero which implies the desired decay +to zero and leaves us with our reconstruction formula. +Now, in order to connect the functions that solve the electrical conductivity equation (1) +and the solutions to the Dirac equation (18), which are exponential growing, we introduce the +following result: +Proposition 3.4. Let Ω be a bounded domain in R3. +Let φ = (φ1, φ2) be a solution of +the Dirac system (18) for a potential q ∈ L∞(Ω) associated with the complex-conductivity +γ ∈ W 1,∞(Ω). +If φ1 = ¯φ2 then there exists a unique solution u of: +� ¯Du = γ−1/2φ1, +Du = γ−1/2φ2. +(30) +Further, this function fulfills the conductivity equation +∇ · (γ∇u) = 0 in Ω. +Let us recall the main theorem, that we are now able to prove with all these pieces we +assembled. +Theorem 1.1 Let Ω ⊂ R3 a bounded Lipschitz domain, γi ∈ W 1,∞(Ω), i = 1, 2 be two +complex-valued conductivities with Re γi ≥ c > 0. +If Λγ1 = Λγ2, then γ1 = γ2. +Proof. Due to Theorem 3.3, one only needs to show that the scattering data h for |k| >> 1 +is uniquely determined by the Dirichlet-to-Neumann map Λγ. Due to the lack of Poincar´e +Lemma in our current framework in quaternionic analysis with the D and ¯D operator, than a +new technique is required to obtain a similar proof to [9], for example. +For such, let us start with two conductivities γ1, γ2 in W 1,∞(Ω) for Ω a bounded domain. +By hypothesis Λγ1 = Λγ2 and thus by [29] we have γ1|∂Ω = γ2|∂Ω. +Further, we can extend γj, j = 1, 2 outside Ω in such a way that in R3 \ Ω and γj − 1 ∈ +W 1,∞ +comp(R3). Let qj, φj, µj, hj, j = 1, 2 be the potential and the solution in (18), the function +in (19), and the scattering data in (26) all associated with the conductivity γj. +Due to the scattering formulation at the boundary ∂Ω, then we just want to know if φ1 = φ2 +on ∂Ω when |k| >> 1. +First, by Proposition 3.4, we know that there exists an u1 such that +φ1 = γ1/2 +1 +( ¯Du1, Du1)T , +which is a solution to +∇ · (γ1∇u1) = 0 in R3. +Now, let us define u2 by +u2 = +� +u1 +in R3 \ Ω, +u +in Ω. +7 + +where ˆu is the solution to the Dirichlet problem +� +∇ · (γ2∇u) = 0 +in Ω, +u = u1 +on ∂Ω. +Let g ∈ C∞ +c (R3). Then, +� +R3 γ2∇u2∇g dx = +� +R3\Ω +γ1∇u1∇g dx + +� +Ω +γ2∇ˆu∇g dx += − +� +∂Ω +Λγ1 +� +u1|∂Ω +� +g dsx + +� +∂Ω +Λγ2 +� +u|∂Ω +� +g dsx += 0. +Hence, u2 is the solution of ∇ · (γ2∇u2) = 0 in R3. Further, the following function +ψ2 = γ1/2 +2 +� ¯Du2, Du2 +�T +is the solution of (18) where the potential is given by γ2. +Furthermore, ψ2 has the asymptotics of φ1 in R3 \ Ω, thus by Lemma 3.1 and 3.2 it will +be the unique solution of the respective integral equation of (18). Thus, ψ2 will be equal φ2 +when |k| > R. Since, on the outside ψ2 ≡ φ1. Then we obtain: +φ1 = φ2 in R3 \ Ω. +In particular, we have equality at the boundary ∂Ω. So, this implies that if the Dirichlet- +to-Neumann maps are equal the respective scattering data will also be the same. Thus, the +Dirichlet-to-Neumann map uniquely determines the potential q. +From the definition of q, we can uniquely determine the conductivity γ up to a constant, +which in the end is defined by γ|∂Ω which is uniquely determined by the Dirichlet-to-Neumann +map Λγ. +4 +Auxiliary Proofs +Proof of Lemma 3.1. +Let us assume, without loss of generality, that f is a scalar function. Further, we present +the proof for M 1, since for M 2 it follows analogously. +Recall, that we choose ζ ∈ C(2) with respect to k ∈ R(2) as +ζ = k⊥ + ik +2 , +k⊥ · k = 0. +In vector form, this leads to ζ − ζc = ik which implies the following deductions: +M 1f(x) = +� +R3 e−w·(ζ−¯ζc) x − w +|x − w|3 +� +R3 ey·(ζ−¯ζc) w − y +|w − y|3 f(y)q2(y) dy q1(w) dw += +� +R3 +� +R3 e−iw·k x − w +|x − w|3 eiy·k w − y +|w − y|3 f(y)q2(y)q1(w) dwdy += +� +R3 A(x, y; k)f(y) dy, +where +A(x, y; k) = +� +R3 e−i(w−y)·k x − y +|x − y|3 +w − y +|w − y|3 q2(y)q1(w) dw. +Due to the compact support of the potential q2, it holds that A has compact support on the +second variable. +Let us now apply the norm in terms of k to it: +∥Mf(x, ·)∥Lp(|k|>R) = +�� +|k|>R +|Mf(x, ζ)|p dσζ +�1/p += +�� +|k|>R +���� +� +Ω +A(x, y; k)f(y) dy +���� +p +dσk +�1/p +≤ +� +Ω +�� +|k|>R +|A(x, y; k)f(y)|p dσk +�1/p +dy +≤ +� +Ω +sup +k +|A(x, y; k)| dy ∥f∥S. +8 + +In order to complete the proof we show that the first integral goes to zero as R → ∞. +Let As be given with the extra factor α(s|x−w|)α(s|w−y|) in the integrand, where α ∈ C∞ +is 1 outside a neighborhood of the origin and 0 inside a smaller neighborhood of it. +Since, +� +B1(0) +� +B1(0) +1 +|w|2 +1 +|w − y|2 dw dy, +it holds that for any ǫ > 0 there exists an s > 0 such that: +� +Ω +|A − As| dy < ǫ. +Further, we denote As0,n the function As0 with potentials q1, q2 replaced by their L1 +smooth approximation Qn +1 , Qn +2 ∈ C∞. Since the other factors are bounded it holds +� +Ω +|As0 − As0,n| dy < ǫ. +Now it is enough to show that As0,n → 0 as |k| → 0 uniformly! +All integrands inside of it will be in C∞ and uniformly bounded, thus by Riemann-Lebesgue +the result follows. +Proof of Lemma 3.2. Once again recall that ζ = +� +k⊥ + i k +2 +� +for k ∈ R3. First we show +that M 11 ∈ S. We have +M 11 = +� +Ω +� +Ω +e−iw·k x − w +|x − w|3 +w − y +|w − y|3 eiy·kq2(y)q1(w) dy dw, +and applying the Lp norm in k followed with Minkowski integral inequality we obtain +�� +|k|>R +|M 11|pdk +�1/p +≤ +� +Ω +|q1(w)| +|x − w|2 +�� +|k|>R +���� +� +Ω +eiy·k w − y +|w − y|3 q2(y)dy +���� +p +dk +�1/p +dw +The inner most integral resembles a Fourier transform, hence, applying the Hausdorff-Young +inequality for p > 2 we have +�� +|k|>R +���� +� +Ω +eiy·k w − y +|w − y|3 q2(y) dy +���� +p +dk +�1/p +≤ +�� +Ω +|q2(y)|p′ +|w − y|2p′ dy +�1/p′ +< C∥q2∥∞, +where the last inequality follows quickly by Young’s convolution inequality and Riesz type +estimate of the kernel. +Therefore, by the same Riesz type estimate it holds: +�� +|k|>R +|M 11|p dk +�1/p +≤ C∥q2∥∞ +� +Ω +|q1(w)| +|x − w|2 dw ≤ C′∥q2∥∞∥q1∥∞. +To complete the proof we need to show statement (25). Similarly, to the above proof, we +have by Hausdorff-Young Inequality, Young’s convolution inequality and a Riesz type estimate +the following: +�� +|k|>R +���� +� +R3 eiy·k x − y +|x − y|3 q2(y) dσy +���� +p +dσk +�1/p +≤ +�� +R3 +���� +x − y +|x − y|3 q2(y) +���� +p′ +dσy +�1/p′ +≤ C∥q2∥∞ +We need the following auxiliary result for the proof of Proposition 3.4. +Lemma 4.1. Let Ω be a bounded Lipschitz domain in R3. +If h is a scalar-valued and harmonic function that fulfills +Vec(S∂Ωh) = 0, +then h|∂Ω is constant. +9 + +Proof. First, note that I + S∂Ω = P∂Ω is a projector and by Proposition 2.5.12 and Corollary +2.5.15 of [16] it holds that P∂Ωh is the boundary value of a monogenic function in Ω. +Since h is a scalar-valued function it holds that +P∂Ωh = Sc(P∂Ωh) + Vec(P∂Ωh) += (h + Sc∂Ωh) + Vec(S∂Ωh). +Let w = (h + Sc∂Ωh) and v = Vec(S∂Ωh). Now, we denote f as the monogenic extension +of P∂Ωh in Ω, as such, the boundary values of f fulfill trf = w + v. Note that by hypothesis +we have that v|∂Ω = 0. +Hence, f is also an harmonic function, which implies that the scalar and vector components +are harmonic. +� +∆(Vecf) = 0, +Vec f|∂Ω = 0. +By a mean value theorem or a maximum principle it holds that Vecf = 0. Due to this +and f being monogenic we obtain that Df = 0 ⇔ D(Ref) = 0. Thus, Ref = c since D is a +quaternionic operator. +Consequently, the boundary values are also constant, which means that w = c in ∂Ω. +Since, Sc(S∂Ωh) is an averaging operator it holds that h = c. +Proof of Proposition 3.4 +Suppose that (u, v) are solutions to the following equations: +� ¯Du = γ−1/2φ1 +Dv = γ−1/2φ2. +From applying the operator D and ¯D to the first and second equation respectively, we +obtain from φ2 = φ +H +1 and q2 = qH +1 the following: +∆u = D(γ−1/2φ1) = D(γ−1/2)φ1 + γ−1/2Dφ1 += −1 +2γ−3/2(Dγφ1) + γ−1/2φ2q1 += γ−1/2 [q2φ1 + φ2q1] = γ−1/2 � +qH +1 φ1 + φ +H +1 q1 +� += γ−1/2Sc (φ +H +1 q1). +and +∆v = ¯D(γ−1/2φ2) = ¯D(γ−1/2)φ2 + γ−1/2 ¯Dφ2 += −1 +2γ−3/2( ¯Dγ)φ2 + γ−1/2φ1q2 += γ−1/2 [q1φ2 + φ1q2] = γ−1/2 � +q1φ +H +1 + φ1qH +1 +� += γ−1/2Sc (φ1qH +1 ). +The first thing to notice is that both equations imply that u and v must be scalar-valued +functions. +Further, notice that +∆(u − v) = γ−1/2 � +Sc (φ +H +1 q1) − Sc (φ1qH +1 ) +� += γ−1/2 +� +Sc (φ +H +1 q1) − Sc (q1φ +H +1 ) +� += 0. +Therefore, h = u − v is an harmonic function. Our objective is to show that h ≡ 0, thus +showing that u = v. +For such, let us consider the theory of integral transforms in quaternionic analysis. We +have +u = ¯T(γ−1/2φ1) + F ∂Ω(γ−1/2φ1) and +u = ¯T(γ−1/2φ1) + F ∂Ω(u), +which implies that +F ∂Ω(γ−1/2φ1) = F ∂Ωu. +10 + +Analogously, we obtain +F∂Ω(γ−1/2φ2) = F∂Ωv. +Here, we can extrapolate from the first equation and from u being scalar-valued that +γ−1/2φ1F∂Ω = F∂Ωu +⇔ γ−1/2φ2F∂Ω = F∂Ωu. +Applying the operator F∂Ω on the other side, we obtain: +F 2 +∂Ωu = F∂Ω(γ−1/2φ2)F∂Ω and +F 2 +∂Ωv = F∂Ω(γ−1/2φ2)F∂Ω +⇒ F 2 +∂Ωh = F 2 +∂Ω(u − v) = 0. +If we take the trace on both sides, the operator becomes a projector thus we obtain +tr F∂Ωh = 0. +Now, through the Sokhotski-Plemelj formula we obtain: +tr F∂Ωh = h|∂Ω + S∂Ωh = 0, at ∂Ω. +Since h is a scalar-valued function that we decompose this formulation with the scalar and +vector part to obtain two conditions: +� +h + Sc(S∂Ωh) = 0 +Vec(S∂Ωh) = 0. +Through the second condition and Lemma 4.1 we have that h is constant over ∂Ω. +Now, given that h is a scalar constant, the first condition reduces to: +h(1 + Sc(S∂Ω1)) = 0 +By [16] we obtain that 1 + Sc(S∂Ω1) = 1/2 in ∂Ω. Therefore, we conclude that h ≡ 0 in +∂Ω. Given that h is harmonic, this immediately implies that h = 0 in Ω. +Therefore, we obtain u = v, and therefore there exists a unique solution to the initial +system through the T and F∂Ω operators in Ω. +To finalize, we only need to show that u fulfills the conductivity equation in Ω. +Bringing the first equation to light +¯Du = γ−1/2φ1, +changing the side of the conductivity we get γ1/2 ¯Du = φ1 and applying the D operator to +both sides now brings +D +� +γ1/2 ¯Du +� += Dφ1 +⇔ +D +� +γ1/2� +¯Du + γ1/2∆u = φ2q1 +⇔ +D +� +γ1/2� +¯Du + γ1/2∆u = γ−1/2Du1 +2 +¯Dγ +γ +⇔ +1 +2γ1/2Dγ ¯Du + γ1/2∆u + 1 +2Du +¯Dγ +γ1/2 = 0 +⇔ +∇γ · ∇u + γ∆u = 0 ⇔ ∇ · (γ∇u) = 0 +As such, we conclude our proof of uniqueness for complex-conductivities in W 1,∞(Ω) from +the Dirichlet-to-Neumann map Λγ. 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Annals of mathematics, 153-169 +13 + diff --git a/GtFAT4oBgHgl3EQftR52/content/tmp_files/load_file.txt b/GtFAT4oBgHgl3EQftR52/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8b64601273b3e3a59acf3e6e8adf09673de19aa --- /dev/null +++ b/GtFAT4oBgHgl3EQftR52/content/tmp_files/load_file.txt @@ -0,0 +1,457 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf,len=456 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='08663v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='AP] 20 Jan 2023 Uniqueness of the inverse conductivity problem once-differentiable complex conductivities in three dimensions Ivan Pombo June 2022 Abstract We prove uniqueness of the inverse conductivity problem in three dimensions for complex conductivities in W 1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We apply quaternionic analysis to transform the inverse problem into an inverse Dirac scattering problem, as established in two dimensions by Brown and Uhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' This is a novel methodology that allows to extend the uniqueness result from once-differentiable real conductivities to complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 1 Introduction Let γ ∈ W 1,∞(Ω) be the complex-valued conductivity defined in a bounded Lipschitz domain Ω ⊂ R3 and given by γ = σ +iωǫ where σ is the electrical conductivity and satisfies σ ≥ c > 0, ǫ is the electrical permittivity and ω is the current frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Given a boundary value f ∈ H1/2(∂Ω) we can determine the respective electrical potential u ∈ H1(Ω) by uniquely solving � ∇ · (γ∇u) = 0, in Ω, u|∂Ω = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (1) This is the so-called conductivity equation which describes the behavior of the electrical potential, u, in a conductive body when a voltage potential is applied on the boundary, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In 1980, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Calder´on, [11] introduced the problem of whether one can recover uniquely a conductivity σ ∈ L∞(Ω) from the boundary measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=', from the Dirichlet-to- Neumann map Λσ : H1/2(∂Ω) → H−1/2(∂Ω), (2) f �→ σ ∂u ∂ν ���� ∂Ω which connects the voltage and electrical current at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The normal derivative exists as an element of H−1/2(∂Ω) by ⟨Λσf, g⟩ = � Ω σ∇u · ∇v dx (3) where v ∈ H1(Ω) with v|∂Ω = g and u solves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In the same paper, Calder´on was able to prove that the linearized problem at constant real conductivities has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Thereafter, many authors extended is work into global uniqueness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Sylvester and Uhlmann [31] used ideas of scattering theory, namely the exponential growing solutions of Faddeev [14] to obtain global uniqueness in dimensions n ≥ 3 for smooth conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Using this foundations the uniqueness for lesser regular conductivities was further generalized for dimensions n ≥ 3 in the works of ([1], [7], [8], [12], [13], [18], [22], [24], [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Currently, the best know result is due to Haberman [17] for conductivities γ ∈ W 1,3(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The reconstruction procedure for n ≥ 3 was obtained in both [22] and [25] independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' As far as we are aware, there seems to be no literature concerning reconstruction for conductivities with less than two derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In two dimensions the problem seems to be of a different nature and tools of complex analy- sis were used to establish uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Nachman [23] obtained uniqueness and a reconstruction method for conductivities with two derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The uniqueness result was soon extend for once-differentiable conductivities in [9] and a corresponding reconstruction method was ob- tained in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In 2006, Astala and P¨aiv¨arinta [3] gave a positive answer Calder´on’s problem 1 for σ ∈ L∞(Ω), σ ≥ c > 0, by providing the uniqueness proof through the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' All of this definitions can be extended to the complex-conductivity case with the assump- tion Re γ ≥ c > 0, in particular, we can define the Dirichlet-to-Neumann as above Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In this scenario, the first works was done in two-dimensions by Francini [15], by extending the work of Brown and Uhlmann [9] in two-dimensions proving uniqueness for small frequencies ω and γ ∈ W 2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Afterwards, Bukgheim influential paper [10] proved the general result in two-dimensions for complex-conductivities in W 2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' He reduced the (1) to a Schr¨odinger equation and shows uniqueness through the stationary phase method (based on is work many extensions followed in two-dimensions [2], [4], [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Recently, by mixing techniques of [9] and [10], Lakshtanov, Tejero and Vainberg obtained [21] uniqueness for Lipschitz complex- conductivities in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In [28], the author followed up their work to show that it is possible to reconstruct complex-conductivity with a jump at least in a certain set of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In three dimensions, the uniqueness results presented in [31] and [24] hold for twice- differentiable complex-conductivities in W 2,∞, but there was no reconstruction process pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Nachman’s reconstruction method in three dimensions [22] was used in [19] to re- construct complex conductivities from boundary measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Even though the Nachman’s proof was presented only for real conductivities, the paper [29] structures the proof in order to show the result holds for complex-conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' As far as we aware, the works with lower regularity require real-conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In this paper our interest resides in Calder´on’s problem for once-differentiable complex- conductivities in three-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The aim is to prove the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let Ω ⊂ R3 a bounded Lipschitz domain, γi ∈ W 1,∞(Ω), i = 1, 2 be two complex-valued conductivities with Re γi ≥ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' If Λγ1 = Λγ2, then γ1 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Our work basis itself on a transformation of (1) into a Dirac system of equation in three- dimensions with the help of quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In this scenario, we obtain a potential q that we want to determine from boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The main ideas follow the work of Brown and Uhlmann [9] for real conductivities in two-dimensions and Lakshtanov, Vainberg and Tejero [21], as well as the authors work [28], for complex-conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In this paper, we provide a novel reconstruction of the bounded potential q from the boundary data, but we are yet to be able to establish a relation between this boundary data and the Dirichlet-to-Neumann map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' This is essentially to answer Calder´on problem for Lipschitz complex conductivities, but the lack of a well-suited Poincar´e lemma that fits the quaternion structure does not allow such a simple work as in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 2 Minimalistic lesson of Quaternions We present the basis of the quaternionic framework we will use for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let R(2) be the real universal Clifford Algebra over R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' By definition, it is generated as an algebra over R by the elements {e0, e1, e2}, where e1, e2 is a basis of R2 with eiej + ejei = −2δij, for i, j = 1, 2 and e0 = 1 is the identity and commutes with the basis elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' This algebra has dimension 4 and is identified with the algebra of the quaternions, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' As such it holds e3 = e1e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In the following, we refer to this algebra as the quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' An element of the quaternions can be written as: x = x0 + x1e1 + x2e2 + x3e3, (4) where xj, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=', 3 are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We define the quaternionic conjugate ¯x of an element x as ¯x = x0 − x1e1 − x2e2 − x3e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (5) Let x, y ∈ H, we write xy for the resulting quaternionic product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The product ¯xy defines a Clifford valued inner product on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Further, we have xy = ¯y¯x and the conjugate of the conjugate of quaternion is that same quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let x ∈ H then Sc x = x0 denotes the scalar of x and Vec x = x − Sc x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The scalar of a Clifford inner product Sc(¯xy) is the usual inner product in R4 for x, y identified as vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' With this inner product H is an Hilbert space and the resulting norm is the usual Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In order to introduce some of the concepts we also extend the real quaternions to complex quaternions C2 = C ⊗ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Here, we use the same generators (e1, e2) as above, with the same 2 multiplication rules, however, the coefficients of the quaternion can be complex-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' That is, λ ∈ C ⊗ H may be written as λ = λ0 + λ1e1 + λ2e2 + λ3e3, λj ∈ C, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=', 3 (6) or still as λ = x + iy, x, y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (7) Due to the complexification we can still take another conjugation, to which we define has Hermitian conjugation and denote it by ·†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Explicitly, for λ ∈ C ⊗ H one has ¯λ† = λc 0 − λc 1e1 − λc 2e2 − λc 3e3, (8) where ·c denotes complex conjugation, or ¯λ† = ¯x − i¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (9) Similarly, one can introduce an associated inner product and norm in C ⊗ H by means of this conjugation: ⟨λ, µ⟩ = Sc � ¯λ†µ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' |λ|C2 = � Sc �¯λ†λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (10) For ease of notation, we also define for λ ∈ C2 the complex conjugation as ¯λc = λc 0 + λc 1e1 + λc 2e2 + λc 3e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (11) Now, we can also introduce Quaternion-valued functions f : R3 → C2 written as f = f0 + f1e1 + f2e2 + f3e3, where fj : R3 → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The Banach spaces Lp, W n,p of C2-valued functions are defines by requiring that each component is in such space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' On L2(Ω) we introduce the C2-valued inner product ⟨f, g⟩ = � Ω ¯f †(x)g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (12) Analogously to the Wirtinger derivatives in complex analysis, we have the Cauchy-Riemann operators under (x0, x1, x2) coordinates of R3 defined as D = ∂0 + e1∂1 + e2∂2, (13) where ∂j is the derivative with respect to the xj, j = 0, 1, 2 variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' and ¯D = ∂0 − e1∂1 − e2∂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (14) The vector part of the Cauchy-Riemann operator is designated as Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' It holds that D ¯D = ∆ where ∆ is the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We designate any function f fulfilling Df = 0 as a monogenic function, analogous to the holomorphic functions in complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1 A bit of Operator Theory Let Ω be a bounded domain and f : Ω → C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' All the results in this subsection were taken out from the classical book on quaternionic analysis of G¨urlebeck and Spr¨ossig [16] The Cauchy-Riemann operator has a right-inverse in the form (T f) (x) = − 1 ω � Ω y − x |y − x|3 f(y) dy, for x ∈ Ω, (15) where E(x, y) = − 1 ω y−x |y−x|3 is the generalized Cauchy kernel and ω = 4π stands for the surface area of the unit sphere in R3, that is, DT f = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' This operator acts from W k,p(Ω) to W k+1,p(Ω) with 1 < p < ∞ and k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Furthermore, we introduce the boundary integral operator for x /∈ ∂Ω (F∂Ωf) (x) = 1 ω � ∂Ω y − x |y − x|3 α(y)f(y) dS(y), (16) where α(y) is the outward pointing normal unit vector to ∂Ω at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We get the well-known Borel-Pompeiu formula (F∂Ωf) (x) + (T Df) (x) = f(x) for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Obviously, DF∂Ω = 0 holds through this formula it it holds that F∂Ω acts from W k− 1 p ,p(∂Ω) into W k,p(Ω), for k ∈ N and 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' One of the other well-known results we will need for our work is the Plemelj-Sokhotzki formula is obtaining by taking the trace of the boundary integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' First we introduce an operator over the boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 3 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' If f ∈ W k,p(∂Ω), then there exists the integral (S∂Ωf) = 1 2π � ∂Ω y − x |y − x|3 α(y)f(y) dS(y) (17) for all points x ∈ Ω in the sense of Cauchy principal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Furthermore, the operator S∂Ω is continuous in W k,p(∂Ω), for 1 < p < ∞, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' From this the Plemelj-Sokhotzki formula is given as: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let f ∈ W k,p(∂Ω) where by taking the non-tangential limit we have: lim x→x0, x∈Ω, x0∈∂Ω (F∂Ωf) (x) = 1 2 (f(x0) + (S∂Ωf) (x0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' One of the corollaries concerns the limit to the boundary acting as a projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' That is, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The operator P∂Ω denoting the projection onto the space of all H−valued functions which may be monogenicaly extended into the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Then, this projection may be represented as P∂Ω = 1 2 (I + S∂Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The proofs of this results and others to follow in our proofs may be found in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Now we are ready to start constructing our work on the inverse conductivity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 3 Inverse Dirac scattering problem Transforming our conductivity equation into another type of equation also changes the in- verse problem we are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We transform it into a system of equations based on the Cauchy-Riemann operator D (also called Dirac operator in some contexts) and thus we need to solve the inverse Dirac scattering problem first and only afterwards we care about the inverse conductivity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let u be a solution to (1) for some boundary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We define φ = γ1/2 � ¯Du, Du �T , remark that γ1/2 is well-defined since it is contained in C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Then, φ solves the system � Dφ1 = φ2q1, ¯Dφ2 = φ1q2, in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (18) where q1 = − 1 2 ¯ Dγ γ and q2 = − 1 2 Dγ γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' This transformation arises as follows: Dφ1 = D � γ1/2 ¯Du � = Dγ1/2 ¯Du + γ1/2∆u = Dγ1/2 ¯Du − γ−1/2∇γ · ∇u = Dγ1/2 ¯Du − 1 2γ−1/2 � Dγ ¯Du + Du ¯Dγ � = −1 2 � γ1/2Du � ¯Dγ γ = φ2q1 Carefully, we can extend our potential to the outside by setting γ ≡ 1 outside of Ω, which lead us to treat the study the equation in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1 Exponentially Growing Solutions We devise new exponentially growing solutions from the classical ones used in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In most literature works, the exponential behavior is defined through the function ex·ζ, with ζ ∈ C3 fulfilling ζ · ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' However, in our scenario this function does not fulfill Deix·ζ = 0, which brings the simplicity in all of the literature works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Since we know that it is harmonic we can generate a monogenic function through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let ζ ∈ C3 such that ζ · ζ := ζ2 0 + ζ2 1 + ζ2 2 = 0, then it holds ∆ex·ζ = 0 ⇔ D � ¯Dex·ζ� = 0 ≡ D � ex·ζ ¯ζ � 4 where now ζ is also defined as a quaternion through ζ = ζ0 + e1ζ1 + e2ζ2 ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Thus the function E(x, ζ) = ex·ζ ¯ζ is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' This also arises from the choice of ζ, since ζ ¯ζ = ζ2 0 + ζ2 1 + ζ2 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We make a clear statement of when ζ is a complex-quaternion or complex-a vector, but in most cases it is clear from context: it is a vector if it is in the exponent and a quaternion otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We assume the following asymptotic behaviour for φ: φ1 = ex·ζ ¯ζµ1, (19) φ2 = ex·¯ζc ¯ζcµ2 (20) Setting ˜µ1 = ¯ζµ1 and ˜µ2 = ¯ζcµ2 we have the equations: � D˜µ1 = e−x·(ζ− ¯ζc)˜µ2q1 ¯D˜µ2 = ex·(ζ− ¯ζc)˜µ1q2 (21) Further, we assume ˜µ → � 1 0 � as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' These system of equations will lead us to an integral equation from which we can extract interesting behaviour for ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The main point of this subsection is to demonstrate how we can obtain the system of integral equations related with (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Here, the approach is similar to [21], but we need to be careful due to the non-commutative nature of quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Recall, that DT = ¯D ¯T = I (in appropriate spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' applying this to (21) it holds: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ˜µ1 = 1 + T � e−x·(ζ−¯ζc)˜µ2q1 � ˜µ2 = T � ex·(ζ−¯ζc)˜µ1q2 � Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' we can obtain two integral equations with respect to their function: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ˜µ1 = 1 + T � e−x·(ζ−¯ζc) ¯T � ex·(ζ−¯ζc)˜µ1q2 � q1 � ˜µ2 = ¯T � ex·(ζ−¯ζc)q2 � + ¯T � ex·(ζ−¯ζc)T � e−x·(ζ−¯ζc)˜µ2q1 � q2 � \uf8f1 \uf8f2 \uf8f3 ˜µ1 = 1 + M 1˜µ1 ˜µ2 = T � e x· � ζ−ζC� q2 � + M 2˜µ2 ⇔ � [I − M 1](˜µ1 − 1) = M 11 [I − M 2](˜µ2) = ¯T � ex·(ζ−¯ζC)q2 � (22) Our objective now is to study the uniqueness and existence of this equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' we approach this task by proving that M j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 2 are contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Instead of working with all possible ζ ∈ C(2) fulfilling ζ ¯ζ = 0, we choose them for k ∈ R3 as ζ = k⊥ + ik 2 , k⊥ · k = 0 and k⊥ can be algorithmically found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We now describe our space of functions in terms of the space variable and k ∈ R3 as S = L∞ x (Lp k(|k| > R)) (23) where R > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In this space we prove that the operators M 1, M 2 are indeed contractions: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Then lim R→∞ ∥M j∥S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' To further study the system (22), we also need to show that the right-hand side is in S for an R large enough: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Then there exists R > 0 such that M 11 ∈ S, (24) ¯T � ex·(ζ−¯ζC)q2 � ∈ S (25) The above Lemmas imply the existence and uniqueness of (˜µ1, ˜µ2) solving the system (22) with respect to the potential q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' This is essential for the reconstruction procedure we show up next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='2 Reconstruction from scattering data In this section, we are mixing ideas from [21] and [22] with quaternionic theory to obtain the potential from the scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Starting from Clifford-Green theorem � Ω � g(x) � ¯Df(x) � + � g(x) ¯D � f(x) � dx = � ∂Ω g(x)η(x)f(x) dSx and using g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' iξ + ζ) = (iξ + ζ)e−x·(iξ+ζ) for ξ ∈ R3 such that (iξ + ζ) · (iξ + ζ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' This implies that g ¯D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Thus we define the scattering data as: h(ξ, ζ) = (iξ + ζ) � ∂Ω e−x·(iξ+ζ)η(x)φ2(x, ζ) dx (26) Applying now Clifford-Green theorem we obtain another form for the scattering data: h(ξ, ζ) = (iξ + ζ) � Ω e−x·(iξ+ζ) ¯Dφ2(x, ζ) dx = (iξ + ζ) � Ω e−ix·ξ � e−x·ζφ1(x, ζ) � q2(x) dx, by Dφ2 = φ1q2 = (iξ + ζ) � Ω e−ix·ξ (ζµ1(x, ζ)) q2(x) dx = iξ � Ω e−ix·ξ ˜µ1(x, ζ)q2(x) dx, since ¯ζζ = 0 = iξˆq2(ξ) + iξ � Ω e−ix·ξ [˜µ1(x, ζ) − 1] q2(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Thus, we have: ˆq2(ξ) = h(ξ, ζ) iξ − � Ω e−ix·ξ [˜µ1(x, ζ) − 1] q2(x) dx (27) This is yet not enough to reconstruct the potential, since the integral acts as a residual in the reconstruction and requires data that we technically do not have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Therefore, we integrate everything over an annulus in k � R<|k|<2R ˆq2(ξ) |k|3 dk = 1 iξ � R<|k|<2R h(ξ, ζ(k)) |k|3 dk − � R<|k|<2R 1 |k|3 � Ω e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx, (28) since the potential does not depend on k it can be taken out of the integral and taking the limit as R → ∞ leads the second integral on the right to decay to zero, obtaining a reconstruction formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let Ω ⊂ R3 a bounded Lipschitz domain, q ∈ L∞(Ω) be a complex-valued po- tential obtained through a conductivity γ ∈ W 1,∞(Ω), Re γ ≥ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Then, we can reconstruct the potential from ˆq2(ξ) = lim R→∞ C iξ � R<|k|<2R h(ξ, ζ(k)) |k|3 dk, (29) where C = 1 4π ln(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The scattering data is defined from the solutions of the Dirac system (22) and therefore it holds that ˜µ1 − 1 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Starting from (28) we obtain by integrating the right-hand side for any ξ ∈ R3: 4π ln 2 ˆq2(ξ) = 1 iξ � R<|k|<2R h(ξ, ζ(k)) |k|3 dk − � R<|k|<2R 1 |k|3 � Ω e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx 6 Let p > 2 and 1/p + 1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We estimate the last integral: ����� � R<|k|<2R 1 |k|3 � Ω e−ix·ξ [˜µ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' ζ(k)) − 1] q2(x) dx ����� ≤ ≤ � R<|k|<2R 1 |k|3 � Ω ���e−ix·ξ [˜µ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' ζ(k)) − 1] q2(x) ��� dx ≤ CΩ∥q∥∞ � R<|k|<2R 1 |k|3 sup x |˜µ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' ζ(k)) − 1| dk ≤ CΩ∥q∥∞ �� R<|k|<2R 1 |k|3q dk �1/q �� R<|k|<2R sup x |˜µ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' ζ(k)) − 1|p dk �1/p ≤ CΩ∥q∥∞∥˜µ1 − 1∥S �� R<|k|<2R 1 |k|3q dk �1/q Taking the limit as R → 0 the integral that is left goes to zero which implies the desired decay to zero and leaves us with our reconstruction formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Now, in order to connect the functions that solve the electrical conductivity equation (1) and the solutions to the Dirac equation (18), which are exponential growing, we introduce the following result: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let Ω be a bounded domain in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let φ = (φ1, φ2) be a solution of the Dirac system (18) for a potential q ∈ L∞(Ω) associated with the complex-conductivity γ ∈ W 1,∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' If φ1 = ¯φ2 then there exists a unique solution u of: � ¯Du = γ−1/2φ1, Du = γ−1/2φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (30) Further, this function fulfills the conductivity equation ∇ · (γ∇u) = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let us recall the main theorem, that we are now able to prove with all these pieces we assembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1 Let Ω ⊂ R3 a bounded Lipschitz domain, γi ∈ W 1,∞(Ω), i = 1, 2 be two complex-valued conductivities with Re γi ≥ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' If Λγ1 = Λγ2, then γ1 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='3, one only needs to show that the scattering data h for |k| >> 1 is uniquely determined by the Dirichlet-to-Neumann map Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Due to the lack of Poincar´e Lemma in our current framework in quaternionic analysis with the D and ¯D operator, than a new technique is required to obtain a similar proof to [9], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' For such, let us start with two conductivities γ1, γ2 in W 1,∞(Ω) for Ω a bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' By hypothesis Λγ1 = Λγ2 and thus by [29] we have γ1|∂Ω = γ2|∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Further, we can extend γj, j = 1, 2 outside Ω in such a way that in R3 \\ Ω and γj − 1 ∈ W 1,∞ comp(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let qj, φj, µj, hj, j = 1, 2 be the potential and the solution in (18), the function in (19), and the scattering data in (26) all associated with the conductivity γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Due to the scattering formulation at the boundary ∂Ω, then we just want to know if φ1 = φ2 on ∂Ω when |k| >> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' First, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='4, we know that there exists an u1 such that φ1 = γ1/2 1 ( ¯Du1, Du1)T , which is a solution to ∇ · (γ1∇u1) = 0 in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Now, let us define u2 by u2 = � u1 in R3 \\ Ω, u in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 7 where ˆu is the solution to the Dirichlet problem � ∇ · (γ2∇u) = 0 in Ω, u = u1 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let g ∈ C∞ c (R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Then, � R3 γ2∇u2∇g dx = � R3\\Ω γ1∇u1∇g dx + � Ω γ2∇ˆu∇g dx = − � ∂Ω Λγ1 � u1|∂Ω � g dsx + � ∂Ω Λγ2 � u|∂Ω � g dsx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Hence, u2 is the solution of ∇ · (γ2∇u2) = 0 in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Further, the following function ψ2 = γ1/2 2 � ¯Du2, Du2 �T is the solution of (18) where the potential is given by γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Furthermore, ψ2 has the asymptotics of φ1 in R3 \\ Ω, thus by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='2 it will be the unique solution of the respective integral equation of (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Thus, ψ2 will be equal φ2 when |k| > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Since, on the outside ψ2 ≡ φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Then we obtain: φ1 = φ2 in R3 \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In particular, we have equality at the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' So, this implies that if the Dirichlet- to-Neumann maps are equal the respective scattering data will also be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Thus, the Dirichlet-to-Neumann map uniquely determines the potential q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' From the definition of q, we can uniquely determine the conductivity γ up to a constant, which in the end is defined by γ|∂Ω which is uniquely determined by the Dirichlet-to-Neumann map Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 4 Auxiliary Proofs Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let us assume, without loss of generality, that f is a scalar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Further, we present the proof for M 1, since for M 2 it follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Recall, that we choose ζ ∈ C(2) with respect to k ∈ R(2) as ζ = k⊥ + ik 2 , k⊥ · k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' In vector form, this leads to ζ − ζc = ik which implies the following deductions: M 1f(x) = � R3 e−w·(ζ−¯ζc) x − w |x − w|3 � R3 ey·(ζ−¯ζc) w − y |w − y|3 f(y)q2(y) dy q1(w) dw = � R3 � R3 e−iw·k x − w |x − w|3 eiy·k w − y |w − y|3 f(y)q2(y)q1(w) dwdy = � R3 A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' k)f(y) dy, where A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' k) = � R3 e−i(w−y)·k x − y |x − y|3 w − y |w − y|3 q2(y)q1(w) dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Due to the compact support of the potential q2, it holds that A has compact support on the second variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let us now apply the norm in terms of k to it: ∥Mf(x, ·)∥Lp(|k|>R) = �� |k|>R |Mf(x, ζ)|p dσζ �1/p = �� |k|>R ���� � Ω A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' k)f(y) dy ���� p dσk �1/p ≤ � Ω �� |k|>R |A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' k)f(y)|p dσk �1/p dy ≤ � Ω sup k |A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' k)| dy ∥f∥S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 8 In order to complete the proof we show that the first integral goes to zero as R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let As be given with the extra factor α(s|x−w|)α(s|w−y|) in the integrand, where α ∈ C∞ is 1 outside a neighborhood of the origin and 0 inside a smaller neighborhood of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Since, � B1(0) � B1(0) 1 |w|2 1 |w − y|2 dw dy, it holds that for any ǫ > 0 there exists an s > 0 such that: � Ω |A − As| dy < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Further, we denote As0,n the function As0 with potentials q1, q2 replaced by their L1 smooth approximation Qn 1 , Qn 2 ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Since the other factors are bounded it holds � Ω |As0 − As0,n| dy < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Now it is enough to show that As0,n → 0 as |k| → 0 uniformly!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' All integrands inside of it will be in C∞ and uniformly bounded, thus by Riemann-Lebesgue the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Once again recall that ζ = � k⊥ + i k 2 � for k ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' First we show that M 11 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We have M 11 = � Ω � Ω e−iw·k x − w |x − w|3 w − y |w − y|3 eiy·kq2(y)q1(w) dy dw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' and applying the Lp norm in k followed with Minkowski integral inequality we obtain �� |k|>R |M 11|pdk �1/p ≤ � Ω |q1(w)| |x − w|2 �� |k|>R ���� � Ω eiy·k w − y |w − y|3 q2(y)dy ���� p dk �1/p dw The inner most integral resembles a Fourier transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' applying the Hausdorff-Young inequality for p > 2 we have �� |k|>R ���� � Ω eiy·k w − y |w − y|3 q2(y) dy ���� p dk �1/p ≤ �� Ω |q2(y)|p′ |w − y|2p′ dy �1/p′ < C∥q2∥∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' where the last inequality follows quickly by Young’s convolution inequality and Riesz type estimate of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Therefore, by the same Riesz type estimate it holds: �� |k|>R |M 11|p dk �1/p ≤ C∥q2∥∞ � Ω |q1(w)| |x − w|2 dw ≤ C′∥q2∥∞∥q1∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' To complete the proof we need to show statement (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Similarly, to the above proof, we have by Hausdorff-Young Inequality, Young’s convolution inequality and a Riesz type estimate the following: �� |k|>R ���� � R3 eiy·k x − y |x − y|3 q2(y) dσy ���� p dσk �1/p ≤ �� R3 ���� x − y |x − y|3 q2(y) ���� p′ dσy �1/p′ ≤ C∥q2∥∞ We need the following auxiliary result for the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let Ω be a bounded Lipschitz domain in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' If h is a scalar-valued and harmonic function that fulfills Vec(S∂Ωh) = 0, then h|∂Ω is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' First, note that I + S∂Ω = P∂Ω is a projector and by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='12 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='15 of [16] it holds that P∂Ωh is the boundary value of a monogenic function in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Since h is a scalar-valued function it holds that P∂Ωh = Sc(P∂Ωh) + Vec(P∂Ωh) = (h + Sc∂Ωh) + Vec(S∂Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Let w = (h + Sc∂Ωh) and v = Vec(S∂Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Now, we denote f as the monogenic extension of P∂Ωh in Ω, as such, the boundary values of f fulfill trf = w + v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Note that by hypothesis we have that v|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Hence, f is also an harmonic function, which implies that the scalar and vector components are harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' � ∆(Vecf) = 0, Vec f|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' By a mean value theorem or a maximum principle it holds that Vecf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Due to this and f being monogenic we obtain that Df = 0 ⇔ D(Ref) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Thus, Ref = c since D is a quaternionic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Consequently, the boundary values are also constant, which means that w = c in ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Since, Sc(S∂Ωh) is an averaging operator it holds that h = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='4 Suppose that (u, v) are solutions to the following equations: � ¯Du = γ−1/2φ1 Dv = γ−1/2φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' From applying the operator D and ¯D to the first and second equation respectively, we obtain from φ2 = φ H 1 and q2 = qH 1 the following: ∆u = D(γ−1/2φ1) = D(γ−1/2)φ1 + γ−1/2Dφ1 = −1 2γ−3/2(Dγφ1) + γ−1/2φ2q1 = γ−1/2 [q2φ1 + φ2q1] = γ−1/2 � qH 1 φ1 + φ H 1 q1 � = γ−1/2Sc (φ H 1 q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' and ∆v = ¯D(γ−1/2φ2) = ¯D(γ−1/2)φ2 + γ−1/2 ¯Dφ2 = −1 2γ−3/2( ¯Dγ)φ2 + γ−1/2φ1q2 = γ−1/2 [q1φ2 + φ1q2] = γ−1/2 � q1φ H 1 + φ1qH 1 � = γ−1/2Sc (φ1qH 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' The first thing to notice is that both equations imply that u and v must be scalar-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Further, notice that ∆(u − v) = γ−1/2 � Sc (φ H 1 q1) − Sc (φ1qH 1 ) � = γ−1/2 � Sc (φ H 1 q1) − Sc (q1φ H 1 ) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Therefore, h = u − v is an harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Our objective is to show that h ≡ 0, thus showing that u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' For such, let us consider the theory of integral transforms in quaternionic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' We have u = ¯T(γ−1/2φ1) + F ∂Ω(γ−1/2φ1) and u = ¯T(γ−1/2φ1) + F ∂Ω(u), which implies that F ∂Ω(γ−1/2φ1) = F ∂Ωu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' 10 Analogously, we obtain F∂Ω(γ−1/2φ2) = F∂Ωv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Here, we can extrapolate from the first equation and from u being scalar-valued that γ−1/2φ1F∂Ω = F∂Ωu ⇔ γ−1/2φ2F∂Ω = F∂Ωu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Applying the operator F∂Ω on the other side, we obtain: F 2 ∂Ωu = F∂Ω(γ−1/2φ2)F∂Ω and F 2 ∂Ωv = F∂Ω(γ−1/2φ2)F∂Ω ⇒ F 2 ∂Ωh = F 2 ∂Ω(u − v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' If we take the trace on both sides, the operator becomes a projector thus we obtain tr F∂Ωh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Now, through the Sokhotski-Plemelj formula we obtain: tr F∂Ωh = h|∂Ω + S∂Ωh = 0, at ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Since h is a scalar-valued function that we decompose this formulation with the scalar and vector part to obtain two conditions: � h + Sc(S∂Ωh) = 0 Vec(S∂Ωh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Through the second condition and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content='1 we have that h is constant over ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Now, given that h is a scalar constant, the first condition reduces to: h(1 + Sc(S∂Ω1)) = 0 By [16] we obtain that 1 + Sc(S∂Ω1) = 1/2 in ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Therefore, we conclude that h ≡ 0 in ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Given that h is harmonic, this immediately implies that h = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Therefore, we obtain u = v, and therefore there exists a unique solution to the initial system through the T and F∂Ω operators in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' To finalize, we only need to show that u fulfills the conductivity equation in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Bringing the first equation to light ¯Du = γ−1/2φ1, changing the side of the conductivity we get γ1/2 ¯Du = φ1 and applying the D operator to both sides now brings D � γ1/2 ¯Du � = Dφ1 ⇔ D � γ1/2� ¯Du + γ1/2∆u = φ2q1 ⇔ D � γ1/2� ¯Du + γ1/2∆u = γ−1/2Du1 2 ¯Dγ γ ⇔ 1 2γ1/2Dγ ¯Du + γ1/2∆u + 1 2Du ¯Dγ γ1/2 = 0 ⇔ ∇γ · ∇u + γ∆u = 0 ⇔ ∇ · (γ∇u) = 0 As such, we conclude our proof of uniqueness for complex-conductivities in W 1,∞(Ω) from the Dirichlet-to-Neumann map Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Notice that (29) even provides a reconstruction formula, but as mentioned in the previous section it is very unstable for computational purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' References [1] Alessandrini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Stable determination of conductivity by boundary measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' Applicable Analysis, 27(1-3), 153-172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' [2] Astala, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=', Faraco, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=', Rogers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'} +page_content=' (2013).' metadata={'source': 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b/KtE0T4oBgHgl3EQfzgKz/content/tmp_files/2301.02674v1.pdf.txt @@ -0,0 +1,2738 @@ +Draft version January 10, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Molecular Mapping of DR Tau’s Protoplanetary Disk, Envelope, Outflow, and Large-Scale Spiral Arm +Jane Huang,1, ∗ Edwin A. Bergin,1 Jaehan Bae,2 Myriam Benisty,3 and Sean M. Andrews4 +1Department of Astronomy, University of Michigan, 323 West Hall, 1085 S. University Avenue, Ann Arbor, MI 48109, United States of +America +2Department of Astronomy, University of Florida, Gainesville, FL 32611, United States of America +3Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France +4Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA +ABSTRACT +DR Tau has been noted for its unusually high variability in comparison with other T Tauri stars. +Although it is one of the most extensively studied pre-main sequence stars, observations with millimeter +interferometry have so far been relatively limited. We present NOEMA images of 12CO, 13CO, C18O, +SO, DCO+, and H2CO toward DR Tau at a resolution of ∼ 0.5′′ (∼ 100 au). In addition to the +protoplanetary disk, CO emission reveals an envelope, a faint asymmetric outflow, and a spiral arm with +a clump. The ∼ 1200 au extent of the CO arm far exceeds that of the spiral arms previously detected +in scattered light, which underlines the necessity of sensitive molecular imaging for contextualizing +the disk environment. The kinematics and compact emission distribution of C18O, SO, DCO+, and +H2CO indicate that they originate primarily from within the Keplerian circumstellar disk. The SO +emission, though, also exhibits an asymmetry that may be due to interaction with infalling material or +unresolved substructure. The complex environment of DR Tau is reminiscent of those of outbursting +FUor sources and some EXor sources, suggesting that DR Tau’s extreme stellar activity could likewise +be linked to disk instabilities promoted by large-scale infall. +Keywords: protoplanetary disks—ISM: molecules—stars: individual (DR Tau) +1. INTRODUCTION +In the classic schema of low-mass star formation, +young stellar objects (YSOs) are divided into four +classes (0, I, II, and III) based on their spectral energy +distributions (e.g., Lada & Wilking 1984; Lada 1987; +Andre et al. 1993). These classes are generally thought +to correspond to different evolutionary stages, such that +a Class 0 YSO has an envelope mass comparable to or +greater than that of the protostar (and its possible disk), +a Class I YSO has an envelope that is less massive than +the protostar but still comparable to its disk, a Class II +YSO has a disk with negligible envelope material, and +a Class III YSO has negligible amounts of remaining +circumstellar material (e.g., Wilking et al. 1989; Andre +& Montmerle 1994; Dunham et al. 2014). The corre- +spondence between SED class, evolutionary stage, and +morphology, though, is known to be imperfect (e.g., Ro- +bitaille et al. 2006). +Corresponding author: Jane Huang +jnhuang@umich.edu +∗ NASA Hubble Fellowship Program Sagan Fellow +Planet formation models often adopt the characteris- +tics of envelope-free Class II disks as a starting point +(e.g., ¨Oberg et al. 2011a; Lambrechts & Johansen 2012; +Zhang et al. 2018). However, scattered light and molec- +ular imaging have yielded identifications of a number of +Class II disks that appear to be interacting either with +(remnant) envelopes or ambient cloud material (e.g., +Grady et al. 1999; Garufi et al. 2020; Ginski et al. 2021; +Huang et al. 2022). The pace of these identifications has +increased with the advent of instruments such as ALMA +and VLT/SPHERE. Detections of gaps and rings in the +millimeter continuum of some Class II disks that ap- +pear to have remnant envelope material, and even a few +embedded Class I disks, offer evidence that planet for- +mation can take place under more dynamically complex +conditions than typically assumed (e.g., ALMA Part- +nership et al. 2015; Segura-Cox et al. 2020; Huang et al. +2021; Kanagawa et al. 2021). Moreover, Currie et al. +(2022) recently detected a protoplanet in the disk of +AB Aur, a system that appears to still be undergo- +ing infall from a remnant envelope (e.g., Tang et al. +2012). Simulations suggest that accretion of cloud or +envelope material by the disk can influence its thermal +structure, surface density profile, stability, and degree +of misalignment (e.g., Bae et al. 2015; Dullemond et al. +arXiv:2301.02674v1 [astro-ph.SR] 6 Jan 2023 + +2 +Huang et al. +2019; Kuznetsova et al. 2022). These disk conditions, in +turn, are expected to influence where, when, and how +planets form and migrate, as well as their composition +(e.g., Stevenson & Lunine 1988; Boss 1997; Kokubo & +Ida 2002). Hence, observations of the immediate envi- +ronments of young stars are essential to establish the +range of circumstances under which planet formation +might proceed. +Recent +observations +of +DR +Tau +(J2000 +04:47:06.215+16:58:42.81), a T Tauri star located at +a distance of 192 ± 1 pc in the Taurus star-forming re- +gion (Gaia Collaboration et al. 2021; Bailer-Jones et al. +2021), have suggested that its disk is being externally +perturbed. Mesa et al. (2022) detected spiral arms in +scattered light images of the DR Tau protoplanetary +disk and hypothesized that one of them was triggered +by infalling material. Meanwhile, Sturm et al. (2022) +detected non-Keplerian emission in ALMA observations +of 13CO and [C I] toward DR Tau, attributing this +component to an infalling stream of gas. +DR Tau is perhaps best known for its unusual degree +of stellar variability. +The star has faded and bright- +ened in B-band by several magnitudes over the course +of almost a century (Chavarria-K. 1979). Most notably, +DR Tau brightened in B-band by about five magni- +tudes between 1970 to 1979, an event that Chavarria- +K. (1979) compared to the outbursts of FUor (also +known as FU Ori) sources. DR Tau also exhibits sig- +nificant short-term spectroscopic and photometric vari- +ability—on timescales of a few days, DR Tau has been +observed to change by up to a couple magnitudes in B- +band and by up to a factor of a few in its optical line +fluxes (e.g., Bertout et al. 1977; Guenther & Hessman +1993; Alencar et al. 2001). DR Tau has a high stellar +mass accretion rate of 4.8 × 10−7 M⊙ yr−1 (McClure +2019). This high accretion level leads to significant con- +tinuum veiling, which poses a challenge for determining +its spectral type (e.g., Cohen & Kuhi 1979). Spectral +type estimates have ranged from M0 to K4 (e.g. Imhoff +& Appenzeller 1987; Herczeg & Hillenbrand 2014; Mc- +Clure 2019; Gangi et al. 2022). +DR Tau was part of the original list of outbursting +EXor variables by Herbig (1989), although it has not +always been included in subsequent compilations of EX- +ors (e.g., Audard et al. 2014). DR Tau is unique among +the EXors listed in the Herbig (1989) catalog in that +the 18-year rise time to its outburst was much longer +than those of the other EXors, which were typically on +the order of a couple hundred days. EXors are usually +distinguished from outbursting FUor sources insofar as +EXor outbursts tend to be more modest in magnitude +and duration, and EXors have T Tauri-like spectra dur- +ing outbursts rather than the supergiant-like spectra +of FUors (e.g., Audard et al. 2014). Several hypothe- +ses have been proposed to account for the outbursts of +young stars, including disk instabilities driven by mass +buildup through infall from envelopes or cloud material, +binary interactions, and stellar flybys (e.g., Bonnell & +Bastien 1992; Vorobyov & Basu 2005; Zhu et al. 2010; +Forgan & Rice 2010; Bae et al. 2014; Dullemond et al. +2019). Because these outbursts affect the disk thermal +structure, they may significantly affect how planet for- +mation proceeds by altering molecular abundances, dust +properties, and snowline locations (e.g., Juh´asz et al. +2012; Banzatti et al. 2012; Cieza et al. 2016; van ’t Hoff +et al. 2018; Jørgensen et al. 2022). The hypothesized +connection between outbursts and environmental inter- +actions further motivates an examination of DR Tau’s +surroundings. +Although DR Tau has been a popular target for obser- +vations ranging from infrared to ultraviolet wavelengths +(e.g., Kenyon et al. 1994; Ardila et al. 2002; Salyk et al. +2008; Pontoppidan et al. 2011; Banzatti et al. 2014, and +references above), relatively few observations with mil- +limeter interferometry have been reported. +The mil- +limeter continuum, which traces the distribution of large +dust grains, has been imaged on several occasions (e.g., +Kitamura et al. 2002; Andrews & Williams 2007; Taz- +zari et al. 2016; Long et al. 2019). The millimeter con- +tinuum emission is fairly compact, with 95% of the flux +contained within a 53 au radius (Long et al. 2019). Al- +though no substructures are immediately apparent in +the highest resolution image to date (tracing scales down +to ∼ 20 au), modeling of the visibilities suggests the +presence of gaps and rings that may be associated with +planet-disk interactions (Jennings et al. 2020). Other +than 13CO, C18O, and [C I] (Braun et al. 2021; Sturm +et al. 2022), no interferometric line observations of DR +Tau have previously been published. +The upgraded wideband capabilities of the Northern +Extended Millimeter Array (NOEMA) provided an op- +portunity to observe a number of lines simultaneously +toward DR Tau. We obtained sensitive observations of +12CO, 13CO, C18O, SO, DCO+, and H2CO at a resolu- +tion of ∼ 0.5′′ (∼ 100 au) to map DR Tau’s structure. +The observations and data reduction are summarized +in Section 2. The molecular detections are analyzed in +Section 3, and the implications of DR Tau’s complex +structures are discussed in Section 4. The summary and +conclusions are presented in Section 5. +2. OBSERVATIONS AND DATA REDUCTION +DR Tau was observed with the NOEMA PolyFiX +correlator in dual polarization mode during program +W20BE (PI: J. Huang). The correlator setup covered +frequencies from 213.9-221.6 GHz and 229.4-237.2 GHz +at a resolution of 2 MHz. Within these frequency ranges, +we placed a series of chunks, each with a resolution of +62.5 kHz and width of 64 MHz, in order to resolve molec- +ular lines of interest (detailed further in Section 3 and +Appendix A). +The first set of observations was executed in C config- +uration on 2021 January 08, with baseline lengths rang- +ing from 24 to 328 m. The second set of observations + +Molecular Mapping of DR Tau +3 +was executed in A configuration on 2021 March 03, with +baseline lengths ranging from 32 to 760 m. Each con- +figuration used eleven antennas. For each set of obser- +vations, LkHα 101 served as the flux calibrator, 3C 84 +served as the bandpass calibrator, and 0446+112 and +0507+179 served as the phase calibrators. The on-source +time was 3.0 hours in C configuration and 3.4 hours in +A configuration. +The raw data were calibrated with the NOEMA +pipeline in CLIC, which is part of the GILDAS package +(Pety 2005; Gildas Team 2013). +Then, the following +steps were performed with the GILDAS MAPPING software. +The calibrated visibilities were written out to separate +uv-tables corresponding to the low spectral resolution, +wide bandwidth data and the high spectral resolution, +narrow bandwidth spectral windows. After flagging of +channels with strong line emission, the wide bandwidth +uv-tables were spectrally averaged to produce contin- +uum uv-tables. +For each of the four basebands, the +continuum was imaged with the CLEAN algorithm and +three phase self-calibration loops were performed using +solution intervals of 180, 90, and 45 seconds. The self- +calibration solutions were then applied to the uv-tables +for the narrow spectral windows that fell within the cor- +responding basebands. Continuum subtraction was per- +formed for each spectral window separately in the uv +plane by fitting a linear baseline. +The self-calibrated, continuum-subtracted uv tables +were converted to measurement sets to enable imag- +ing with the Common Astronomy Software Applica- +tions (CASA) 6.4 (CASA Team et al. 2022). Because +GILDAS outputs frequencies in the rest frame of the +source (i.e., the frequency that corresponds to the source +systemic velocity input by the observer is the rest fre- +quency of the line of interest), we had to manually cor- +rect the frequencies in the measurement sets so that +CASA would output image cubes with the appropri- +ate LSRK velocities. +Each line was imaged with the +tclean implementation of the multi-scale CLEAN al- +gorithm (Rau & Cornwell 2011). +We set the robust +value to 0.5 and and the image cube channel spacing +to 0.2 km s−1. +To accommodate the irregular mor- +phology of the 12CO and 13CO J = 2 − 1 emission, +we employed the auto-multithresh algorithm (Kep- +ley et al. 2020) to define the CLEAN masks, choos- +ing the following parameter values after some experi- +mentation: sidelobethreshold=2.0, noisethreshold=4.0, +minbeamfrac=0.3, and negativethreshold=7.0. +Initial +imaging tests yielded prominent striping artifacts due +to the poor uv sampling of the spatially extended cloud +emission, so we re-imaged these lines without baselines +shorter than 20 kλ. +For the other molecules, where +only compact emission was detected, we used a circu- +lar CLEAN mask with a radius of 2.6′′ and included all +baselines. A Gaussian uv taper of 1.0′′ was used to in- +crease sensitivity to weaker lines (i.e., lines other than +12CO, 13CO, C18O, SO, DCO+, and H2CO JKaKc = +303 − 202). After CLEANing, a primary beam correc- +tion was applied to each image cube. +Calibrated visibilities and images can be down- +loaded +at +https://zenodo.org/record/7370498# +.Y7U-qezMKeB. +3. RESULTS +3.1. Overview of Line Observations +The primary line targets were 12CO, 13CO, C18O, SO, +DCO+, and H2CO. The CO isotopologues serve as gas +tracers, SO is a potential shock tracer (e.g., Pineau des +Forˆets et al. 1993), and H2CO and DCO+ are common +cold disk gas tracers (e.g., Huang et al. 2017; Pegues +et al. 2020). The synthesized beam and per-channel rms +(estimated from line-free channels) for the primary line +targets are listed in Table 1, and channel maps are pre- +sented in Appendix B. Spectra for the detected lines, +which were extracted using circular masks with diam- +eters listed in Table 1, are shown in Figure 1. +Since +the spatial extent of 12CO and 13CO are ambiguous due +to spatial filtering and cloud contamination, we used +extraction masks approximately equal to the primary +beam FWHM at 1.3 mm (21′′). The mask sizes for the +other lines were chosen based on the approximate radial +extent of the 3σ emission in the image cubes. Fluxes +were measured by integrating each spectrum within the +velocity ranges listed in Table 1. The velocity integra- +tion ranges for the CO isotopologues were selected based +on where emission above the 3σ level is detected. For +the weaker lines, the C18O velocity integration range +was adopted. The 1σ flux uncertainties were estimated +as ∆v × +√ +N × σspec, where ∆v is the channel width (in +km s−1), N is the number of channels spanned by the +line, and σspec is the standard deviation (in Jy) mea- +sured from a signal-free portion of the spectrum (this is +not to be confused with the per-channel rms value listed +in Table 1 (in mJy beam−1), which is calculated from the +image cube). However, the statistical uncertainties do +not capture the true uncertainty of the fluxes for 12CO +and 13CO, which are affected by cloud contamination +and spatial filtering. +We categorize a line as detected if emission is above +the 5σ level within 2′′ of DR Tau in at least one channel +of the image cube and above the 3σ level in at least two +adjacent channels. By these criteria, 12CO, 13CO, C18O, +SO, DCO+, and H2CO 303 − 202 are firmly detected. +While H2CO 321 − 220 does not meet these criteria, its +integrated flux is ⪆ 4σ when extracted over the same ve- +locity range and emitting region as the strong 303 − 202 +transition, so this line is considered to be tentatively de- +tected. The channel maps for the 322 − 221 transition +(Appendix B) show 4σ emission at 10.2 kms−1 that is +cospatial with the stronger 303 − 202 transition, but the +velocity-integrated flux from the spectrum is < 2σ. Fur- +thermore, the peak of the spectrum occurs at a velocity + +4 +Huang et al. +Table 1. Imaging Summary for Primary Line Targets +Transition +Synthesized beam +Per-channel RMS noisea +Velocity rangeb +Extraction Mask Diameter +Fluxc +(arcsec × arcsec (◦)) +(mJy beam−1) +(km s−1) +(arcsec) +(mJy km s−1) +12CO J = 2 − 1 +0.79 × 0.47 (18.2◦) +7 +[−2, 17] +21 +37900 ± 200d +13CO J = 2 − 1 +0.84 × 0.49 (17.3◦) +6 +[7.6, 11.4] +21 +5730 ± 70d +C18O J = 2 − 1 +0.85 × 0.50 (17.3◦) +6 +[9.0, 10.8] +4 +622 ± 10 +SO JN = 65 − 54 +0.85 × 0.50 (17.2◦) +6 +[9.0, 10.8] +3 +195 ± 9 +SO JN = 55 − 44 +0.86 × 0.50 (17.1◦) +6 +[9.0, 10.8] +3 +96 ± 10 +DCO+ J = 3 − 2 +0.86 × 0.50 (17.1◦) +6 +[9.0, 10.8] +3 +40 ± 10 +H2CO JKaKc = 303 − 202 +0.86 × 0.50 (17.2◦) +6 +[9.0, 10.8] +3 +248 ± 11 +H2CO JKaKc = 322 − 221 +1.20 × 0.93 (17.1◦) +7 +[9.0, 10.8] +3 +< 30 +H2CO JKaKc = 321 − 220 +1.20 × 0.93 (17.1◦) +7 +[9.0, 10.8] +3 +38 ± 8 +aWith channel widths of 0.2 km s−1. +b LSRK velocity range over which moment maps are produced and the flux is estimated. +c The 1σ error bars do not include the systematic flux uncertainty (∼ 10%). +dThese lines are significantly affected by spatial filtering, so the statistical uncertainty does not reflect the true uncertainty in the fluxes. +well offset from the peak of the 303 − 202 and 321 − 220 +lines. Therefore, we do not consider the 322 − 221 tran- +sition to be detected. +Integrated intensity maps of the primary line targets +are presented in Figure 2, using the velocity integration +ranges listed in Table 1. The intensity-weighted veloc- +ity maps of the stronger lines are presented in Figure +3. For 12CO and 13CO, the integrated intensity maps +excluded pixels in the image cube below the 3σ level +and the intensity-weighted velocity map excluded pix- +els below the 6σ level in order to reduce contributions +from cloud contamination and artifacts from spatial fil- +tering. For all other lines, no clipping was used for the +integrated intensity maps, and a 4σ clip was adopted for +the intensity-weighed velocity maps. +A summary of the auxiliary line observations (none of +which yielded a detection) is presented in Appendix C. +3.2. Structures traced by CO isotopologues +Due to their differing optical depths, the three de- +tected CO isotopologues reveal different components of +the DR Tau system, including the circumstellar disk, an +arm, an envelope, and an outflow. +An overhead car- +toon schematic of the system is shown in Figure 4. We +describe each component in further detail below. +3.2.1. The circumstellar disk +C18O is the least optically thick of the three detected +CO isotopologues and therefore best traces the Keple- +rian rotation of the circumstellar disk (see Figure 3). +The southern side is blueshifted and the northern side is +redshifted relative to the systemic velocity, which Braun +et al. (2021) estimated to be vsys = 9.9+0.08 +−0.09 km s−1 from +ALMA observations of 13CO and C18O J = 2−1. Signs +of Keplerian rotation are visible in the inner regions of +the NOEMA 13CO intensity-weighted velocity map and +coincide with the bright, compact emission component +in the integrated intensity map, but the disk edge is +not well-defined due to the presence of extended, non- +Keplerian emission. From visual inspection of the 13CO +emission, we estimate that the Keplerian disk has a ra- +dial extent of ∼ 300 au, but this should only be consid- +ered a lower bound for the disk size because the abun- +dance of 13CO is generally too low in the outer disk +to recover the disk size robustly (e.g., Trapman et al. +2019). Finally, 12CO is dominated by large-scale, non- +Keplerian structures. +The NOEMA observations do not strongly constrain +the disk orientation, since the C18O emission is spanned +by only a few synthesized beams. However, Long et al. +(2019) measured a position angle (P.A.) of 3.4+8.2 +−8.0 de- +grees east of north and an inclination angle of 5.4+2.1 +−2.6 +degrees from ALMA millimeter continuum observations +at an angular resolution of ∼ 0.1′′, which corresponds +to ∼ 20 au. +We adopt these values for our analysis. +Although our new 12CO and 13CO observations show +significant non-disk emission, Sturm et al. (2022) found +that their ALMA C18O observations could be largely re- +produced by a Keplerian disk model employing the disk +orientation derived from Long et al. (2019). Because the +disk is nearly face-on, the C18O spectrum only exhibits +a single peak at the systemic velocity rather than the +double-peak characteristic of more inclined disks. + +Molecular Mapping of DR Tau +5 +5 +0 +5 +10 +15 +20 +25 +LSRK Velocity (km s +1) +0 +5 +10 +15 +20 +25 +Flux (Jy) +12CO 2-1 +5.0 +7.5 +10.0 +12.5 +15.0 +LSRK Velocity (km s +1) +0.0 +2.5 +5.0 +7.5 +Flux (Jy) +13CO 2-1 +5.0 +7.5 +10.0 +12.5 +15.0 +LSRK Velocity (km s +1) +0.00 +0.25 +0.50 +0.75 +Flux (Jy) +C18O 2-1 +5.0 +7.5 +10.0 +12.5 +15.0 +LSRK Velocity (km s +1) +0.0 +0.1 +0.2 +Flux (Jy) +SO 65 +54 +5.0 +7.5 +10.0 +12.5 +15.0 +LSRK Velocity (km s +1) +0.05 +0.00 +0.05 +0.10 +0.15 +Flux (Jy) +SO 55 +44 +5.0 +7.5 +10.0 +12.5 +15.0 +LSRK Velocity (km s +1) +0.04 +0.00 +0.04 +0.08 +Flux (Jy) +DCO + 3 +2 +5.0 +7.5 +10.0 +12.5 +15.0 +LSRK Velocity (km s +1) +0.0 +0.1 +0.2 +0.3 +0.4 +Flux (Jy) +H2CO 303 +202 +5.0 +7.5 +10.0 +12.5 +15.0 +LSRK Velocity (km s +1) +0.050 +0.025 +0.000 +0.025 +0.050 +Flux (Jy) +H2CO 322 +221 +5.0 +7.5 +10.0 +12.5 +15.0 +LSRK Velocity (km s +1) +0.025 +0.000 +0.025 +0.050 +Flux (Jy) +H2CO 321 +220 +Figure 1. Source-integrated spectra of the primary line targets toward DR Tau. The vertical red dotted line marks the systemic +velocity. The gray bars denote regions where cloud contamination is apparent. + +6 +Huang et al. +9 +6 +3 +0 +3 +6 +9 +9 +6 +3 +0 +3 +6 +9 +12CO 2 +1 +200 au +5 100 300 +900 2100 +9 +6 +3 +0 +3 +6 +9 +9 +6 +3 +0 +3 +6 +9 +13CO 2 +1 +200 au +5 +25 +75 +150 300 +Integrated Intensity (mJy beam +1 km s +1) +9 +6 +3 +0 +3 +6 +9 +9 +6 +3 +0 +3 +6 +9 +C18O 2 +1 +200 au +0 +25 50 +100 +160 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +SO 65 +54 +200 au +0 +10 20 30 40 50 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +SO 55 +44 +200 au +0 +7 +14 +21 +28 +35 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +DCO + 3 +2 +200 au +0 +5 +10 +15 +20 +25 +2 +1 +0 +1 +2 + [''] +2 +1 +0 +1 +2 + [''] +H2CO 303 +202 +200 au +0 +15 +30 +45 +60 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +H2CO 322 +221 +200 au +0 +4 +8 +12 +16 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +H2CO 321 +220 +200 au +0 +6 +12 +18 +24 +Figure 2. Integrated intensity maps of primary line targets observed toward DR Tau. The synthesized beam is drawn in the +lower left corner of each panel. Black crosses mark the disk center. The axes show offsets from the disk center in arcseconds. +For the CO isotopologues, the color scale uses an arcsinh stretch to make faint extended features more visible. Note that the +size scales are different between the top row and the other rows. + +Molecular Mapping of DR Tau +7 +8 +4 +0 +4 +8 +8 +4 +0 +4 +8 +12CO 2 +1 +200 au +8.4 +9.4 +10.4 +11.4 +8 +4 +0 +4 +8 +8 +4 +0 +4 +8 +13CO 2 +1 +200 au +8.4 +9.4 +10.4 +11.4 +Intensity-weighted velocity +(km s +1) +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +C18O 2 +1 +200 au +9.3 +9.6 +9.9 10.2 10.5 +2 +1 +0 +1 +2 + [''] +2 +1 +0 +1 +2 + [''] +SO 65 +54 +200 au +9.3 +9.6 +9.9 10.2 10.5 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +SO 55 +44 +200 au +9.3 +9.6 +9.9 10.2 10.5 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +H2CO 303 +202 +200 au +9.3 +9.6 +9.9 10.2 10.5 +Figure 3. Intensity-weighted velocity maps of strong lines detected toward DR Tau. The synthesized beam is drawn in the lower +left corner of each panel. The purple cross denotes the disk center. The axes show offsets from the disk center in arcseconds. +Note that the velocity ranges and size scales are not the same for all imags. +3.2.2. Blueshifted spiral arm +The intensity-weighted velocity maps for 12CO and +13CO (Figure 3) both show an arm that is blueshifted +with respect to the systemic velocity. +To isolate the +emission from the CO arm, we produced integrated in- +tensity maps between 8.4 and 9.0 km s−1 (Figure 5). +The arm is connected to the south side of the disk and +curves around the western side, terminating at a pro- +jected distance of ∼ 1200 au from DR Tau at a P.A. of +∼ 330◦. The 12CO emission also shows a clump along +the arm at a projected distance of ∼ 500 au southwest +from DR Tau. This arm was not detected in previously +published high-resolution ALMA 13CO images of DR +Tau (Sturm et al. 2022), presumably due to some com- +bination of lack of sensitivity and spatial filtering. How- +ever, low angular resolution ALMA ACA observations of +[C I] from Sturm et al. (2022) show extended blueshifted +emission, which may originate from the arm traced by +CO in our NOEMA observations. +In order to estimate the pitch angle of the arm, we +transformed the integrated intensity map of the arm into +a polar coordinate map (i.e., as a function of deprojected +radius R and polar angle θ), assuming that the arm is in +the plane of the disk (Figure 6). We then measured the +position of the spiral arm by searching for local radial +maxima in the polar coordinate map for fixed values of θ +in steps of 8◦ from 124◦ to 260◦. The arm was modelled +as an Archimedean spiral of the form R(θ) = a + cθ3, +where θ is in radians. (We found that logarithmic spirals +and Archimedean spirals with smaller exponents did not +fit the data well). The log-likelihood function was spec- +ified as log L = −0.5 � +n +� +(Rdata−Rmodel)2 +σ2 ++ log(2πσ2) +� +, +where σ is the standard deviation of the major axis of +the synthesized beam. Uniform priors of [0, 2000] and +[−2000, 0] were used for a and c, respectively. Posteriors +were explored using the affine-invariant sampler emcee +(Goodman & Weare 2010; Foreman-Mackey et al. 2013) +with 40 walkers and 1000 steps. After discarding the +first 500 steps as burn-in, we computed the 50th per- + +8 +Huang et al. +Outflow +Disk +Blueshifted +arm +Line of Sight +EAST +WEST +Envelope +Figure 4. A proposed cartoon schematic of the DR Tau system from an overhead perspective (i.e., perpendicular to the line +of sight). The components are not drawn to scale. Note that while the disk is drawn such that the east side is tilted toward the +observer in order to show that the disk is slightly inclined, the observations do not constrain which side is closer to the observer. +The 3-dimensional orientations of the envelope and the arm are not known in detail, but the former is drawn in front of the +disk and the latter is drawn behind the disk (from the perspective of the observer) under the assumption of infalling motion. +However, the observations may also be explained by other configurations of the structures. +6 +3 +0 +3 +6 + [''] +6 +3 +0 +3 +6 + [''] +12CO J = 2 +1 (8.4-9.0 km s +1) +Clump +200 au +0.0 +50.0 +100.0 +150.0 +200.0 +250.0 +6 +3 +0 +3 +6 +6 +3 +0 +3 +6 +13CO J = 2 +1 (8.4-9.0 km s +1) +200 au +0.0 +6.0 +12.0 +18.0 +24.0 +Integrated Intensity +(mJy beam +1 km s +1) +Figure 5. Integrated intensity maps of 12CO (left) and 13CO, summed up between 8.4 and 9.0 km s−1 to highlight DR Tau’s +blueshifted spiral arm. The blue cross marks the center of the disk. The synthesized beam is shown as a white ellipse in the +lower left corner of each panel. The 12CO color scale is saturated in order to show the fainter arm emission more clearly. + +Molecular Mapping of DR Tau +9 +0 +300 +600 +900 +1200 +Deprojected radius (au) +0 +60 +120 +180 +240 +300 +360 + (degrees) +6 +4 +2 +0 +2 +4 +6 + [''] +6 +4 +2 +0 +2 +4 +6 + [''] +120 150 180 210 240 270 + (degrees) +0 +15 +30 +45 +60 +75 +Pitch angle (degrees) +Figure 6. Left: Integrated intensity map of the 12CO arm, replotted as a function of deprojected radius and polar angle θ. +Center: Integrated intensity map of the 12CO arm, overplotted with the spiral function defined by the posterior median values +of the spiral parameters. Right: Pitch angle of the arm as a function of the polar angle θ. The black curve corresponds to the +values derived from the median of the spiral arm parameter posteriors, while the blue curves correspond to 1000 random draws +from the posterior. + +10 +Huang et al. +centile of the marginal posterior distribution to obtain a +point estimate and the 16th and 84th percentiles to ob- +tain error estimates: a = 1060±30 au and c = −7.8±0.6 +au. We computed the pitch angles (φ = arctan +��� 1 +R +dR +dθ +��) +� +corresponding to the median values of a and c, then +also computed pitch angles for spiral curves defined by +1000 random draws of (a, c) from the posterior. Figure +6 shows the median spiral plotted over the integrated +intensity map and a plot of the derived pitch angles as +a function of polar angle θ. The pitch angles range from +6 to 56 degrees between polar angle values of 124 to +260 degrees (corresponding to deprojected radius values +between 980 and 330 au). +In other words, the pitch +angle appears to decrease with distance from the star, +although the true values may differ if the assumption +that the arm is in the plane of the disk is incorrect. +We computed the escape velocity, vesc = +� +2GM∗ +r +, at +the tip of the arm to assess whether it is gravitationally +bound to DR Tau. +The dynamical mass of DR Tau +has been measured to be 1.2 M⊙ (Braun et al. 2021). +Emission from the arm is detected up to r ∼ 1200 au +at an LSRK velocity of 8.8 km s−1, which is offset from +the systemic velocity by 1.1 km s−1. The corresponding +escape velocity at r = 1200 au is 1.3 km s−1. Thus the +arm appears to be compatible with being gravitationally +bound to DR Tau, but not definitively so, since there +may also be a transverse velocity component. +Mesa et al. (2022) recently identified two spiral arms +in SPHERE H-band Qφ observations of DR Tau. Figure +7 compares the arms identified in the SPHERE image +to the CO arm. The CO arm is much more extended +than the scattered light arms, which are only detected +up to ∼ 220 au in projection from the star. Because the +NOEMA synthesized beam is comparable in scale to the +SPHERE spiral arms, it is not clear whether the CO arm +is an extension of one of the arms detected in scattered +light or a separate structure. Mesa et al. (2022) mea- +sured pitch angles of 11◦ and 26◦ for the two scattered +light arms, which are smaller than the pitch angle mea- +sured for the inner region of the CO arm. +However, +since the pitch angles appear to change along the arm, +the differing values do not necessarily imply that they +are separate structures. Mesa et al. (2022) also noted +that the northeastern spiral in the SPHERE image had +a clump-like feature, which they hypothesized was asso- +ciated with a protoplanet embedded in a dusty envelope. +While this compact feature is well below the resolution +limits of our NOEMA observations, the presence of a dif- +ferent clump in the 12CO arm suggests that the clumps +could be intrinsic features of the arms themselves. +3.2.3. Envelope +DR Tau shows envelope emission in 12CO up to ∼ 5′′ +(1000 au) in projection from the star (Figure 8). En- +velope emission is detected between 9.8 and 12 km s−1, +i.e., mostly redshifted with respect to the systemic ve- +locity. In most of these channels, the envelope emission +is more spatially extended and brighter on the northern +side. +As with 12CO, the 13CO emission is more extended +north of the star compared to south of the star for +LSRK velocities above 10.4 km s−1. +In contrast to +12CO, though, the 13CO maps show features that ap- +pear more streamer-like than envelope-like. +However, +since this is the velocity range where cloud contamina- +tion is most significant, spatial filtering of large-scale +emission may be artificially creating the appearance of +streamers. Sturm et al. (2022) identified a possible in- +falling stream in ALMA observations of 13CO toward +DR Tau, but those observations were likewise affected +by spatial filtering. Observations of other lines that are +bright but less susceptible to cloud contamination (e.g., +species with higher critical densities like HCO+ or CO +transitions with higher upper energy levels) might help +to clarify the nature of these apparent streamers. +The channels where envelope emission is detected in +12CO overlap with the channels where [C I] exhibits a +redshifted non-Keplerian component that Sturm et al. +(2022) attributed to an infalling stream. However, since +the beam FWHM of the [C I] observations is ∼ 3′′, most +of the emission is spatially unresolved. Given the simi- +lar velocities to the 12CO envelope, it is likely that the +redshifted non-Keplerian [C I] emission also originates +from the envelope. +3.2.4. Outflow +DR Tau’s 12CO spectrum (Figure 1) exhibits a faint +blueshifted line wing without a corresponding redshifted +line wing, suggesting the presence of an asymmetric out- +flow. +The channel maps (Figure 10.1) show compact +emission at LSRK velocities lower than 8.4 km s−1. To +highlight this compact outflow emission more clearly, we +extracted a new 12CO spectrum using a smaller circu- +lar aperture with a diameter of 4′′ (Figure 9). Because +DR Tau is nearly face-on, it is not straightforward to +separate the outflow emission from the line wings of the +Keplerian disk. However, the asymmetry of the line pro- +file allows us to estimate the velocities at which outflow +emission dominates by mirroring the outflow spectrum +about the systemic velocity and taking the ratio of the +original and mirrored spectrum. We assume that the +outflow emission on the blueshifted side dominates when +the ratio exceeds 10, which occurs at 7.4 km s−1. +While the outflow emission is weak in individual +channels (Figure 10.1), the spatial distribution of the +blueshifted side can be better seen by producing an in- +tegrated intensity map between −2.0 and 7.4 km s−1. +The lower bound of the velocity integration range was +determined by where the emission in individual chan- +nels drops below 3σ. For comparison, we also produced +an integrated intensity map from 12.4 to 21.8 km s−1, +corresponding to the redshifted channels at the opposing +offsets from the systemic velocity. The two integrated in- + +Molecular Mapping of DR Tau +11 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 + [′′] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 + [′′] +Southern spiral +Northeastern +spiral +H-band Q +Clump +H-band Q +50 au +6 +4 +2 +0 +2 +4 +6 +6 +4 +2 +0 +2 +4 +6 +H-band Q vs. 12CO arm +200 au +Figure 7. A comparison between the SPHERE H-band Qφ image of DR Tau from Mesa et al. (2022) and the 12CO NOEMA +observations from this work. Left: H-band Qφ image of DR Tau. The arrows point to the northeastern spiral, southern spiral, +and clump identified in Mesa et al. (2022). The gray circle shows the extent of the SPHERE coronagraph. Right: A contour +plot of the 12CO arm overlaid atop the H-band Qφ image. Note that the size scale is different from the image on the left. The +contours, drawn at 50, 100, and 150 mJy beam−1 km s−1, correspond to the 12CO integrated intensity map from Figure 5. +tensity maps are presented in Figure 9. The blueshifted +map shows relatively compact emission with a radial ex- +tent of ∼ 2′′ (∼ 400 au). Although the opening angle +of the outflow cannot be computed because the disk is +nearly face-on, the compactness of the emission suggests +that the outflow is quite collimated. The redshifted map +shows emission near the stellar position, but given that +the redshifted emission is fainter and much more com- +pact than the blueshifted outflow, it seems likely that +the compact redshifted emission originates from the line +wing of the Keplerian disk emission. While a redshifted +outflow component is not readily visible in the 12CO +spectrum, the redshifted map shows a faint ring with a +radius of ∼ 4.5′′ (∼ 900 au), which is significantly wider +than the blueshifted outflow component. +3.3. SO, DCO+, and H2CO emission +SO, DCO+, and H2CO emission all originate from a +relatively compact region within 300 au of DR Tau. The +SO 65 − 54, SO 55 − 44, and H2CO 303 − 202 intensity- +weighted velocity maps (Figure 3) show velocity gradi- +ents similar to that of C18O, indicating that they like- +wise (largely) originate from the Keplerian disk. The +kinematics of DCO+ are not well-defined due to the low +signal-to-noise ratio, but the compactness of the emis- +sion suggests that it also primarily traces the Keplerian +disk. +That said, whereas C18O and H2CO 303−202 both ex- +hibit relatively axisymmetric emission in the integrated +intensity maps (Figure 2) and line profiles that are sym- +metric about the systemic velocity (Figure 1), SO 65−54 +and 55 − 44 are both asymmetric. +Their emission is +stronger on the northern (redshifted) side of the disk. +In addition, their spectra both peak at an LSRK veloc- +ity of 10.2 km s−1, which is redshifted by 0.3 km s−1 +with respect to the systemic velocity. The DCO+ spec- +trum also appears stronger on the redshifted side, but +given that its SNR is lower than that of the SO lines, +more sensitive observations will be necessary to deter- +mine whether the DCO+ asymmetry is genuine. +4. DISCUSSION +4.1. The evolutionary stage of DR Tau +DR Tau is traditionally considered to have a Class II +SED, with stellar age estimates ranging from 0.9 to 3.2 +Myr (Kenyon & Hartmann 1995; McClure 2019; Long +et al. 2019). The presence of the envelope, if primordial, +would suggest that the younger end of the age range is +more likely. DR Tau’s chemistry also appears to point +to a younger age. Although SO is detected in DR Tau, +it has otherwise rarely been detected in Class II disks, + +12 +Huang et al. +9.8 +200 au +10.0 +10.2 +10.4 +10.6 +10.8 +6 +3 +0 +3 +6 + [′′] +6 +3 +0 +3 +6 + [′′] +11.0 +11.2 +11.4 +11.6 +11.8 +12.0 +0 +50 +100 +250 +500 +12CO J = 2 +1 intensity (mJy beam +1) +9.8 +200 au +10.0 +10.2 +10.4 +10.6 +10.8 +6 +3 +0 +3 +6 + [′′] +6 +3 +0 +3 +6 + [′′] +11.0 +11.2 +11.4 +11.6 +11.8 +12.0 +0 +50 +100 +200 +13CO J = 2 +1 intensity (mJy beam +1) +Figure 8. Channel maps of 12CO J = 2 − 1 and 13CO J = 2 − 1 over the velocity range where envelope emission is present. +The black contours denote the 5, 15, 25, and 35σ contours of C18O J = 2 − 1 to serve as a visual reference for the kinematics of +the Keplerian disk. (Note that because C18O is less abundant than 12CO and 13CO, the Keplerian line wings of C18O are not +detected out to as high velocities as the other two isotopologues). +especially those hosted by T Tauri stars (e.g., Guilloteau +et al. 2016; Semenov et al. 2018; Le Gal et al. 2021). It +is commonly detected, though, in younger, embedded +Class 0 and I systems (e.g., Sakai et al. 2014; Le Gal +et al. 2020; Garufi et al. 2022; Mercimek et al. 2022). +In addition, Sturm et al. (2022) found that gas-phase +carbon is not as severely depleted in DR Tau as other +Class II disks that have been observed, although it is +still more depleted than Class 0/I systems. +However, simulations have suggested that pre-main se- +quence stars might be able to form second-generation en- +velopes through interaction with cloud material, a pro- +cess sometimes referred to as “late infall” (e.g., Dulle- +mond et al. 2019; Kuffmeier et al. 2020). Indeed, Mesa +et al. (2022) hypothesized that DR Tau was undergo- +ing late infall based on the detection of spiral arms +in scattered light. +Given the range of ages estimated +for DR Tau, it is ambiguous whether infall onto DR +Tau should be considered “late.” As noted above, DR + +Molecular Mapping of DR Tau +13 +0 +10 +20 +LSRK Velocity (km s +1) +2 +0 +2 +4 +6 +8 +10 +12 +Flux (Jy) +12CO inner 4′′ +9 +6 +3 +0 +3 +6 +9 + [''] +9 +6 +3 +0 +3 +6 +9 + [''] +-2.0 to 7.4 km s +1 +0 +100 +200 300400 +Integrated Intensity (mJy beam +1 km s +1) +9 +6 +3 +0 +3 +6 +9 +9 +6 +3 +0 +3 +6 +9 +12.4 to 21.8 km s +1 +0 +50 +100 +150 +Figure 9. Overview of DR Tau’s outflow emission. Left: 12CO spectrum extracted from a circular aperture with a 4′′ diameter, +showing a blueshifted outflow wing. The approximate velocity range of the blueshifted outflow is shaded in blue. The purple +dotted line marks the system velocity. Middle: +12CO integrated intensity map covering velocities from −2.0 to 7.4 km s−1. +Compact emission from the blueshifted side of the outflow is visible. An arcsinh stretch is used on the color scale to make faint +emission more readily visible. The faint vertical striping is due to the sidelobes of the point spread function. The pink cross +marks the position of the disk center. Right: 12CO integrated intensity map covering velocities from 12.4 to 21.8 km s−1. The +map shows a faint ring with a radius of ∼ 4.5′′ (∼ 900 au) and compact emission located at the stellar position. The redshifted +compact emission may be from a line wing of the Keplerian disk rather than the outflow. +Tau’s chemistry seems to suggest that the disk is rela- +tively young. This appearance of chemical youthfulness, +though, stems from comparisons of Class 0/I sources to +isolated Class II disks. Based on molecular observations +of GM Aur, a Class II disk with large-scale spiral arms +suggestive of ongoing late infall, Huang et al. (2021) +speculated that accretion of cloud material could par- +tially reset disk chemistry such that it bears greater +resemblance to that of Class 0/I sources. The impact +of late infall on disk chemistry will need to be examined +through astrochemical modeling to determine the extent +to which chemical properties can be used to sort disks +by relative age. +Given that DR Tau is commonly included in sur- +veys of Class II disks because of its relatively large +disk mass and bright line emission (e.g., Salyk et al. +2011; Long et al. 2019; Arulanantham et al. 2020; Sturm +et al. 2022), an erroneous classification of its evolution- +ary stage may skew interpretations of disk observations. +Huang et al. (2022) remarked that a similar problem +exists for DO Tau, another commonly observed Class II +disk that also shows signatures of being partially embed- +ded. Interestingly, among the twelve single star systems +that Long et al. (2019) identified as having “smooth” +disks in millimeter continuum emission, at least three +of them (DR Tau, DO Tau, and Haro 6-13) exhibit ev- +idence of an envelope in spatially resolved CO emission +(e.g., Fern´andez-L´opez et al. 2020; Garufi et al. 2021; +Huang et al. 2022, and this work). Most of the remain- +ing sources (including both the “smooth” and structured +disks) lack high quality interferometric CO observations, +so it is unknown whether they might be embedded as +well. +Analyses of where and when disk substructures +tend to emerge will require sensitive, spatially resolved +molecular line observations to provide context about the +evolutionary stages of the objects being studied. +4.2. Origin of SO in DR Tau +The detection of SO in DR Tau is notable given that +SO detections have thus far been uncommon in Class II +disks, which has been attributed to high gas-phase C/O +ratios (> 1) disfavoring SO production (e.g., Guilloteau +et al. 2016; Semenov et al. 2018; Le Gal et al. 2021). In +two of the disks where SO has been detected, AB Aur +and Oph IRS 48, the gas-phase C/O ratio has been esti- +mated to be less than 1 (Rivi`ere-Marichalar et al. 2020; +Booth et al. 2021). Based on thermochemical model- +ing of [C I] and CO isotopologue emission, Sturm et al. +(2022) estimated that DR Tau has a gas-phase C/O ra- +tio of 0.47. The detection of SO toward DR Tau is thus +qualitatively consistent with SO production in disks be- +ing favored in gas with C/O ratios less than 1. +DR Tau’s SO emission exhibits a mild asymmetry that +is not seen in C18O. This suggests that the SO asym- +metry is not merely tracing the underlying gas surface +density, but could instead be due to some dynamical +process locally favoring SO production. +SO has been +proposed to be enhanced by outflows, winds, gravita- + +14 +Huang et al. +tional instabilities, or accretion shocks (e.g., Pineau des +Forˆets et al. 1993; Sakai et al. 2014; Tabone et al. 2017; +Ilee et al. 2017). We discuss these possibilities in turn +for DR Tau. +Our NOEMA observations have shown that DR Tau +has a molecular outflow, and past CO ro-vibrational +spectroscopy indicates that DR Tau has a wide-angle +molecular wind (Pontoppidan et al. 2011). An outflow +shock does not appear to be a likely major contributor +to SO in DR Tau, since SO is not detected at the same +high velocities as CO. At the spatial and spectral resolu- +tion of our NOEMA observations, it is unclear whether +the SO kinematics are consistent with those expected +for a disk wind (e.g. Haworth & Owen 2020), but DR +Tau’s bright emission makes it an excellent target for +more detailed follow-up. +Chemical modeling of gravitationally unstable disks +suggests that gas-phase SO can be enhanced either by +spiral shocks or within warm disk fragments (Ilee et al. +2011, 2017). While DR Tau does feature spiral struc- +ture, Mesa et al. (2022) argued that DR Tau’s spiral +arms are unlikely to be due to gravitational instability +given that its disk-to-stellar mass ratio and stellar accre- +tion rate are a factor of a few lower than hydrodynamical +simulations suggest would be necessary to induce grav- +itational instabilities. Nevertheless, disk mass is notori- +ously difficult to measure (e.g., Miotello et al. 2022, and +references therein), and studies of other spiral-armed +disks have often disagreed on whether they are massive +enough to be gravitationally unstable (e.g., P´erez et al. +2016; Cleeves et al. 2016; Veronesi et al. 2019; Sierra +et al. 2021). +Higher spatial resolution would help to +determine if the SO asymmetry traces spiral structure +and/or a disk fragment. +Accretion shocks in protoplanetary disks might oc- +cur due to cloud or envelope material being accreted +by the disk or disk material being accreted by an em- +bedded planet (e.g., Bodenheimer 1974; Boss & Graham +1993; Yorke & Bodenheimer 1999; Szul´agyi & Mordasini +2017). +Accretion streamers traced by SO have been +observed in several Class I protostellar systems (e.g., +Garufi et al. 2022; Artur de la Villarmois et al. 2022). +Given that DR Tau is now known to be partially embed- +ded, its asymmetric SO emission may arise in a manner +similar to Class I systems. Booth et al. (2022) proposed +that an SO asymmetry in the HD 100546 disk could +be due to shocks from gas accreting onto an embedded +planet. This likely does not account for DR Tau’s SO +asymmetry, since high-contrast imaging from Mesa et al. +(2022) rules out the presence of a companion above sev- +eral Jupiter masses at separations greater than 50 au +from DR Tau. +4.3. Origin of DR Tau’s molecular spiral arm +Mesa et al. (2022) hypothesized that the northeastern +spiral arm detected in scattered light toward DR Tau is +due to planet-disk interactions, while the southern arm +is due to infall from cloud material. As noted in Section +3.2.2, the angular resolution of our NOEMA observa- +tions does not allow us to determine whether the CO +spiral arm is an extension of either scattered light spiral +arm, but the very large extent of the molecular arm sug- +gests that it is unlikely to be generated by interactions +with a bound planet. Mesa et al. (2022) placed an upper +limit of several Jupiter masses on any companion farther +out than 50 au from DR Tau. Given that the millimeter +continuum appears smooth down to a resolution of 20 +au (Long et al. 2018), it is unlikely that the disk harbors +massive (super-Jovian) companions within 50 au. More- +over, hydrodynamical simulations indicate that external +companions exceeding several MJ should create a pair +of (nearly) symmetric spiral arms (e.g., Zhu et al. 2015; +Dong et al. 2016), contrary to what is observed for DR +Tau. Thus, a stellar companion is also unlikely to be +responsible for the arm. +An infalling stream is a plausible explanation for the +molecular arm, given that similar large-scale structures +have been detected in association with a number of em- +bedded Class 0/I sources as well as Class II disks pro- +posed to be undergoing late infall (e.g., Tang et al. 2012; +Yen et al. 2019; Pineda et al. 2020; Huang et al. 2021; +Garufi et al. 2022; Valdivia-Mena et al. 2022). One pos- +sible difference of note is that the structures proposed +to be infalling streams in the other systems have tended +to be open, whereas DR Tau’s pitch angle exhibits a +marked decrease with distance from the star. However, +this apparent difference may simply be a projection ef- +fect, since we do not know their three-dimensional ori- +entations. +As noted in the previous subsection, it is uncertain +whether DR Tau is gravitationally unstable. The possi- +bility that DR Tau’s arm arises from gravitational in- +stabilities remains intriguing given that clumpy arms +are a hallmark of simulations of fragmenting disks (e.g., +Zhu et al. 2012; Basu & Vorobyov 2012). Furthermore, +migration of clumps onto stars has been proposed as +a trigger for FUor outbursts (e.g., Boley et al. 2010). +Clump migration might likewise explain DR Tau’s ex- +treme brightening event in the 1970s. If DR Tau’s disk +mass has been estimated correctly, then the presence +of a clump along DR Tau’s arm raises the question of +whether fragmentation can occur under less stringent +conditions than models demand. +Close stellar encounters can also generate large-scale +arm-like structures with pitch angles comparable to that +observed for the DR Tau molecular arm (e.g., Dai et al. +2015; Cuello et al. 2019, 2020). However, Shuai et al. +(2022) inferred from an analysis of Gaia EDR3 data +(Gaia Collaboration et al. 2021) that the closest ex- +pected approach between DR Tau and a neighboring +star in the past 10,000 years is ∼ 105 au, which would +be too distant to meaningfully perturb the known cir- +cumstellar environment of DR Tau. Mesa et al. (2022) +found that DQ Tau may have passed within 5100 au of + +Molecular Mapping of DR Tau +15 +DR Tau 0.23 Myr ago, but considered such an encounter +unlikely to be responsible for DR Tau’s spiral arms be- +cause flyby-induced arms are only expected to survive on +timescales of several thousand years (e.g., Cuello et al. +2022). +4.4. Connections to EXor and FUor phenomena +In recent years, the circumstellar environments of a +number of FUors and EXors have been spatially resolved +with millimeter interferometry and high-contrast scat- +tered light imaging. +FUors are often associated with +envelopes, outflows, and arm-like structures (e.g., Liu +et al. 2016; Zurlo et al. 2017; Ru´ız-Rodr´ıguez et al. 2017; +K´osp´al et al. 2017), similar to the structures associated +with DR Tau. +In FUor systems, the envelopes sup- +ply infalling material that may help to activate gravita- +tional instabilities (and then possibly magnetorotational +instabilities), the arms may form as a consequence of +gravitational instabilities, and instabilities may trigger +outbursts that subsequently drive outflows (e.g., Evans +et al. 1994; Vorobyov & Basu 2005; Zhu et al. 2010). +With the exception of EX Lup and V1647 Ori, the latter +of which is sometimes considered to be an FUor source, +the EXor sources imaged so far have generally lacked +analogous features (e.g., Principe et al. 2018; Hales et al. +2018; Cieza et al. 2018; Hales et al. 2020). However, DR +Tau exhibits striking similarities to EX Lup in that they +both feature outflows, non-Keplerian spiral-like struc- +tures, and (remnant) envelopes. The circumstellar en- +vironments of DR Tau and EX Lup also share similar- +ities with that of RU Lup, which is not classified as +an EXor source but is nevertheless an exceptionally ac- +tive T Tauri star (Joy 1945; Gahm et al. 1974; Huang +et al. 2020). Hales et al. (2018) suggested that the pres- +ence of complex structures associated with EX Lup but +not other EXors is an indication that EX Lup occupies +an intermediate evolutionary stage between FUors and +most EXors. The same may hold true for DR Tau (and +perhaps RU Lup). Alternatively, the differences in EXor +circumstellar environments may indicate that EXors are +a heterogeneous group of objects, only some of which are +closely related to the FUor phenomenon. In any case, +the observations of EX Lup, DR Tau, and RU Lup moti- +vate more spatially resolved imaging of extremely active +T Tauri stars to elucidate the connection between cir- +cumstellar environments and stellar properties. +4.5. A changing view of Class II disks +In the past decade, the introduction of high angular +resolution imaging at millimeter wavelengths has trans- +formed our understanding of planet formation by show- +ing that dust substructures on scales of several au are +common (e.g., ALMA Partnership et al. 2015; Andrews +et al. 2018). Meanwhile, sensitive molecular imaging at +more modest resolution has highlighted a deficit in our +understanding of disk environments on scales of tens to +thousands of au. With single-dish telescopes and ear- +lier generations of interferometers, signs of large-scale +non-Keplerian emission towards Class II disks were of- +ten ascribed to foreground contamination (e.g., Thi et al. +2001; Hughes et al. 2009; ¨Oberg et al. 2011b). Even in +the era of more powerful millimeter interferometers, in- +sufficient integration times or insufficient uv coverage +at larger spatial scales can lead to key structures being +missed. +High-quality molecular mapping, though, has demon- +strated that there are indeed large-scale tails, spirals, +streams, and/or remnant envelopes associated with a +number of Class II systems (e.g., Akiyama et al. 2019; +Huang et al. 2021; Paneque-Carre˜no et al. 2021; Huang +et al. 2022). +Scattered light imaging has also played +an important role in uncovering examples of Class II +disks that appear to be interacting with surrounding +material (e.g., Grady 2004; Garufi et al. 2018; Ginski +et al. 2021), although as demonstrated by the examples +of DR Tau from this work and RU Lup from Huang et al. +(2020), molecular observations can reveal structures far +beyond the detected extent of scattered light features. +Infall from these larger-scale structures is increasingly +being invoked to explain certain disk structures observed +at smaller scales, such as misalignments or spiral arms +(e.g., Ginski et al. 2021; Paneque-Carre˜no et al. 2021; +Mesa et al. 2022). These observations thus imply an in- +triguing link between dynamical processes operating on +disparate size scales. +For individual systems, though, +infall is only one of several possible explanations for the +observed disk phenomena. More systematic molecular +line observations will be key for establishing patterns of +association between large- and small-scale properties. +5. SUMMARY +We present new NOEMA observations of 12CO, 13CO, +C18O, SO, DCO+, and H2CO toward the T Tauri star +DR Tau, representing the highest-quality millimeter line +observations of this source to date. Our findings are as +follows: +1. CO emission shows that the DR Tau protoplane- +tary disk is associated with an envelope, a faint +asymmetric outflow, and a large non-Keplerian +spiral arm with a clump. +2. The molecular spiral arm resembles a scaled-up +version of the spiral arms detected in scattered +light, although the angular resolution of NOEMA +is not sufficient to determine whether the molecu- +lar arm is an extension of one of the scattered light +arms or a separate feature. Whereas the scattered +light arms are only detected up to ∼ 220 au in +projection from DR Tau, the molecular arm is de- +tected up to ∼ 1200 au in projection from the star. +3. We report detections of SO, DCO+, and H2CO in +the DR Tau disk for the first time. Their kinemat- + +16 +Huang et al. +ics and compact emission extent suggest that they +primarily trace the Keplerian circumstellar disk. +4. SO emission is stronger on the northern, redshifted +side of the disk. This asymmetry might be linked +to infall from an asymmetric envelope or to un- +resolved spiral substructure associated with the +arms detected in scattered light. Higher angular +resolution observations of SO will be needed to +clarify the origins of the asymmetry. +DR Tau’s envelope, outflow, and arm are reminiscent +of the structures that have been observed in association +with various FUor sources as well as the EXor source EX +Lup. Given that FUor and EXor outbursts have been +linked to instabilities driven by envelope accretion, a +similar mechanism may account for DR Tau’s dramatic +stellar brightness changes. The NOEMA observations +of DR Tau highlight the utility of sensitive, spatially re- +solved molecular line observations for providing context +about the conditions under which young stars and their +protoplanetary disks evolve. +This work is based on observations carried out un- +der project number W20BE with the IRAM NOEMA +Interferometer. +IRAM is supported by INSU/CNRS +(France), MPG (Germany) and IGN (Spain). This work +is also based on observations collected at the Euro- +pean Southern Observatory under ESO programme(s) +0102.C-0453(A). We thank our NOEMA local contact, +Ana Lopez-Sepulcre, for setting up the observing scripts +and assisting with data reduction. We also thank Arthur +Bosman, Ke Zhang, Joel Bregman, Lee Hartmann, +Merel van’t Hoff, Ardjan Sturm, Melissa McClure, and +Ewine van Dishoeck for helpful discussions. We thank +the referee, Ruobing Dong, for helpful comments im- +proving the clarity of the manuscript. Support for J. +H. was provided by NASA through the NASA Hubble +Fellowship grant #HST-HF2-51460.001-A awarded by +the Space Telescope Science Institute, which is operated +by the Association of Universities for Research in As- +tronomy, Inc., for NASA, under contract NAS5-26555. +This project has received funding from the European +Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme (grant +agreement No. 101002188). +Facilities: NOEMA +Software: analysisUtils (https://casaguides.nrao. +edu/index.php/Analysis Utilities), +AstroPy +(Astropy +Collaboration et al. 2013), CASA (CASA Team et al. +2022), cmasher (van der Velden 2020), emcee (Foreman- +Mackey +et +al. +2013), +GILDAS +(Pety +2005; +Gildas +Team 2013), matplotlib (Hunter 2007), pandas (pan- +das development team 2022; Wes McKinney 2010), +scikit-image (van der Walt et al. 2014), SciPy (Vir- +tanen et al. 2020) +APPENDIX +A. SPECTROSCOPIC PARAMETERS OF TARGETED LINES +The spectroscopic parameters of the targeted lines, taken from the Cologne Database for Molecular Spectroscopy +(M¨uller et al. 2001, 2005) via Splatalogue1, are listed in Table 2. Primary line targets are marked in bold. +1 https://splatalogue.online// + +Molecular Mapping of DR Tau +17 +Table 2. +Spectroscopic Parameters of All Targeted +Lines +Transition +Rest frequency +Eu +(GHz) +(K) +13C17O J = 2 − 1 +214.5738730 +15.4 +SO JN = 55 − 44 +215.2206530 +44.1 +DCO+ J = 3 − 2 +216.1125822 +20.7 +H2S JKaKc = 220 − 211 +216.7104365 +84.0 +c-C3H2 JKaKc = 330 − 221 +216.2787560 +19.5 +SiO J = 5 − 4 +217.1049190 +31.3 +DCN J = 3 − 2 +217.2385378 +20.9 +c-C3H2 JKaKc = 514 − 423 +217.9400460 +35.4 +H2CO JKaKc = 303 − 202 +218.2221920 +21.0 +HC3N J = 24 − 23 +218.3247230 +131.0 +H2CO JKaKc = 322 − 221 +218.4756320 +68.1 +H2CO JKaKc = 321 − 220 +218.7600660 +68.1 +C18O J = 2 − 1 +219.5603541 +15.8 +SO JN = 65 − 54 +219.9494420 +35.0 +13CO J = 2 − 1 +220.3986842 +15.9 +12CO J = 2 − 1 +230.5380000 +16.6 +OCS J = 19 − 18 +231.0609934 +110.9 +N2D+ J = 3 − 2 +231.3218283 +22.2 +13CS J = 5 − 4 +231.2206852 +33.3 +C2S JN = 1918 − 1817 +233.9384580 +109.6 +PN J = 5 − 4 +234.9356940 +33.8 +HC3N J = 26 − 25 +236.5127888 +153.2 +H2CS JKaKc = 717 − 616 +236.7270204 +58.6 +B. CHANNEL MAPS +Channel maps of the primary line targets (listed in +Table 1) are presented in Figure 10.1. +C. AUXILIARY LINE TARGETS +Table 3 lists the beam sizes, per-channel rms of the +image cubes, and 3σ flux upper limits for the auxiliary +line targets. The flux upper limits were estimated as- +suming the same velocity range and aperture used to +measure the C18O flux (see Table 1). Flux upper lim- +its may be underestimated if the molecule is primarily +present in the envelope or outflow rather than the disk. +REFERENCES +Akiyama, E., Vorobyov, E. I., Baobabu Liu, H., et al. 2019, +AJ, 157, 165, doi: 10.3847/1538-3881/ab0ae4 +Alencar, S. H. P., Johns-Krull, C. M., & Basri, G. 2001, +AJ, 122, 3335, doi: 10.1086/323914 +ALMA Partnership, Brogan, C. L., P´erez, L. 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Contours are drawn in pink at the 3, 5σ levels. +9.0 +9.2 +9.4 +9.6 +9.8 +2 +0 +2 + [′′] +2 +0 +2 + [′′] +10.0 +200 au +10.2 +10.4 +10.6 +10.8 +0 +9 +18 +27 +36 +DCO + J = 3 +2 intensity (mJy beam +1) +Figure 10.6. Channel maps of DCO+ J = 3 − 2 toward DR Tau. Contours are drawn in pink at the 3, 5σ levels. + +Molecular Mapping of DR Tau +25 +9.0 +9.2 +9.4 +9.6 +9.8 +2 +0 +2 + [′′] +2 +0 +2 + [′′] +10.0 +200 au +10.2 +10.4 +10.6 +10.8 +0 +25 +50 +75 +100 +H2CO JKaKc = 303 +202 intensity (mJy beam +1) +Figure 10.7. Channel maps of H2CO JKaKc = 303 −202 toward DR Tau. Contours are drawn in pink at the 3, 5, 10, 15σ levels. +9.0 +9.2 +9.4 +9.6 +9.8 +2 +0 +2 + [′′] +2 +0 +2 + [′′] +10.0 +200 au +10.2 +10.4 +10.6 +10.8 +0 +10 +20 +30 +H2CO JKaKc = 322 +221 intensity (mJy beam +1) +Figure 10.8. Channel maps of H2CO JKaKc = 322 − 221 toward DR Tau. Contours are drawn in pink at the 3, 4σ levels. + +26 +Huang et al. +9.0 +9.2 +9.4 +9.6 +9.8 +2 +0 +2 + [′′] +2 +0 +2 + [′′] +10.0 +200 au +10.2 +10.4 +10.6 +10.8 +0 +5 +10 +15 +20 +25 +H2CO JKaKc = 321 +220 intensity (mJy beam +1) +Figure 10.9. Channel maps of H2CO JKaKc = 321 − 220 toward DR Tau. Contours are drawn in pink at the 3σ level. +Table 3. Imaging Summary for Auxiliary Line Targets +Transition +Synthesized beam +Per-channel RMS noisea +3σ Flux Upper Limit +(arcsec × arcsec (◦)) +(mJy beam−1) +(mJy km s−1) +13C17O J = 2 − 1 +1.21 × 0.93 (18.0◦) +8 +< 40 +H2S JKaKc = 220 − 211 +1.21 × 0.93 (16.6◦) +7 +< 40 +c-C3H2 JKaKc = 330 − 221 +1.21 × 0.93 (16.7◦) +7 +< 30 +SiO J = 5 − 4 +1.21 × 0.93 (16.4◦) +7 +< 30 +DCN J = 3 − 2 +1.21 × 0.93 (16.5◦) +7 +< 40 +c-C3H2 JKaKc = 514 − 423 +1.21 × 0.93 (16.7◦) +7 +< 30 +HC3N J = 24 − 23 +1.20 × 0.94 (17.0◦) +7 +< 30 +OCS J = 19 − 18 +1.18 × 0.91 (17.3◦) +7 +< 30 +N2D+ J = 3 − 2 +1.18 × 0.91 (17.3◦) +10 +< 70 +13CS J = 5 − 4 +1.18 × 0.91 (17.3◦) +8 +< 50 +C2S JN = 1918 − 1817 +1.17 × 0.90 (16.6◦) +10 +< 50 +PN J = 5 − 4 +1.17 × 0.90 (16.8◦) +9 +< 40 +HC3N J = 26 − 25 +1.17 × 0.90 (16.7◦) +10 +< 50 +H2CS JKaKc = 717 − 616 +1.17 × 0.90 (16.6◦) +11 +< 80 +aFor channel widths of 0.2 km s−1. + +Molecular Mapping of DR Tau +27 +Cuello, N., Louvet, F., Mentiplay, D., et al. 2020, MNRAS, +491, 504, doi: 10.1093/mnras/stz2938 +Currie, T., Lawson, K., Schneider, G., et al. 2022, Nature +Astronomy, 6, 751, doi: 10.1038/s41550-022-01634-x +Dai, F., Facchini, S., Clarke, C. 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P., et al. 2017, +MNRAS, 465, 834, doi: 10.1093/mnras/stw2845 + diff --git a/KtE0T4oBgHgl3EQfzgKz/content/tmp_files/load_file.txt b/KtE0T4oBgHgl3EQfzgKz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..563353cb65b0507769ff3ec0218610641586bd4d --- /dev/null +++ b/KtE0T4oBgHgl3EQfzgKz/content/tmp_files/load_file.txt @@ -0,0 +1,2455 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf,len=2454 +page_content='Draft version January 10, 2023 Typeset using LATEX twocolumn style in AASTeX631 Molecular Mapping of DR Tau’s Protoplanetary Disk, Envelope, Outflow, and Large-Scale Spiral Arm Jane Huang,1, ∗ Edwin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Bergin,1 Jaehan Bae,2 Myriam Benisty,3 and Sean M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Andrews4 1Department of Astronomy, University of Michigan, 323 West Hall, 1085 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' University Avenue, Ann Arbor, MI 48109, United States of America 2Department of Astronomy, University of Florida, Gainesville, FL 32611, United States of America 3Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 4Center for Astrophysics | Harvard & Smithsonian, 60 Garden St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Cambridge, MA 02138, USA ABSTRACT DR Tau has been noted for its unusually high variability in comparison with other T Tauri stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Although it is one of the most extensively studied pre-main sequence stars, observations with millimeter interferometry have so far been relatively limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We present NOEMA images of 12CO, 13CO, C18O, SO, DCO+, and H2CO toward DR Tau at a resolution of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5′′ (∼ 100 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In addition to the protoplanetary disk, CO emission reveals an envelope, a faint asymmetric outflow, and a spiral arm with a clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The ∼ 1200 au extent of the CO arm far exceeds that of the spiral arms previously detected in scattered light, which underlines the necessity of sensitive molecular imaging for contextualizing the disk environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The kinematics and compact emission distribution of C18O, SO, DCO+, and H2CO indicate that they originate primarily from within the Keplerian circumstellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The SO emission, though, also exhibits an asymmetry that may be due to interaction with infalling material or unresolved substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The complex environment of DR Tau is reminiscent of those of outbursting FUor sources and some EXor sources, suggesting that DR Tau’s extreme stellar activity could likewise be linked to disk instabilities promoted by large-scale infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Keywords: protoplanetary disks—ISM: molecules—stars: individual (DR Tau) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' INTRODUCTION In the classic schema of low-mass star formation, young stellar objects (YSOs) are divided into four classes (0, I, II, and III) based on their spectral energy distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Lada & Wilking 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Lada 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Andre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' These classes are generally thought to correspond to different evolutionary stages, such that a Class 0 YSO has an envelope mass comparable to or greater than that of the protostar (and its possible disk), a Class I YSO has an envelope that is less massive than the protostar but still comparable to its disk, a Class II YSO has a disk with negligible envelope material, and a Class III YSO has negligible amounts of remaining circumstellar material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Wilking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Andre & Montmerle 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The corre- spondence between SED class, evolutionary stage, and morphology, though, is known to be imperfect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Ro- bitaille et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Corresponding author: Jane Huang jnhuang@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='edu ∗ NASA Hubble Fellowship Program Sagan Fellow Planet formation models often adopt the characteris- tics of envelope-free Class II disks as a starting point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', ¨Oberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2011a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Lambrechts & Johansen 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, scattered light and molec- ular imaging have yielded identifications of a number of Class II disks that appear to be interacting either with (remnant) envelopes or ambient cloud material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Grady et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The pace of these identifications has increased with the advent of instruments such as ALMA and VLT/SPHERE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Detections of gaps and rings in the millimeter continuum of some Class II disks that ap- pear to have remnant envelope material, and even a few embedded Class I disks, offer evidence that planet for- mation can take place under more dynamically complex conditions than typically assumed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', ALMA Part- nership et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Segura-Cox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Kanagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Moreover, Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) recently detected a protoplanet in the disk of AB Aur, a system that appears to still be undergo- ing infall from a remnant envelope (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Simulations suggest that accretion of cloud or envelope material by the disk can influence its thermal structure, surface density profile, stability, and degree of misalignment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='02674v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='SR] 6 Jan 2023 2 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Kuznetsova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' These disk conditions, in turn, are expected to influence where, when, and how planets form and migrate, as well as their composition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Stevenson & Lunine 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Boss 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Kokubo & Ida 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Hence, observations of the immediate envi- ronments of young stars are essential to establish the range of circumstances under which planet formation might proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Recent observations of DR Tau (J2000 04:47:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='215+16:58:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='81), a T Tauri star located at a distance of 192 ± 1 pc in the Taurus star-forming re- gion (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021), have suggested that its disk is being externally perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) detected spiral arms in scattered light images of the DR Tau protoplanetary disk and hypothesized that one of them was triggered by infalling material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Meanwhile, Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) detected non-Keplerian emission in ALMA observations of 13CO and [C I] toward DR Tau, attributing this component to an infalling stream of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DR Tau is perhaps best known for its unusual degree of stellar variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The star has faded and bright- ened in B-band by several magnitudes over the course of almost a century (Chavarria-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Most notably, DR Tau brightened in B-band by about five magni- tudes between 1970 to 1979, an event that Chavarria- K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (1979) compared to the outbursts of FUor (also known as FU Ori) sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DR Tau also exhibits sig- nificant short-term spectroscopic and photometric vari- ability—on timescales of a few days, DR Tau has been observed to change by up to a couple magnitudes in B- band and by up to a factor of a few in its optical line fluxes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Bertout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Guenther & Hessman 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Alencar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DR Tau has a high stellar mass accretion rate of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 × 10−7 M⊙ yr−1 (McClure 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This high accretion level leads to significant con- tinuum veiling, which poses a challenge for determining its spectral type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Cohen & Kuhi 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Spectral type estimates have ranged from M0 to K4 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Imhoff & Appenzeller 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Herczeg & Hillenbrand 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mc- Clure 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Gangi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DR Tau was part of the original list of outbursting EXor variables by Herbig (1989), although it has not always been included in subsequent compilations of EX- ors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Audard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DR Tau is unique among the EXors listed in the Herbig (1989) catalog in that the 18-year rise time to its outburst was much longer than those of the other EXors, which were typically on the order of a couple hundred days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' EXors are usually distinguished from outbursting FUor sources insofar as EXor outbursts tend to be more modest in magnitude and duration, and EXors have T Tauri-like spectra dur- ing outbursts rather than the supergiant-like spectra of FUors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Audard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Several hypothe- ses have been proposed to account for the outbursts of young stars, including disk instabilities driven by mass buildup through infall from envelopes or cloud material, binary interactions, and stellar flybys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Bonnell & Bastien 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Vorobyov & Basu 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Forgan & Rice 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Because these outbursts affect the disk thermal structure, they may significantly affect how planet for- mation proceeds by altering molecular abundances, dust properties, and snowline locations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Juh´asz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Banzatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Cieza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' van ’t Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Jørgensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The hypothesized connection between outbursts and environmental inter- actions further motivates an examination of DR Tau’s surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Although DR Tau has been a popular target for obser- vations ranging from infrared to ultraviolet wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Kenyon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Ardila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Salyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Pontoppidan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Banzatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2014, and references above), relatively few observations with mil- limeter interferometry have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The mil- limeter continuum, which traces the distribution of large dust grains, has been imaged on several occasions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Kitamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Andrews & Williams 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Taz- zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The millimeter con- tinuum emission is fairly compact, with 95% of the flux contained within a 53 au radius (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Al- though no substructures are immediately apparent in the highest resolution image to date (tracing scales down to ∼ 20 au), modeling of the visibilities suggests the presence of gaps and rings that may be associated with planet-disk interactions (Jennings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Other than 13CO, C18O, and [C I] (Braun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022), no interferometric line observations of DR Tau have previously been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The upgraded wideband capabilities of the Northern Extended Millimeter Array (NOEMA) provided an op- portunity to observe a number of lines simultaneously toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We obtained sensitive observations of 12CO, 13CO, C18O, SO, DCO+, and H2CO at a resolu- tion of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5′′ (∼ 100 au) to map DR Tau’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The observations and data reduction are summarized in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The molecular detections are analyzed in Section 3, and the implications of DR Tau’s complex structures are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The summary and conclusions are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION DR Tau was observed with the NOEMA PolyFiX correlator in dual polarization mode during program W20BE (PI: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The correlator setup covered frequencies from 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9-221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 GHz and 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4-237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 GHz at a resolution of 2 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Within these frequency ranges, we placed a series of chunks, each with a resolution of 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 kHz and width of 64 MHz, in order to resolve molec- ular lines of interest (detailed further in Section 3 and Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The first set of observations was executed in C config- uration on 2021 January 08, with baseline lengths rang- ing from 24 to 328 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The second set of observations Molecular Mapping of DR Tau 3 was executed in A configuration on 2021 March 03, with baseline lengths ranging from 32 to 760 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Each con- figuration used eleven antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For each set of obser- vations, LkHα 101 served as the flux calibrator, 3C 84 served as the bandpass calibrator, and 0446+112 and 0507+179 served as the phase calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The on-source time was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 hours in C configuration and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 hours in A configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The raw data were calibrated with the NOEMA pipeline in CLIC, which is part of the GILDAS package (Pety 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Gildas Team 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Then, the following steps were performed with the GILDAS MAPPING software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The calibrated visibilities were written out to separate uv-tables corresponding to the low spectral resolution, wide bandwidth data and the high spectral resolution, narrow bandwidth spectral windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' After flagging of channels with strong line emission, the wide bandwidth uv-tables were spectrally averaged to produce contin- uum uv-tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For each of the four basebands, the continuum was imaged with the CLEAN algorithm and three phase self-calibration loops were performed using solution intervals of 180, 90, and 45 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The self- calibration solutions were then applied to the uv-tables for the narrow spectral windows that fell within the cor- responding basebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Continuum subtraction was per- formed for each spectral window separately in the uv plane by fitting a linear baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The self-calibrated, continuum-subtracted uv tables were converted to measurement sets to enable imag- ing with the Common Astronomy Software Applica- tions (CASA) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 (CASA Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Because GILDAS outputs frequencies in the rest frame of the source (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', the frequency that corresponds to the source systemic velocity input by the observer is the rest fre- quency of the line of interest), we had to manually cor- rect the frequencies in the measurement sets so that CASA would output image cubes with the appropri- ate LSRK velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Each line was imaged with the tclean implementation of the multi-scale CLEAN al- gorithm (Rau & Cornwell 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We set the robust value to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 and and the image cube channel spacing to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' To accommodate the irregular mor- phology of the 12CO and 13CO J = 2 − 1 emission, we employed the auto-multithresh algorithm (Kep- ley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020) to define the CLEAN masks, choos- ing the following parameter values after some experi- mentation: sidelobethreshold=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, noisethreshold=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, minbeamfrac=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3, and negativethreshold=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Initial imaging tests yielded prominent striping artifacts due to the poor uv sampling of the spatially extended cloud emission, so we re-imaged these lines without baselines shorter than 20 kλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For the other molecules, where only compact emission was detected, we used a circu- lar CLEAN mask with a radius of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6′′ and included all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' A Gaussian uv taper of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0′′ was used to in- crease sensitivity to weaker lines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', lines other than 12CO, 13CO, C18O, SO, DCO+, and H2CO JKaKc = 303 − 202).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' After CLEANing, a primary beam correc- tion was applied to each image cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Calibrated visibilities and images can be down- loaded at https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='org/record/7370498# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='Y7U-qezMKeB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Overview of Line Observations The primary line targets were 12CO, 13CO, C18O, SO, DCO+, and H2CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The CO isotopologues serve as gas tracers, SO is a potential shock tracer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Pineau des Forˆets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1993), and H2CO and DCO+ are common cold disk gas tracers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Pegues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The synthesized beam and per-channel rms (estimated from line-free channels) for the primary line targets are listed in Table 1, and channel maps are pre- sented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Spectra for the detected lines, which were extracted using circular masks with diam- eters listed in Table 1, are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Since the spatial extent of 12CO and 13CO are ambiguous due to spatial filtering and cloud contamination, we used extraction masks approximately equal to the primary beam FWHM at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 mm (21′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The mask sizes for the other lines were chosen based on the approximate radial extent of the 3σ emission in the image cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Fluxes were measured by integrating each spectrum within the velocity ranges listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The velocity integra- tion ranges for the CO isotopologues were selected based on where emission above the 3σ level is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For the weaker lines, the C18O velocity integration range was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The 1σ flux uncertainties were estimated as ∆v × √ N × σspec, where ∆v is the channel width (in km s−1), N is the number of channels spanned by the line, and σspec is the standard deviation (in Jy) mea- sured from a signal-free portion of the spectrum (this is not to be confused with the per-channel rms value listed in Table 1 (in mJy beam−1), which is calculated from the image cube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, the statistical uncertainties do not capture the true uncertainty of the fluxes for 12CO and 13CO, which are affected by cloud contamination and spatial filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We categorize a line as detected if emission is above the 5σ level within 2′′ of DR Tau in at least one channel of the image cube and above the 3σ level in at least two adjacent channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' By these criteria, 12CO, 13CO, C18O, SO, DCO+, and H2CO 303 − 202 are firmly detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' While H2CO 321 − 220 does not meet these criteria, its integrated flux is ⪆ 4σ when extracted over the same ve- locity range and emitting region as the strong 303 − 202 transition, so this line is considered to be tentatively de- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The channel maps for the 322 − 221 transition (Appendix B) show 4σ emission at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 kms−1 that is cospatial with the stronger 303 − 202 transition, but the velocity-integrated flux from the spectrum is < 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Fur- thermore, the peak of the spectrum occurs at a velocity 4 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Imaging Summary for Primary Line Targets Transition Synthesized beam Per-channel RMS noisea Velocity rangeb Extraction Mask Diameter Fluxc (arcsec × arcsec (◦)) (mJy beam−1) (km s−1) (arcsec) (mJy km s−1) 12CO J = 2 − 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='79 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='47 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2◦) 7 [−2, 17] 21 37900 ± 200d 13CO J = 2 − 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='84 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='49 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3◦) 6 [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4] 21 5730 ± 70d C18O J = 2 − 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='85 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='50 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3◦) 6 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8] 4 622 ± 10 SO JN = 65 − 54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='85 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='50 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2◦) 6 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8] 3 195 ± 9 SO JN = 55 − 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='86 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='50 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1◦) 6 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8] 3 96 ± 10 DCO+ J = 3 − 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='86 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='50 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1◦) 6 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8] 3 40 ± 10 H2CO JKaKc = 303 − 202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='86 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='50 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2◦) 6 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8] 3 248 ± 11 H2CO JKaKc = 322 − 221 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='20 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='93 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1◦) 7 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8] 3 < 30 H2CO JKaKc = 321 − 220 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='20 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='93 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1◦) 7 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8] 3 38 ± 8 aWith channel widths of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' b LSRK velocity range over which moment maps are produced and the flux is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' c The 1σ error bars do not include the systematic flux uncertainty (∼ 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' dThese lines are significantly affected by spatial filtering, so the statistical uncertainty does not reflect the true uncertainty in the fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' well offset from the peak of the 303 − 202 and 321 − 220 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Therefore, we do not consider the 322 − 221 tran- sition to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Integrated intensity maps of the primary line targets are presented in Figure 2, using the velocity integration ranges listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The intensity-weighted veloc- ity maps of the stronger lines are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For 12CO and 13CO, the integrated intensity maps excluded pixels in the image cube below the 3σ level and the intensity-weighted velocity map excluded pix- els below the 6σ level in order to reduce contributions from cloud contamination and artifacts from spatial fil- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For all other lines, no clipping was used for the integrated intensity maps, and a 4σ clip was adopted for the intensity-weighed velocity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' A summary of the auxiliary line observations (none of which yielded a detection) is presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Structures traced by CO isotopologues Due to their differing optical depths, the three de- tected CO isotopologues reveal different components of the DR Tau system, including the circumstellar disk, an arm, an envelope, and an outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' An overhead car- toon schematic of the system is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We describe each component in further detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The circumstellar disk C18O is the least optically thick of the three detected CO isotopologues and therefore best traces the Keple- rian rotation of the circumstellar disk (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The southern side is blueshifted and the northern side is redshifted relative to the systemic velocity, which Braun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2021) estimated to be vsys = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='09 km s−1 from ALMA observations of 13CO and C18O J = 2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Signs of Keplerian rotation are visible in the inner regions of the NOEMA 13CO intensity-weighted velocity map and coincide with the bright, compact emission component in the integrated intensity map, but the disk edge is not well-defined due to the presence of extended, non- Keplerian emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' From visual inspection of the 13CO emission, we estimate that the Keplerian disk has a ra- dial extent of ∼ 300 au, but this should only be consid- ered a lower bound for the disk size because the abun- dance of 13CO is generally too low in the outer disk to recover the disk size robustly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Trapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Finally, 12CO is dominated by large-scale, non- Keplerian structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The NOEMA observations do not strongly constrain the disk orientation, since the C18O emission is spanned by only a few synthesized beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2019) measured a position angle (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=') of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 de- grees east of north and an inclination angle of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 degrees from ALMA millimeter continuum observations at an angular resolution of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1′′, which corresponds to ∼ 20 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We adopt these values for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Although our new 12CO and 13CO observations show significant non-disk emission, Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) found that their ALMA C18O observations could be largely re- produced by a Keplerian disk model employing the disk orientation derived from Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Because the disk is nearly face-on, the C18O spectrum only exhibits a single peak at the systemic velocity rather than the double-peak characteristic of more inclined disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Molecular Mapping of DR Tau 5 5 0 5 10 15 20 25 LSRK Velocity (km s 1) 0 5 10 15 20 25 Flux (Jy) 12CO 2-1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 LSRK Velocity (km s 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 Flux (Jy) 13CO 2-1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 LSRK Velocity (km s 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='75 Flux (Jy) C18O 2-1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 7.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='H2CO 321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='200 au ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Integrated intensity maps of primary line targets observed toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The synthesized beam is drawn in the lower left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Black crosses mark the disk center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The axes show offsets from the disk center in arcseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For the CO isotopologues, the color scale uses an arcsinh stretch to make faint extended features more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Note that the size scales are different between the top row and the other rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Molecular Mapping of DR Tau 7 8 4 0 4 8 8 4 0 4 8 12CO 2 1 200 au 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 8 4 0 4 8 8 4 0 4 8 13CO 2 1 200 au 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 Intensity-weighted velocity (km s 1) 2 1 0 1 2 2 1 0 1 2 C18O 2 1 200 au 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content="5 2 1 0 1 2 [''] 2 1 0 1 2 [''] SO 65 54 200 au 9." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 2 1 0 1 2 2 1 0 1 2 SO 55 44 200 au 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 2 1 0 1 2 2 1 0 1 2 H2CO 303 202 200 au 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Intensity-weighted velocity maps of strong lines detected toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The synthesized beam is drawn in the lower left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The purple cross denotes the disk center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The axes show offsets from the disk center in arcseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Note that the velocity ranges and size scales are not the same for all imags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Blueshifted spiral arm The intensity-weighted velocity maps for 12CO and 13CO (Figure 3) both show an arm that is blueshifted with respect to the systemic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' To isolate the emission from the CO arm, we produced integrated in- tensity maps between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 km s−1 (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The arm is connected to the south side of the disk and curves around the western side, terminating at a pro- jected distance of ∼ 1200 au from DR Tau at a P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' of ∼ 330◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The 12CO emission also shows a clump along the arm at a projected distance of ∼ 500 au southwest from DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This arm was not detected in previously published high-resolution ALMA 13CO images of DR Tau (Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022), presumably due to some com- bination of lack of sensitivity and spatial filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' How- ever, low angular resolution ALMA ACA observations of [C I] from Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) show extended blueshifted emission, which may originate from the arm traced by CO in our NOEMA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In order to estimate the pitch angle of the arm, we transformed the integrated intensity map of the arm into a polar coordinate map (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', as a function of deprojected radius R and polar angle θ), assuming that the arm is in the plane of the disk (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We then measured the position of the spiral arm by searching for local radial maxima in the polar coordinate map for fixed values of θ in steps of 8◦ from 124◦ to 260◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The arm was modelled as an Archimedean spiral of the form R(θ) = a + cθ3, where θ is in radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (We found that logarithmic spirals and Archimedean spirals with smaller exponents did not fit the data well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The log-likelihood function was spec- ified as log L = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 � n � (Rdata−Rmodel)2 σ2 + log(2πσ2) � , where σ is the standard deviation of the major axis of the synthesized beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Uniform priors of [0, 2000] and [−2000, 0] were used for a and c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Posteriors were explored using the affine-invariant sampler emcee (Goodman & Weare 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2013) with 40 walkers and 1000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' After discarding the first 500 steps as burn-in, we computed the 50th per- 8 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Outflow Disk Blueshifted arm Line of Sight EAST WEST Envelope Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' A proposed cartoon schematic of the DR Tau system from an overhead perspective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', perpendicular to the line of sight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The components are not drawn to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Note that while the disk is drawn such that the east side is tilted toward the observer in order to show that the disk is slightly inclined, the observations do not constrain which side is closer to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The 3-dimensional orientations of the envelope and the arm are not known in detail, but the former is drawn in front of the disk and the latter is drawn behind the disk (from the perspective of the observer) under the assumption of infalling motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, the observations may also be explained by other configurations of the structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=" 6 3 0 3 6 [''] 6 3 0 3 6 [''] 12CO J = 2 1 (8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 km s 1) Clump 200 au 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 6 3 0 3 6 6 3 0 3 6 13CO J = 2 1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 km s 1) 200 au 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 Integrated Intensity (mJy beam 1 km s 1) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Integrated intensity maps of 12CO (left) and 13CO, summed up between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 km s−1 to highlight DR Tau’s blueshifted spiral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The blue cross marks the center of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The synthesized beam is shown as a white ellipse in the lower left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The 12CO color scale is saturated in order to show the fainter arm emission more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=" Molecular Mapping of DR Tau 9 0 300 600 900 1200 Deprojected radius (au) 0 60 120 180 240 300 360 (degrees) 6 4 2 0 2 4 6 [''] 6 4 2 0 2 4 6 [''] 120 150 180 210 240 270 (degrees) 0 15 30 45 60 75 Pitch angle (degrees) Figure 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Left: Integrated intensity map of the 12CO arm, replotted as a function of deprojected radius and polar angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Center: Integrated intensity map of the 12CO arm, overplotted with the spiral function defined by the posterior median values of the spiral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Right: Pitch angle of the arm as a function of the polar angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The black curve corresponds to the values derived from the median of the spiral arm parameter posteriors, while the blue curves correspond to 1000 random draws from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 10 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' centile of the marginal posterior distribution to obtain a point estimate and the 16th and 84th percentiles to ob- tain error estimates: a = 1060±30 au and c = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We computed the pitch angles (φ = arctan ��� 1 R dR dθ ��) � corresponding to the median values of a and c, then also computed pitch angles for spiral curves defined by 1000 random draws of (a, c) from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Figure 6 shows the median spiral plotted over the integrated intensity map and a plot of the derived pitch angles as a function of polar angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The pitch angles range from 6 to 56 degrees between polar angle values of 124 to 260 degrees (corresponding to deprojected radius values between 980 and 330 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In other words, the pitch angle appears to decrease with distance from the star, although the true values may differ if the assumption that the arm is in the plane of the disk is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We computed the escape velocity, vesc = � 2GM∗ r , at the tip of the arm to assess whether it is gravitationally bound to DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The dynamical mass of DR Tau has been measured to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 M⊙ (Braun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Emission from the arm is detected up to r ∼ 1200 au at an LSRK velocity of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 km s−1, which is offset from the systemic velocity by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The corresponding escape velocity at r = 1200 au is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Thus the arm appears to be compatible with being gravitationally bound to DR Tau, but not definitively so, since there may also be a transverse velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) recently identified two spiral arms in SPHERE H-band Qφ observations of DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Figure 7 compares the arms identified in the SPHERE image to the CO arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The CO arm is much more extended than the scattered light arms, which are only detected up to ∼ 220 au in projection from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Because the NOEMA synthesized beam is comparable in scale to the SPHERE spiral arms, it is not clear whether the CO arm is an extension of one of the arms detected in scattered light or a separate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) mea- sured pitch angles of 11◦ and 26◦ for the two scattered light arms, which are smaller than the pitch angle mea- sured for the inner region of the CO arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, since the pitch angles appear to change along the arm, the differing values do not necessarily imply that they are separate structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) also noted that the northeastern spiral in the SPHERE image had a clump-like feature, which they hypothesized was asso- ciated with a protoplanet embedded in a dusty envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' While this compact feature is well below the resolution limits of our NOEMA observations, the presence of a dif- ferent clump in the 12CO arm suggests that the clumps could be intrinsic features of the arms themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Envelope DR Tau shows envelope emission in 12CO up to ∼ 5′′ (1000 au) in projection from the star (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' En- velope emission is detected between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 and 12 km s−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', mostly redshifted with respect to the systemic ve- locity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In most of these channels, the envelope emission is more spatially extended and brighter on the northern side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' As with 12CO, the 13CO emission is more extended north of the star compared to south of the star for LSRK velocities above 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In contrast to 12CO, though, the 13CO maps show features that ap- pear more streamer-like than envelope-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, since this is the velocity range where cloud contamina- tion is most significant, spatial filtering of large-scale emission may be artificially creating the appearance of streamers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) identified a possible in- falling stream in ALMA observations of 13CO toward DR Tau, but those observations were likewise affected by spatial filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Observations of other lines that are bright but less susceptible to cloud contamination (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', species with higher critical densities like HCO+ or CO transitions with higher upper energy levels) might help to clarify the nature of these apparent streamers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The channels where envelope emission is detected in 12CO overlap with the channels where [C I] exhibits a redshifted non-Keplerian component that Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) attributed to an infalling stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, since the beam FWHM of the [C I] observations is ∼ 3′′, most of the emission is spatially unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Given the simi- lar velocities to the 12CO envelope, it is likely that the redshifted non-Keplerian [C I] emission also originates from the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Outflow DR Tau’s 12CO spectrum (Figure 1) exhibits a faint blueshifted line wing without a corresponding redshifted line wing, suggesting the presence of an asymmetric out- flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The channel maps (Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1) show compact emission at LSRK velocities lower than 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' To highlight this compact outflow emission more clearly, we extracted a new 12CO spectrum using a smaller circu- lar aperture with a diameter of 4′′ (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Because DR Tau is nearly face-on, it is not straightforward to separate the outflow emission from the line wings of the Keplerian disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, the asymmetry of the line pro- file allows us to estimate the velocities at which outflow emission dominates by mirroring the outflow spectrum about the systemic velocity and taking the ratio of the original and mirrored spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We assume that the outflow emission on the blueshifted side dominates when the ratio exceeds 10, which occurs at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' While the outflow emission is weak in individual channels (Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1), the spatial distribution of the blueshifted side can be better seen by producing an in- tegrated intensity map between −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The lower bound of the velocity integration range was determined by where the emission in individual chan- nels drops below 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For comparison, we also produced an integrated intensity map from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 km s−1, corresponding to the redshifted channels at the opposing offsets from the systemic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The two integrated in- Molecular Mapping of DR Tau 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 [′′] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 [′′] Southern spiral Northeastern spiral H-band Q Clump H-band Q 50 au 6 4 2 0 2 4 6 6 4 2 0 2 4 6 H-band Q vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 12CO arm 200 au Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' A comparison between the SPHERE H-band Qφ image of DR Tau from Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) and the 12CO NOEMA observations from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Left: H-band Qφ image of DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The arrows point to the northeastern spiral, southern spiral, and clump identified in Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The gray circle shows the extent of the SPHERE coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Right: A contour plot of the 12CO arm overlaid atop the H-band Qφ image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Note that the size scale is different from the image on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The contours, drawn at 50, 100, and 150 mJy beam−1 km s−1, correspond to the 12CO integrated intensity map from Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' tensity maps are presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The blueshifted map shows relatively compact emission with a radial ex- tent of ∼ 2′′ (∼ 400 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Although the opening angle of the outflow cannot be computed because the disk is nearly face-on, the compactness of the emission suggests that the outflow is quite collimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The redshifted map shows emission near the stellar position, but given that the redshifted emission is fainter and much more com- pact than the blueshifted outflow, it seems likely that the compact redshifted emission originates from the line wing of the Keplerian disk emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' While a redshifted outflow component is not readily visible in the 12CO spectrum, the redshifted map shows a faint ring with a radius of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5′′ (∼ 900 au), which is significantly wider than the blueshifted outflow component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' SO, DCO+, and H2CO emission SO, DCO+, and H2CO emission all originate from a relatively compact region within 300 au of DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The SO 65 − 54, SO 55 − 44, and H2CO 303 − 202 intensity- weighted velocity maps (Figure 3) show velocity gradi- ents similar to that of C18O, indicating that they like- wise (largely) originate from the Keplerian disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The kinematics of DCO+ are not well-defined due to the low signal-to-noise ratio, but the compactness of the emis- sion suggests that it also primarily traces the Keplerian disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' That said, whereas C18O and H2CO 303−202 both ex- hibit relatively axisymmetric emission in the integrated intensity maps (Figure 2) and line profiles that are sym- metric about the systemic velocity (Figure 1), SO 65−54 and 55 − 44 are both asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Their emission is stronger on the northern (redshifted) side of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In addition, their spectra both peak at an LSRK veloc- ity of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 km s−1, which is redshifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 km s−1 with respect to the systemic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The DCO+ spec- trum also appears stronger on the redshifted side, but given that its SNR is lower than that of the SO lines, more sensitive observations will be necessary to deter- mine whether the DCO+ asymmetry is genuine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The evolutionary stage of DR Tau DR Tau is traditionally considered to have a Class II SED, with stellar age estimates ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 Myr (Kenyon & Hartmann 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' McClure 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The presence of the envelope, if primordial, would suggest that the younger end of the age range is more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DR Tau’s chemistry also appears to point to a younger age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Although SO is detected in DR Tau, it has otherwise rarely been detected in Class II disks, 12 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 6 3 0 3 6 [′′] 6 3 0 3 6 [′′] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 0 50 100 250 500 12CO J = 2 1 intensity (mJy beam 1) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 6 3 0 3 6 [′′] 6 3 0 3 6 [′′] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 0 50 100 200 13CO J = 2 1 intensity (mJy beam 1) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of 12CO J = 2 − 1 and 13CO J = 2 − 1 over the velocity range where envelope emission is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The black contours denote the 5, 15, 25, and 35σ contours of C18O J = 2 − 1 to serve as a visual reference for the kinematics of the Keplerian disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (Note that because C18O is less abundant than 12CO and 13CO, the Keplerian line wings of C18O are not detected out to as high velocities as the other two isotopologues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' especially those hosted by T Tauri stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Guilloteau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Semenov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Le Gal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' It is commonly detected, though, in younger, embedded Class 0 and I systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Sakai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Le Gal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mercimek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In addition, Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) found that gas-phase carbon is not as severely depleted in DR Tau as other Class II disks that have been observed, although it is still more depleted than Class 0/I systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, simulations have suggested that pre-main se- quence stars might be able to form second-generation en- velopes through interaction with cloud material, a pro- cess sometimes referred to as “late infall” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Dulle- mond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Indeed, Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) hypothesized that DR Tau was undergo- ing late infall based on the detection of spiral arms in scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=" Given the range of ages estimated for DR Tau, it is ambiguous whether infall onto DR Tau should be considered “late.” As noted above, DR Molecular Mapping of DR Tau 13 0 10 20 LSRK Velocity (km s 1) 2 0 2 4 6 8 10 12 Flux (Jy) 12CO inner 4′′ 9 6 3 0 3 6 9 [''] 9 6 3 0 3 6 9 [''] 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 km s 1 0 100 200 300400 Integrated Intensity (mJy beam 1 km s 1) 9 6 3 0 3 6 9 9 6 3 0 3 6 9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 km s 1 0 50 100 150 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Overview of DR Tau’s outflow emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Left: 12CO spectrum extracted from a circular aperture with a 4′′ diameter, showing a blueshifted outflow wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The approximate velocity range of the blueshifted outflow is shaded in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The purple dotted line marks the system velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Middle: 12CO integrated intensity map covering velocities from −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Compact emission from the blueshifted side of the outflow is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' An arcsinh stretch is used on the color scale to make faint emission more readily visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The faint vertical striping is due to the sidelobes of the point spread function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The pink cross marks the position of the disk center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Right: 12CO integrated intensity map covering velocities from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The map shows a faint ring with a radius of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5′′ (∼ 900 au) and compact emission located at the stellar position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The redshifted compact emission may be from a line wing of the Keplerian disk rather than the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Tau’s chemistry seems to suggest that the disk is rela- tively young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This appearance of chemical youthfulness, though, stems from comparisons of Class 0/I sources to isolated Class II disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Based on molecular observations of GM Aur, a Class II disk with large-scale spiral arms suggestive of ongoing late infall, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2021) speculated that accretion of cloud material could par- tially reset disk chemistry such that it bears greater resemblance to that of Class 0/I sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The impact of late infall on disk chemistry will need to be examined through astrochemical modeling to determine the extent to which chemical properties can be used to sort disks by relative age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Given that DR Tau is commonly included in sur- veys of Class II disks because of its relatively large disk mass and bright line emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Salyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Arulanantham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022), an erroneous classification of its evolution- ary stage may skew interpretations of disk observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) remarked that a similar problem exists for DO Tau, another commonly observed Class II disk that also shows signatures of being partially embed- ded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Interestingly, among the twelve single star systems that Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2019) identified as having “smooth” disks in millimeter continuum emission, at least three of them (DR Tau, DO Tau, and Haro 6-13) exhibit ev- idence of an envelope in spatially resolved CO emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Fern´andez-L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022, and this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Most of the remain- ing sources (including both the “smooth” and structured disks) lack high quality interferometric CO observations, so it is unknown whether they might be embedded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Analyses of where and when disk substructures tend to emerge will require sensitive, spatially resolved molecular line observations to provide context about the evolutionary stages of the objects being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Origin of SO in DR Tau The detection of SO in DR Tau is notable given that SO detections have thus far been uncommon in Class II disks, which has been attributed to high gas-phase C/O ratios (> 1) disfavoring SO production (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Guilloteau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Semenov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Le Gal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In two of the disks where SO has been detected, AB Aur and Oph IRS 48, the gas-phase C/O ratio has been esti- mated to be less than 1 (Rivi`ere-Marichalar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Booth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Based on thermochemical model- ing of [C I] and CO isotopologue emission, Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) estimated that DR Tau has a gas-phase C/O ra- tio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The detection of SO toward DR Tau is thus qualitatively consistent with SO production in disks be- ing favored in gas with C/O ratios less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DR Tau’s SO emission exhibits a mild asymmetry that is not seen in C18O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This suggests that the SO asym- metry is not merely tracing the underlying gas surface density, but could instead be due to some dynamical process locally favoring SO production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' SO has been proposed to be enhanced by outflows, winds, gravita- 14 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' tional instabilities, or accretion shocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Pineau des Forˆets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Sakai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Tabone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Ilee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We discuss these possibilities in turn for DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Our NOEMA observations have shown that DR Tau has a molecular outflow, and past CO ro-vibrational spectroscopy indicates that DR Tau has a wide-angle molecular wind (Pontoppidan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' An outflow shock does not appear to be a likely major contributor to SO in DR Tau, since SO is not detected at the same high velocities as CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' At the spatial and spectral resolu- tion of our NOEMA observations, it is unclear whether the SO kinematics are consistent with those expected for a disk wind (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Haworth & Owen 2020), but DR Tau’s bright emission makes it an excellent target for more detailed follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Chemical modeling of gravitationally unstable disks suggests that gas-phase SO can be enhanced either by spiral shocks or within warm disk fragments (Ilee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2011, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' While DR Tau does feature spiral struc- ture, Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) argued that DR Tau’s spiral arms are unlikely to be due to gravitational instability given that its disk-to-stellar mass ratio and stellar accre- tion rate are a factor of a few lower than hydrodynamical simulations suggest would be necessary to induce grav- itational instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Nevertheless, disk mass is notori- ously difficult to measure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Miotello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022, and references therein), and studies of other spiral-armed disks have often disagreed on whether they are massive enough to be gravitationally unstable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', P´erez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Cleeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Veronesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Sierra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Higher spatial resolution would help to determine if the SO asymmetry traces spiral structure and/or a disk fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Accretion shocks in protoplanetary disks might oc- cur due to cloud or envelope material being accreted by the disk or disk material being accreted by an em- bedded planet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Bodenheimer 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Boss & Graham 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Yorke & Bodenheimer 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Szul´agyi & Mordasini 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Accretion streamers traced by SO have been observed in several Class I protostellar systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Artur de la Villarmois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Given that DR Tau is now known to be partially embed- ded, its asymmetric SO emission may arise in a manner similar to Class I systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Booth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) proposed that an SO asymmetry in the HD 100546 disk could be due to shocks from gas accreting onto an embedded planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This likely does not account for DR Tau’s SO asymmetry, since high-contrast imaging from Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) rules out the presence of a companion above sev- eral Jupiter masses at separations greater than 50 au from DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Origin of DR Tau’s molecular spiral arm Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) hypothesized that the northeastern spiral arm detected in scattered light toward DR Tau is due to planet-disk interactions, while the southern arm is due to infall from cloud material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' As noted in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2, the angular resolution of our NOEMA observa- tions does not allow us to determine whether the CO spiral arm is an extension of either scattered light spiral arm, but the very large extent of the molecular arm sug- gests that it is unlikely to be generated by interactions with a bound planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) placed an upper limit of several Jupiter masses on any companion farther out than 50 au from DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Given that the millimeter continuum appears smooth down to a resolution of 20 au (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018), it is unlikely that the disk harbors massive (super-Jovian) companions within 50 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' More- over, hydrodynamical simulations indicate that external companions exceeding several MJ should create a pair of (nearly) symmetric spiral arms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2016), contrary to what is observed for DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Thus, a stellar companion is also unlikely to be responsible for the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' An infalling stream is a plausible explanation for the molecular arm, given that similar large-scale structures have been detected in association with a number of em- bedded Class 0/I sources as well as Class II disks pro- posed to be undergoing late infall (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Yen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Valdivia-Mena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' One pos- sible difference of note is that the structures proposed to be infalling streams in the other systems have tended to be open, whereas DR Tau’s pitch angle exhibits a marked decrease with distance from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, this apparent difference may simply be a projection ef- fect, since we do not know their three-dimensional ori- entations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' As noted in the previous subsection, it is uncertain whether DR Tau is gravitationally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The possi- bility that DR Tau’s arm arises from gravitational in- stabilities remains intriguing given that clumpy arms are a hallmark of simulations of fragmenting disks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Basu & Vorobyov 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Furthermore, migration of clumps onto stars has been proposed as a trigger for FUor outbursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Boley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Clump migration might likewise explain DR Tau’s ex- treme brightening event in the 1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' If DR Tau’s disk mass has been estimated correctly, then the presence of a clump along DR Tau’s arm raises the question of whether fragmentation can occur under less stringent conditions than models demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Close stellar encounters can also generate large-scale arm-like structures with pitch angles comparable to that observed for the DR Tau molecular arm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Cuello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, Shuai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) inferred from an analysis of Gaia EDR3 data (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021) that the closest ex- pected approach between DR Tau and a neighboring star in the past 10,000 years is ∼ 105 au, which would be too distant to meaningfully perturb the known cir- cumstellar environment of DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2022) found that DQ Tau may have passed within 5100 au of Molecular Mapping of DR Tau 15 DR Tau 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='23 Myr ago, but considered such an encounter unlikely to be responsible for DR Tau’s spiral arms be- cause flyby-induced arms are only expected to survive on timescales of several thousand years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Cuello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Connections to EXor and FUor phenomena In recent years, the circumstellar environments of a number of FUors and EXors have been spatially resolved with millimeter interferometry and high-contrast scat- tered light imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' FUors are often associated with envelopes, outflows, and arm-like structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Zurlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Ru´ız-Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2017), similar to the structures associated with DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In FUor systems, the envelopes sup- ply infalling material that may help to activate gravita- tional instabilities (and then possibly magnetorotational instabilities), the arms may form as a consequence of gravitational instabilities, and instabilities may trigger outbursts that subsequently drive outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Vorobyov & Basu 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' With the exception of EX Lup and V1647 Ori, the latter of which is sometimes considered to be an FUor source, the EXor sources imaged so far have generally lacked analogous features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Principe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Cieza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' However, DR Tau exhibits striking similarities to EX Lup in that they both feature outflows, non-Keplerian spiral-like struc- tures, and (remnant) envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The circumstellar en- vironments of DR Tau and EX Lup also share similar- ities with that of RU Lup, which is not classified as an EXor source but is nevertheless an exceptionally ac- tive T Tauri star (Joy 1945;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Gahm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2018) suggested that the pres- ence of complex structures associated with EX Lup but not other EXors is an indication that EX Lup occupies an intermediate evolutionary stage between FUors and most EXors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The same may hold true for DR Tau (and perhaps RU Lup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Alternatively, the differences in EXor circumstellar environments may indicate that EXors are a heterogeneous group of objects, only some of which are closely related to the FUor phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' In any case, the observations of EX Lup, DR Tau, and RU Lup moti- vate more spatially resolved imaging of extremely active T Tauri stars to elucidate the connection between cir- cumstellar environments and stellar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' A changing view of Class II disks In the past decade, the introduction of high angular resolution imaging at millimeter wavelengths has trans- formed our understanding of planet formation by show- ing that dust substructures on scales of several au are common (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', ALMA Partnership et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Meanwhile, sensitive molecular imaging at more modest resolution has highlighted a deficit in our understanding of disk environments on scales of tens to thousands of au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' With single-dish telescopes and ear- lier generations of interferometers, signs of large-scale non-Keplerian emission towards Class II disks were of- ten ascribed to foreground contamination (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Thi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' ¨Oberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2011b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Even in the era of more powerful millimeter interferometers, in- sufficient integration times or insufficient uv coverage at larger spatial scales can lead to key structures being missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' High-quality molecular mapping, though, has demon- strated that there are indeed large-scale tails, spirals, streams, and/or remnant envelopes associated with a number of Class II systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Paneque-Carre˜no et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Scattered light imaging has also played an important role in uncovering examples of Class II disks that appear to be interacting with surrounding material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Grady 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021), although as demonstrated by the examples of DR Tau from this work and RU Lup from Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' (2020), molecular observations can reveal structures far beyond the detected extent of scattered light features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Infall from these larger-scale structures is increasingly being invoked to explain certain disk structures observed at smaller scales, such as misalignments or spiral arms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Paneque-Carre˜no et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' These observations thus imply an in- triguing link between dynamical processes operating on disparate size scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' For individual systems, though, infall is only one of several possible explanations for the observed disk phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' More systematic molecular line observations will be key for establishing patterns of association between large- and small-scale properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' SUMMARY We present new NOEMA observations of 12CO, 13CO, C18O, SO, DCO+, and H2CO toward the T Tauri star DR Tau, representing the highest-quality millimeter line observations of this source to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Our findings are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' CO emission shows that the DR Tau protoplane- tary disk is associated with an envelope, a faint asymmetric outflow, and a large non-Keplerian spiral arm with a clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The molecular spiral arm resembles a scaled-up version of the spiral arms detected in scattered light, although the angular resolution of NOEMA is not sufficient to determine whether the molecu- lar arm is an extension of one of the scattered light arms or a separate feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Whereas the scattered light arms are only detected up to ∼ 220 au in projection from DR Tau, the molecular arm is de- tected up to ∼ 1200 au in projection from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We report detections of SO, DCO+, and H2CO in the DR Tau disk for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Their kinemat- 16 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' ics and compact emission extent suggest that they primarily trace the Keplerian circumstellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' SO emission is stronger on the northern, redshifted side of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This asymmetry might be linked to infall from an asymmetric envelope or to un- resolved spiral substructure associated with the arms detected in scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Higher angular resolution observations of SO will be needed to clarify the origins of the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' DR Tau’s envelope, outflow, and arm are reminiscent of the structures that have been observed in association with various FUor sources as well as the EXor source EX Lup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Given that FUor and EXor outbursts have been linked to instabilities driven by envelope accretion, a similar mechanism may account for DR Tau’s dramatic stellar brightness changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The NOEMA observations of DR Tau highlight the utility of sensitive, spatially re- solved molecular line observations for providing context about the conditions under which young stars and their protoplanetary disks evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This work is based on observations carried out un- der project number W20BE with the IRAM NOEMA Interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This work is also based on observations collected at the Euro- pean Southern Observatory under ESO programme(s) 0102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='C-0453(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We thank our NOEMA local contact, Ana Lopez-Sepulcre, for setting up the observing scripts and assisting with data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We also thank Arthur Bosman, Ke Zhang, Joel Bregman, Lee Hartmann, Merel van’t Hoff, Ardjan Sturm, Melissa McClure, and Ewine van Dishoeck for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' We thank the referee, Ruobing Dong, for helpful comments im- proving the clarity of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Support for J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' was provided by NASA through the NASA Hubble Fellowship grant #HST-HF2-51460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in As- tronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', for NASA, under contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 101002188).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Facilities: NOEMA Software: analysisUtils (https://casaguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' edu/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='php/Analysis Utilities), AstroPy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2013), CASA (CASA Team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022), cmasher (van der Velden 2020), emcee (Foreman- Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2013), GILDAS (Pety 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Gildas Team 2013), matplotlib (Hunter 2007), pandas (pan- das development team 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Wes McKinney 2010), scikit-image (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2014), SciPy (Vir- tanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020) APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' SPECTROSCOPIC PARAMETERS OF TARGETED LINES The spectroscopic parameters of the targeted lines, taken from the Cologne Database for Molecular Spectroscopy (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2001, 2005) via Splatalogue1, are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Primary line targets are marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1 https://splatalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='online// Molecular Mapping of DR Tau 17 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Spectroscopic Parameters of All Targeted Lines Transition Rest frequency Eu (GHz) (K) 13C17O J = 2 − 1 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5738730 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 SO JN = 55 − 44 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2206530 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1 DCO+ J = 3 − 2 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1125822 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='7 H2S JKaKc = 220 − 211 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='7104365 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 c-C3H2 JKaKc = 330 − 221 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2787560 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5 SiO J = 5 − 4 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1049190 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 DCN J = 3 − 2 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2385378 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9 c-C3H2 JKaKc = 514 − 423 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9400460 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 H2CO JKaKc = 303 − 202 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2221920 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 HC3N J = 24 − 23 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3247230 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 H2CO JKaKc = 322 − 221 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4756320 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1 H2CO JKaKc = 321 − 220 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='7600660 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1 C18O J = 2 − 1 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5603541 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 SO JN = 65 − 54 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9494420 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 13CO J = 2 − 1 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3986842 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9 12CO J = 2 − 1 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5380000 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 OCS J = 19 − 18 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0609934 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9 N2D+ J = 3 − 2 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3218283 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 13CS J = 5 − 4 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2206852 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3 C2S JN = 1918 − 1817 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9384580 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 PN J = 5 − 4 234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9356940 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 HC3N J = 26 − 25 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5127888 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 H2CS JKaKc = 717 − 616 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='7270204 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' CHANNEL MAPS Channel maps of the primary line targets (listed in Table 1) are presented in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' AUXILIARY LINE TARGETS Table 3 lists the beam sizes, per-channel rms of the image cubes, and 3σ flux upper limits for the auxiliary line targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The flux upper limits were estimated as- suming the same velocity range and aperture used to measure the C18O flux (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Flux upper lim- its may be underestimated if the molecule is primarily present in the envelope or outflow rather than the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' REFERENCES Akiyama, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Vorobyov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Johns-Krull, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', & Basri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2001, AJ, 122, 3335, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1086/323914 ALMA Partnership, Brogan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1994, ApJ, 420, 837, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1086/173608 Andre, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Ward-Thompson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', & Barsony, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 1993, ApJ, 406, 122, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1086/172425 Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', & Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2007, ApJ, 659, 705, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1086/511741 18 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 8 4 0 4 8 [′′] 8 4 0 4 8 [′′] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 50 100 250 500 700 12CO J = 2 1 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of 12CO J = 2 − 1 toward DR Tau, part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The top right of each panel is labelled with the LSRK velocity (km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The synthesized beam is drawn in the lower left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The purple crosses denote the disk center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Offsets from the disk center (in arcseconds) are marked on the axes in the lower left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' The color scale uses an arcsinh stretch to make faint extended features more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Molecular Mapping of DR Tau 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 4.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 50 100 250 500 700 12CO J = 2 1 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of 12CO J = 2 − 1 toward DR Tau, part 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 8 4 0 4 8 [′′] 8 4 0 4 8 [′′] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 50 100 250 500 700 12CO J = 2 1 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of 12CO J = 2 − 1 toward DR Tau, part 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', P´erez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2018, ApJL, 869, L41, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3847/2041-8213/aaf741 Ardila, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Basri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Walter, F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 8 4 0 4 8 [′′] 8 4 0 4 8 [′′] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 50 100 250 500 700 12CO J = 2 1 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of 12CO J = 2 − 1 toward DR Tau, part 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 8 4 0 4 8 [′′] 8 4 0 4 8 [′′] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 0 50 100 200 300 13CO J = 2 1 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of 13CO J = 2 − 1 toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Banzatti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='09752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='09752 Cuello, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Dipierro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Mentiplay, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2019, MNRAS, 483, 4114, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='1093/mnras/sty3325 Molecular Mapping of DR Tau 23 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 2 0 2 [′′] 2 0 2 [′′] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 50 100 150 200 C18O J = 2 1 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of C18O J = 2 − 1 toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Contours are drawn in pink at the 3, 5, 10, 15, 20, 30σ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 2 0 2 [′′] 2 0 2 [′′] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 15 30 45 60 SO JN = 65 54 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of SO JN = 65 − 54 toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Contours are drawn in pink at the 3, 5, 10σ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 24 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 2 0 2 [′′] 2 0 2 [′′] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 15 30 45 SO JN = 55 44 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of SO JN = 55 − 44 toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Contours are drawn in pink at the 3, 5σ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 2 0 2 [′′] 2 0 2 [′′] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 9 18 27 36 DCO + J = 3 2 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of DCO+ J = 3 − 2 toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Contours are drawn in pink at the 3, 5σ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Molecular Mapping of DR Tau 25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 2 0 2 [′′] 2 0 2 [′′] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 25 50 75 100 H2CO JKaKc = 303 202 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of H2CO JKaKc = 303 −202 toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Contours are drawn in pink at the 3, 5, 10, 15σ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 2 0 2 [′′] 2 0 2 [′′] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 10 20 30 H2CO JKaKc = 322 221 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of H2CO JKaKc = 322 − 221 toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Contours are drawn in pink at the 3, 4σ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 26 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 2 0 2 [′′] 2 0 2 [′′] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0 200 au 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8 0 5 10 15 20 25 H2CO JKaKc = 321 220 intensity (mJy beam 1) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Channel maps of H2CO JKaKc = 321 − 220 toward DR Tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Contours are drawn in pink at the 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Imaging Summary for Auxiliary Line Targets Transition Synthesized beam Per-channel RMS noisea 3σ Flux Upper Limit (arcsec × arcsec (◦)) (mJy beam−1) (mJy km s−1) 13C17O J = 2 − 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='21 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='93 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0◦) 8 < 40 H2S JKaKc = 220 − 211 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='21 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='93 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6◦) 7 < 40 c-C3H2 JKaKc = 330 − 221 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='21 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='93 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='7◦) 7 < 30 SiO J = 5 − 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='21 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='93 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='4◦) 7 < 30 DCN J = 3 − 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='21 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='93 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='5◦) 7 < 40 c-C3H2 JKaKc = 514 − 423 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='21 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='93 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='7◦) 7 < 30 HC3N J = 24 − 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='20 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='94 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='0◦) 7 < 30 OCS J = 19 − 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='18 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='91 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3◦) 7 < 30 N2D+ J = 3 − 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='18 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='91 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3◦) 10 < 70 13CS J = 5 − 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='18 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='91 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='3◦) 8 < 50 C2S JN = 1918 − 1817 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='17 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='90 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6◦) 10 < 50 PN J = 5 − 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='17 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='90 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='8◦) 9 < 40 HC3N J = 26 − 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='17 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='90 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='7◦) 10 < 50 H2CS JKaKc = 717 − 616 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='17 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='90 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='6◦) 11 < 80 aFor channel widths of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content='2 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' Molecular Mapping of DR Tau 27 Cuello, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Louvet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', Mentiplay, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfzgKz/content/2301.02674v1.pdf'} +page_content=' 2020, MNRAS, 491, 504, doi: 10.' metadata={'source': 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Russell, David Beaty, Fiona Thiessen, +Jessica Barnes, Lydie Bonal, John Bridges, Thomas Bristow, John Eiler, Ludovic Ferrière, Teresa +Fornaro, Jérôme Gattacceca, Beda Hoffman, Emmanuelle J. Javaux, Thorsten Kleine, Harry Y. +McSween, Manika Prasad, Liz Rampe, Mariek Schmidt, Blair Schoene, Kirsten L. Siebach, Jennifer +Stern, Nicolas Tosca. + + + + + +Requestor: NASA-ESA MCSG1 team +Date: January 11, 2023 + + + + +Citation to this report: +Dauphas N., Russell S.S., Beaty D., Thiessen F., Barnes J., Bonal L., Bridges J., Bristow T., Eiler J., +Ferrière L., Fornaro T., Gattacceca J., Hoffman B., Javaux E.J., Kleine T., McSween H.Y., Prasad M., +Rampe L., Schmidt M., Schoene B., Siebach K.L., Stern J., Tosca N. (2023) Science priorities for the +extraction of the solid MSR samples from their sample tubes. NASA-ESA Mars Rock Team Report 1 + + + + +2 +Background: The NASA-ESA Mars Rock Team is an outgrowth of the MCSG1 team. It is composed of +scientists with expertise in handling and analyses of both terrestrial and extraterrestrial samples, rock +physics, and contamination mitigation. Two online meetings were organized in the Fall of 2022 where +Oscar Rendon Perez (JPL) and Paulo Younse (JPL) described the engineering options for opening the +tubes that will contain the samples returned from Mars' Jezero crater. This prompted discussions +between the Rock Team members (during online meetings and through emails). The Rock Team +leadership met online with the team focused on gas analysis (Gas Team) to understand their +constraints and make sure that the solutions envisioned for headspace gas extraction would not +compromise solid core retrieval. This report summarizes the consensus view of the Rock Team. It was +written by the Rock Team leadership with input from all team members. +Summary: Preservation of the chemical and structural integrity of samples that will be brought back +from Mars is paramount to achieving the scientific objectives of MSR. Given our knowledge of the +nature of the samples retrieved at Jezero by Perseverance, at least two options need to be tested for +opening the sample tubes: (1) One or two radial cuts at the end of the tube to slide the sample out. +(2) Two radial cuts at the ends of the tube and two longitudinal cuts to lift the upper half of the tube +and access the sample. Strategy 1 will likely minimize contamination but incurs the risk of affecting +the physical integrity of weakly consolidated samples. Strategy 2 will be optimal for preserving the +physical integrity of the samples but increases the risk of contamination and mishandling of the sample +as more manipulations and additional equipment will be needed. A flexible approach to opening the +sample tubes is therefore required, and several options need to be available, depending on the nature +of the rock samples returned. Both opening strategies 1 and 2 may need to be available when the +samples are returned to handle different sample types (e.g., loosely bound sediments vs. indurated +magmatic rocks). This question should be revisited after engineering tests are performed on analogue +samples. The MSR sample tubes will have to be opened under stringent BSL4 conditions and this +aspect needs to be integrated into the planning. +Introduction: NASA-ESA are planning to collect and transport from Mars to Earth a set of samples of +martian materials for the purpose of scientific investigation (Kminek et al. 2022). The samples are +currently collected by the Perseverance Rover (Farley and Stack, 2022) and consist of rocks, regolith, +and at least one dedicated sample of atmospheric gas. In addition, for the rock and regolith samples, +the process of sealing the sample tubes at the martian surface will result in the volume above the solid +samples (referred to as the head space) being occupied by martian atmospheric gas. The samples will +be contained within titanium sample tubes, which will be sealed at the martian surface with a +compression-style cap. +The rocks sampled thus far by the Perseverance Rover comprise magmatic rocks like basalt and olivine +cumulates that experienced various degrees of secondary water alteration, water-laid detrital +sedimentary rocks that show various levels of induration, and unconsolidated Mars regolith that could +contain grains from afar transported to the Jezero crater. Two main considerations weigh on the +strategy that should be adopted for opening the samples: +(1) Important information is contained in the vertical successions and textural characteristics of layers +in sediments, which can provide important clues for interpreting the depositional setting (Fig. 1). For +example, in terrestrial lakes, vertical gradation in grain size can reflect the relative density of +depositional and lacustrine fluids or gradations in organic matter content can reflect seasonal changes +in biological productivity. Fine laminations can sometimes reflect the presence of microbial mats. The +method used for opening the tubes must imperatively preserve those fine structures. + + +3 + +Fig. 1. Examples of possible fine-scale laminations in terrestrial environments (left; seasonal varves from Lake +Belau, Northern Germany; Dörfler et al. 2012; right Microbially-Induced Sedimentary Structures-MISS in the +middle neoproterozoic Chuar Group, Grand Canyon, Arizona; Bohacs and Junium 2007). +(2) Some critical measurements are sensitive to contamination either from the tube, the apparatus +used for cutting the tubes, or surrounding contaminants present in the isolator. Organic matter is of +particular concern given the high stakes involved in any claim for the presence of any form of biotic or +prebiotic chemistry on Mars. Inorganic trace element isotopes may provide dates on when Mars was +habitable, and these are also prone to contamination. +Beginning in 2022, an engineering team was tasked with developing the processes needed to open +the sample tubes and to extract the solid and gaseous samples. The engineering team was asked to +develop engineering priorities associated with this process. Two science teams were asked to develop +parallel science priorities: A group we call the “Gas Team” evaluated the priorities related to the +science associated with all returned gaseous sample, and a second group called the “Rock Team” (the +authors of this report) evaluated the priorities associated with solid materials contained within the +sample tubes. Both the "Gas Team" and "Rock Team" work under the oversight of a third committee, +the Mars Campaign Science Group (MCSG1). +The solid samples returned from the martian surface are certain to include sedimentary rocks (most +important for the search for biosignatures), igneous rocks, and regolith, and they may also include +other kinds of rocks, such as hydrothermal rocks, or impact breccia. The samples will be the basis for +answering the main scientific questions of Mars Sample Return (iMOST, 2018). +The rock samples at Mars will all have been collected from various outcrops (or perhaps very large +blocks of coherent rock). However, at least some of the rocks are relatively weak (i.e. a low +compressive strength), and are vulnerable to fracturing during drilling and during several dynamic +events associated with spacecraft operations during the return phase (most importantly, at Earth +landing). It is anticipated that the mechanical state of each sample, as received in the laboratory on +Earth, will be assessed by a method like computer tomography (CT) scanning prior to opening. The +decision on how to open each sample tube can therefore be based on geological data from the field +(collected by the M2020 science team), tests done on analogue samples, as well as the penetrative +imaging data obtained on Earth. +The engineering team has proposed a 2-phase process for opening the sample tubes: First, puncture +the tube in a way that will allow any gas present to be extracted and captured, then second, cut the +metal of the tube in a way that would allow the solid materials to be removed. Regarding cutting the +metal of the tubes, three primary mechanisms have been proposed (Fig. 2): +• +A single radial cut to the end of the tube, so that the sample could be tipped out. +• +A radial cut at each end of the tube, which would enable the sample to be pushed out from +one end + +9 +belowtopof core segment +10 +12 +(cm) +13 +14 +151cm +4 +• +Two radial cuts and two longitudinal cuts, to reveal the whole sample during cutting. +An option frequently used on Earth to access core samples, for example used with deep sea drill cores, +is to cut the core tube and the core together with something like a band saw. This is not an option for +samples returned from Mars as this would have the effect of driving contamination from both the +metallic core tube and band saw into the interior of the rock core. + + + +Figure 2. Proposed protocols for opening the sample tubes. Drawings courtesy of Oscar Rendon Perez. In the one +radial cut approach, a sharp hard metal wheel shears through the tube by slowly rotating and tightening it +around the tube (bottom panel; left). The sample is extracted from the tube by inclining it and controlling the +rate of descent with a piston. The second approach involves doing a second cut to push the sample outwards. A +virtue of this approach is that it allows for a more controlled extraction, and it minimizes the risk of the sample +getting jammed in the tube. Both options 1 and 2 involve the sample sliding out of the tube and incur the risk of +losing the chemical and structural layering of the sample. The third approach involves doing two longitudinal +cuts on the side of the tube to expose the whole sample within the tube. It is least likely to disturb the physical +integrity of the sample, which stays in place in the tube, but it involves cutting the tube along its length through +a white alumina coating (deposited on the tubes to reduce their heat absorption while seating on Mars' surface) +possibly using a circular blade (bottom panel; right). The chance of contamination is higher with this third +approach as more tube manipulations are involved, more tube material is cut, and the setup to remove or cut +the alumina coating will be more involved than the wheel cutter used in approaches 1 and 2. +Approach: +The issue of how to open the tubes was discussed by the team over two telecons. Presentations by +engineers Oscar Rendon Perez and Paolo Younse were delivered to explain the design of the tubes +and different options for opening them (Fig. 2). +The Rock Sample Team concluded there are three main considerations: +• +Need to minimise (and have knowledge of) contamination +• +Need to preserve stratigraphy and other textural relationships +• +Need to maximise the amount of sample material that ends up in a scientifically useful state +from the tubes. For some samples like the detrital sediments or the regolith sample, each + +BUEHLER +DIAMOND +WAFERING BID +BUEHILER +5 +small grain may provide a unique record of Mars' surface history, so dust adhering to the tube +surface should be recovered to the greatest extent possible. However, such dust will likely +represent a small fraction of the total mass and its retrieval could be done later. Or it could be +used for quickly surveying the petrography and mineralogy of the core as part of a preliminary +examination phase as this material will be of lesser value for other tasks and could be sterilized. +Minimal cutting (i.e., a single radial cut) was considered optimal to minimise potential contamination +of trace elements, especially metals, and organic material from the tubes and cutting tools. The +structural integrity of the sample would, however, be best preserved with radial and longitudinal cuts; +this is considered especially important for sedimentary rocks that may be friable but contain internal +stratigraphic structures. The yield may be maximised by at least two radial cuts. These considerations +may conflict with each other and the approach to be used will depend on the exact nature of each +returned sample. Magnetic contamination should also be minimized during cutting operation and +sample handling. +The preferred opening strategies are summarized in Table 1, which ponders each criterion (structure +integrity, chemical integrity, and yield) for three categories of samples (consolidated rocks, friable +rocks, and loose regolith). We summarize the Rock Team recommendations at the bottom of each +column. The rationale for each entry is summarized below: +Consolidated rocks (example microgabbro). To minimize the risk of contamination, one radial cut is +preferred as cutting by shearing with a hard metal solid wheel will generate little dust, cause little +heating, involve no use of fluid, and involve the least amount of tube material of all considered options. +To get the sample out of the tube, putting it on a vertical incline and lowering the sample in a +controlled manner with a piston would preserve the structural integrity of the sample. One radial cut +is likely to preserve the structural integrity of the sample. The cutting wheel will create a metal lip that +will protrude in the tube, so provision should be ready to straighten that lip so that the sample can be +extracted without rubbing against the lip. With a consolidated sample, there is however a concern +that jamming could occur, as a fragment might be trapped in compression between the solid core and +the tube wall. A second cut might be needed to push/pull the sample from the other side and free it +from such entrapment. Fine dust adhering to the inner tube surface might be difficult to retrieve with +a single radial cut. A second radial cut would allow one to get the fine dust out by pushing it out with +an appropriate instrument. The Rock Team favours 1 radial cut, with 2 radial cuts possibly needed for +sample retrieval in case of jamming and to recover fine dust adhering to the interior tube surface. +Friable rocks (example detrital sediments). These rocks are the ones for which preserving the +stratigraphy is of upmost importance. The rationale is the same as with consolidated rocks that a single +radial cut would be preferred from the point of avoiding contamination. To extract the sample, a single +radial cut might be sufficient as the less consolidated nature of those rocks means that they are less +likely to be hard jammed in the tube. A possible approach would be to put place a piston against the +sample on the opening side with the tube horizontal. The sample tube and piston would then be +rotated to a vertical position, and the piston would be lowered in a controlled manner to transfer the +sample core in a transparent sample holder (quartz or sapphire) with predesigned longitudinal +openings. The reason to transfer the sample vertically is to minimize shear on the tube surface. After +vertical transfer of the sample from the tube to the holder, the holder would be rotated back to +horizontal to be then opened, giving access to the sample. +Alternatively, it might be possible to 2 radial cuts, and one piston to push the sample out in a slightly +inclined orientation and another piston at the open side against the sample to prevent collapse, so +the sample keeps its integrity but we can avoid the longitudinal cuts to avoid more risk of + + +6 +contamination. If too friable, the sample could be gently pushed this way into a transparent sample +holder with predesigned longitudinal openings, allowing visible inspection of the enclosed protected +sample +Letting the sample slide out from one side incurs the risk however that rock fragments will be moved +out of sequence, that the sample will disaggregate, and that important chemical features be smeared +throughout the core. The latter point could include, for instance, organic distribution. If a layer is highly +enriched in organics, sliding the whole sample along the sides may smear the signature throughout +the entire core surface. For preserving the stratigraphy, it may therefore be advantageous to make 2 +radial cuts and 2 longitudinal cuts to access the core without disturbing it. The constraints on fine dust +recovery are the same as with other sample types. +Regolith. There is no stratigraphic information to preserve in that sample and little risk of jamming, +so a single radial cut is preferred as this minimizes the risk of contamination. The fine dust in the +sample may come from afar and each grain will likely tell a story, so complete recovery of dust +adhering to the tube inner surface is important. +Table 1. Preferred opening strategies depending on rock cohesiveness and criteria considered. + + +The Rock Sample Team finds that a single approach will not be appropriate for all the rock samples +returned, but instead a flexible and bespoke approach will be needed for each sample tube opening, +with all three of the above options available. As a general principle, minimal cutting is favoured as +this will also minimise potential contamination issues. However, an overriding consideration is that +Consolidated rocks +Example: microgabbro +Friable rocks +Example: detrital +sediments, igneous +cumulate rocks +Regolith +Trace element and +organic contamination +1 radial cut +1 radial cut +1 radial cut +Structural integrity of +the sample +1 radial cut likely OK +Maybe 2 radial cuts in +case of jamming +1 radial cut or +2 radial cuts and 2 +longitudinal cuts +1 radial cut +Complete retrieval of the +sample (including dust) +1 or 2 radial cuts +1 or 2 radial cuts +1 or 2 radial cuts +Rock Team +recommendation +1 OR 2 radial cuts +1 radial cut OR +2 radial cuts and 2 +longitudinal cuts +1 OR 2 radial cuts +FINDING: There is not one single approach for opening the sample tubes that will +work sufficiently well for all MSR rock samples. Multiple options need to be available. + + +7 +the structural integrity of the rock sample is key to understanding its petrology, and this should remain +intact, even if this requires more processing. +For regolith samples, a single radial cut followed by tipping out the grains is likely to be appropriate, +since this will minimise contamination and there is no need to preserve spatial relationships within +the tube. For well consolidated (e.g., some igneous rock) samples, a radial cut perhaps followed by a +second radial cut may be required to extract the sample completely. For sedimentary rocks, and any +friable igneous rocks, the decision is more complex because a longitudinal cut may be necessary to +observe and preserve structural relationships, but this must be weighed against potentially +contributing more contamination. One possible solution to test for sedimentary samples could be to +make one or two radial cuts, then push the sample or let it slide down while keeping its stratigraphy +in place (possibly with high inclination to minimize shear along tube surface, with a sliding stopper +against the sample to control the sliding rate) into another tube with a closed longitudinal aperture +that allows longitudinal opening later. +The physical state of each core (consolidated or friable) will not be known for certain until the samples +are bought back to Earth, where CT-scanning will reveal the fine structure of the samples and guide +the strategy that adopted for tube opening. +Future Work: +The team suggests areas which require more work prior to sample return. These include: +• +Investigate how/whether analogue sedimentary samples and aqueously altered cumulate +rocks can be removed in a manner that preserves their structural integrity with only one radial +cut. +• +Investigate ways to efficiently remove the fines left behind after core extraction. +• +Impurities in all tube materials, coatings, and opening contraption (e.g., materials used in the +saw) must be characterized with appropriate techniques (e.g., ICP-MS). We suggest that a task +group be established to undertake an in-depth contaminant characterization campaign. +• +Investigate if it is possible to remove the alumina coating without compromising the sample, +and without causing damage (e.g. by vibration) to the martian sample inside the core tube. +• +Investigate the degree to which the different cutting protocols can introduce contamination. +• +Integrate these studies with CT and related scanning techniques. +• +Investigate how the cutting and related techniques can be performed in a Biological Hazard +Level BSL4 environment. + +A concept that is not discussed in this report, but that has been considered elsewhere, is that the +opportunity exists to do penetrative imaging/mineralogical characterization of the sample-bearing +Mars sample tubes once they make it to Earth, so that we can obtain data on the mechanical state of +each sample as received prior to tube opening. This eliminates the need to make guesses based on +pre-sampling field data, or accelerations measured by the return spacecraft, etc. That imaging data +will give us the opportunity to help make decisions on how to open each tube. We know that for +samples with different kinds of mechanical integrity, different tube-opening strategies may be +required to avoid the risk of damage that unnecessarily affects the scientific usefulness of the sample. +A component of the technology program is needed to develop the datasets for what happens when +tubes containing samples with different degrees of mechanical integrity are opened by each of the +three methods described. This will become the basis for future decision-making. We also need data + + +8 +on the real contamination implications of making the horizontal cuts, and what kind of science is +affected by such contamination. + +References. +iMOST (2018), The Potential Science and Engineering Value of Samples Delivered to Earth by Mars +Sample Return, (co-chairs D. W. Beaty, M. M. Grady, H. Y. McSween, E. Sefton-Nash; documentarian +B.L. Carrier; plus 66 co-authors), 186 p. white paper. Posted August, 2018 by MEPAG at +https://mepag.jpl.nasa.gov/reports.cfm. +Bohacs K.M., Junium (2007) Microbial mat sedimentary structures and their relation to organic-carbon +burial in the Middle Neoproterozoic Chuar Group, Grand Canyon, Arizona, USA Microbial-Induced +Sedimentary Structures-MISS in the middle neoproterozoic Chuar Group, Grand Canyon, Arizona. In +Atltas of microbial mat features within the clastic rock record, Schieber J. et al. (Eds). Elsevier 208-213. +Dörfler W et al. (2012) A high-quality annually laminated sequence from Lake Belau, Northern +Germany: revised chronology and its implications for palynological and tephrochronological studies. +The Holocene 22, 1413-1426. +Farley KA, Stack K. (2022) Mars 2020 Initial Reports, Crater Floor Campaign, August 11, 2022. +Kminek G., Meyer M.A., Beaty D.W., Carrier B.L., Haltigin T., Hays L.E. (2022) Mars Sample Return +(MSR): +Planning +for +Returned +Sample +Science. +Astrobiology.Jun +2022.S-1-S- +4.http://doi.org/10.1089/ast.2021.0198 + + diff --git a/KtE3T4oBgHgl3EQfvgte/content/tmp_files/load_file.txt b/KtE3T4oBgHgl3EQfvgte/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8190d53d62285d068c491dfb352a63dbca552c6 --- /dev/null +++ b/KtE3T4oBgHgl3EQfvgte/content/tmp_files/load_file.txt @@ -0,0 +1,336 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf,len=335 +page_content='1 Science Priorities for the Extraction of the Solid MSR Samples from their Sample Tubes NASA-ESA Mars Rock Team Nicolas Dauphas, Sara S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Russell, David Beaty, Fiona Thiessen, Jessica Barnes, Lydie Bonal, John Bridges, Thomas Bristow, John Eiler, Ludovic Ferrière, Teresa Fornaro, Jérôme Gattacceca, Beda Hoffman, Emmanuelle J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Javaux, Thorsten Kleine, Harry Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' McSween, Manika Prasad, Liz Rampe, Mariek Schmidt, Blair Schoene, Kirsten L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Siebach, Jennifer Stern, Nicolas Tosca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Requestor: NASA ESA MCSG1 team Date: January 11, 2023 Citation to this report: Dauphas N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Russell S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Beaty D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Thiessen F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Barnes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Bonal L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Bridges J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Bristow T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Eiler J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Ferrière L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Fornaro T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Gattacceca J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Hoffman B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Javaux E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Kleine T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', McSween H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Prasad M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Rampe L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Schmidt M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Schoene B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Siebach K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Stern J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', Tosca N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' (2023) Science priorities for the extraction of the solid MSR samples from their sample tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' NASA-ESA Mars Rock Team Report 1 2 Background: The NASA-ESA Mars Rock Team is an outgrowth of the MCSG1 team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' It is composed of scientists with expertise in handling and analyses of both terrestrial and extraterrestrial samples, rock physics, and contamination mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=" Two online meetings were organized in the Fall of 2022 where Oscar Rendon Perez (JPL) and Paulo Younse (JPL) described the engineering options for opening the tubes that will contain the samples returned from Mars' Jezero crater." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' This prompted discussions between the Rock Team members (during online meetings and through emails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The Rock Team leadership met online with the team focused on gas analysis (Gas Team) to understand their constraints and make sure that the solutions envisioned for headspace gas extraction would not compromise solid core retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' This report summarizes the consensus view of the Rock Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' It was written by the Rock Team leadership with input from all team members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Summary: Preservation of the chemical and structural integrity of samples that will be brought back from Mars is paramount to achieving the scientific objectives of MSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Given our knowledge of the nature of the samples retrieved at Jezero by Perseverance, at least two options need to be tested for opening the sample tubes: (1) One or two radial cuts at the end of the tube to slide the sample out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' (2) Two radial cuts at the ends of the tube and two longitudinal cuts to lift the upper half of the tube and access the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Strategy 1 will likely minimize contamination but incurs the risk of affecting the physical integrity of weakly consolidated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Strategy 2 will be optimal for preserving the physical integrity of the samples but increases the risk of contamination and mishandling of the sample as more manipulations and additional equipment will be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' A flexible approach to opening the sample tubes is therefore required, and several options need to be available, depending on the nature of the rock samples returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Both opening strategies 1 and 2 may need to be available when the samples are returned to handle different sample types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', loosely bound sediments vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' indurated magmatic rocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' This question should be revisited after engineering tests are performed on analogue samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The MSR sample tubes will have to be opened under stringent BSL4 conditions and this aspect needs to be integrated into the planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Introduction: NASA-ESA are planning to collect and transport from Mars to Earth a set of samples of martian materials for the purpose of scientific investigation (Kminek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The samples are currently collected by the Perseverance Rover (Farley and Stack, 2022) and consist of rocks, regolith, and at least one dedicated sample of atmospheric gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' In addition, for the rock and regolith samples, the process of sealing the sample tubes at the martian surface will result in the volume above the solid samples (referred to as the head space) being occupied by martian atmospheric gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The samples will be contained within titanium sample tubes, which will be sealed at the martian surface with a compression-style cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The rocks sampled thus far by the Perseverance Rover comprise magmatic rocks like basalt and olivine cumulates that experienced various degrees of secondary water alteration, water-laid detrital sedimentary rocks that show various levels of induration, and unconsolidated Mars regolith that could contain grains from afar transported to the Jezero crater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Two main considerations weigh on the strategy that should be adopted for opening the samples: (1) Important information is contained in the vertical successions and textural characteristics of layers in sediments, which can provide important clues for interpreting the depositional setting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' For example, in terrestrial lakes, vertical gradation in grain size can reflect the relative density of depositional and lacustrine fluids or gradations in organic matter content can reflect seasonal changes in biological productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Fine laminations can sometimes reflect the presence of microbial mats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The method used for opening the tubes must imperatively preserve those fine structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Examples of possible fine-scale laminations in terrestrial environments (left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' seasonal varves from Lake Belau, Northern Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Dörfler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' right Microbially-Induced Sedimentary Structures-MISS in the middle neoproterozoic Chuar Group, Grand Canyon, Arizona;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Bohacs and Junium 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' (2) Some critical measurements are sensitive to contamination either from the tube, the apparatus used for cutting the tubes, or surrounding contaminants present in the isolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Organic matter is of particular concern given the high stakes involved in any claim for the presence of any form of biotic or prebiotic chemistry on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Inorganic trace element isotopes may provide dates on when Mars was habitable, and these are also prone to contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Beginning in 2022, an engineering team was tasked with developing the processes needed to open the sample tubes and to extract the solid and gaseous samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The engineering team was asked to develop engineering priorities associated with this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Two science teams were asked to develop parallel science priorities: A group we call the “Gas Team” evaluated the priorities related to the science associated with all returned gaseous sample, and a second group called the “Rock Team” (the authors of this report) evaluated the priorities associated with solid materials contained within the sample tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Both the "Gas Team" and "Rock Team" work under the oversight of a third committee, the Mars Campaign Science Group (MCSG1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The solid samples returned from the martian surface are certain to include sedimentary rocks (most important for the search for biosignatures), igneous rocks, and regolith, and they may also include other kinds of rocks, such as hydrothermal rocks, or impact breccia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The samples will be the basis for answering the main scientific questions of Mars Sample Return (iMOST, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The rock samples at Mars will all have been collected from various outcrops (or perhaps very large blocks of coherent rock).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' However, at least some of the rocks are relatively weak (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' a low compressive strength), and are vulnerable to fracturing during drilling and during several dynamic events associated with spacecraft operations during the return phase (most importantly, at Earth landing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' It is anticipated that the mechanical state of each sample, as received in the laboratory on Earth, will be assessed by a method like computer tomography (CT) scanning prior to opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The decision on how to open each sample tube can therefore be based on geological data from the field (collected by the M2020 science team), tests done on analogue samples, as well as the penetrative imaging data obtained on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The engineering team has proposed a 2-phase process for opening the sample tubes: First, puncture the tube in a way that will allow any gas present to be extracted and captured, then second, cut the metal of the tube in a way that would allow the solid materials to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Regarding cutting the metal of the tubes, three primary mechanisms have been proposed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' 2): • A single radial cut to the end of the tube, so that the sample could be tipped out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' • A radial cut at each end of the tube, which would enable the sample to be pushed out from one end 9 belowtopof core segment 10 12 (cm) 13 14 151cm 4 • Two radial cuts and two longitudinal cuts, to reveal the whole sample during cutting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' An option frequently used on Earth to access core samples, for example used with deep sea drill cores, is to cut the core tube and the core together with something like a band saw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' This is not an option for samples returned from Mars as this would have the effect of driving contamination from both the metallic core tube and band saw into the interior of the rock core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Proposed protocols for opening the sample tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Drawings courtesy of Oscar Rendon Perez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' In the one radial cut approach, a sharp hard metal wheel shears through the tube by slowly rotating and tightening it around the tube (bottom panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The sample is extracted from the tube by inclining it and controlling the rate of descent with a piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The second approach involves doing a second cut to push the sample outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' A virtue of this approach is that it allows for a more controlled extraction, and it minimizes the risk of the sample getting jammed in the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Both options 1 and 2 involve the sample sliding out of the tube and incur the risk of losing the chemical and structural layering of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The third approach involves doing two longitudinal cuts on the side of the tube to expose the whole sample within the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=" It is least likely to disturb the physical integrity of the sample, which stays in place in the tube, but it involves cutting the tube along its length through a white alumina coating (deposited on the tubes to reduce their heat absorption while seating on Mars' surface) possibly using a circular blade (bottom panel;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The chance of contamination is higher with this third approach as more tube manipulations are involved, more tube material is cut, and the setup to remove or cut the alumina coating will be more involved than the wheel cutter used in approaches 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Approach: The issue of how to open the tubes was discussed by the team over two telecons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Presentations by engineers Oscar Rendon Perez and Paolo Younse were delivered to explain the design of the tubes and different options for opening them (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The Rock Sample Team concluded there are three main considerations: • Need to minimise (and have knowledge of) contamination • Need to preserve stratigraphy and other textural relationships • Need to maximise the amount of sample material that ends up in a scientifically useful state from the tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=" For some samples like the detrital sediments or the regolith sample, each BUEHLER DIAMOND WAFERING BID BUEHILER 5 small grain may provide a unique record of Mars' surface history, so dust adhering to the tube surface should be recovered to the greatest extent possible." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' However, such dust will likely represent a small fraction of the total mass and its retrieval could be done later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Or it could be used for quickly surveying the petrography and mineralogy of the core as part of a preliminary examination phase as this material will be of lesser value for other tasks and could be sterilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Minimal cutting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', a single radial cut) was considered optimal to minimise potential contamination of trace elements, especially metals, and organic material from the tubes and cutting tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The structural integrity of the sample would, however, be best preserved with radial and longitudinal cuts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' this is considered especially important for sedimentary rocks that may be friable but contain internal stratigraphic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The yield may be maximised by at least two radial cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' These considerations may conflict with each other and the approach to be used will depend on the exact nature of each returned sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Magnetic contamination should also be minimized during cutting operation and sample handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The preferred opening strategies are summarized in Table 1, which ponders each criterion (structure integrity, chemical integrity, and yield) for three categories of samples (consolidated rocks, friable rocks, and loose regolith).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' We summarize the Rock Team recommendations at the bottom of each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The rationale for each entry is summarized below: Consolidated rocks (example microgabbro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' To minimize the risk of contamination, one radial cut is preferred as cutting by shearing with a hard metal solid wheel will generate little dust, cause little heating, involve no use of fluid, and involve the least amount of tube material of all considered options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' To get the sample out of the tube, putting it on a vertical incline and lowering the sample in a controlled manner with a piston would preserve the structural integrity of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' One radial cut is likely to preserve the structural integrity of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The cutting wheel will create a metal lip that will protrude in the tube, so provision should be ready to straighten that lip so that the sample can be extracted without rubbing against the lip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' With a consolidated sample, there is however a concern that jamming could occur, as a fragment might be trapped in compression between the solid core and the tube wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' A second cut might be needed to push/pull the sample from the other side and free it from such entrapment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Fine dust adhering to the inner tube surface might be difficult to retrieve with a single radial cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' A second radial cut would allow one to get the fine dust out by pushing it out with an appropriate instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The Rock Team favours 1 radial cut, with 2 radial cuts possibly needed for sample retrieval in case of jamming and to recover fine dust adhering to the interior tube surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Friable rocks (example detrital sediments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' These rocks are the ones for which preserving the stratigraphy is of upmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The rationale is the same as with consolidated rocks that a single radial cut would be preferred from the point of avoiding contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' To extract the sample, a single radial cut might be sufficient as the less consolidated nature of those rocks means that they are less likely to be hard jammed in the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' A possible approach would be to put place a piston against the sample on the opening side with the tube horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The sample tube and piston would then be rotated to a vertical position, and the piston would be lowered in a controlled manner to transfer the sample core in a transparent sample holder (quartz or sapphire) with predesigned longitudinal openings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The reason to transfer the sample vertically is to minimize shear on the tube surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' After vertical transfer of the sample from the tube to the holder, the holder would be rotated back to horizontal to be then opened, giving access to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Alternatively, it might be possible to 2 radial cuts, and one piston to push the sample out in a slightly inclined orientation and another piston at the open side against the sample to prevent collapse, so the sample keeps its integrity but we can avoid the longitudinal cuts to avoid more risk of 6 contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' If too friable, the sample could be gently pushed this way into a transparent sample holder with predesigned longitudinal openings, allowing visible inspection of the enclosed protected sample Letting the sample slide out from one side incurs the risk however that rock fragments will be moved out of sequence, that the sample will disaggregate, and that important chemical features be smeared throughout the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The latter point could include, for instance, organic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' If a layer is highly enriched in organics, sliding the whole sample along the sides may smear the signature throughout the entire core surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' For preserving the stratigraphy, it may therefore be advantageous to make 2 radial cuts and 2 longitudinal cuts to access the core without disturbing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The constraints on fine dust recovery are the same as with other sample types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' There is no stratigraphic information to preserve in that sample and little risk of jamming, so a single radial cut is preferred as this minimizes the risk of contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The fine dust in the sample may come from afar and each grain will likely tell a story, so complete recovery of dust adhering to the tube inner surface is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Preferred opening strategies depending on rock cohesiveness and criteria considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The Rock Sample Team finds that a single approach will not be appropriate for all the rock samples returned, but instead a flexible and bespoke approach will be needed for each sample tube opening, with all three of the above options available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' As a general principle, minimal cutting is favoured as this will also minimise potential contamination issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' an overriding consideration is that Consolidated rocks Example: microgabbro Friable rocks Example: detrital sediments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='igneous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='cumulate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='rocks ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='longitudinal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='cuts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='OR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='radial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='cuts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='FINDING: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='There is not one single approach for opening the sample tubes that will work sufficiently well for all MSR rock samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Multiple options need to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' 7 the structural integrity of the rock sample is key to understanding its petrology, and this should remain intact, even if this requires more processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' For regolith samples, a single radial cut followed by tipping out the grains is likely to be appropriate, since this will minimise contamination and there is no need to preserve spatial relationships within the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' For well consolidated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', some igneous rock) samples, a radial cut perhaps followed by a second radial cut may be required to extract the sample completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' For sedimentary rocks, and any friable igneous rocks, the decision is more complex because a longitudinal cut may be necessary to observe and preserve structural relationships, but this must be weighed against potentially contributing more contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' One possible solution to test for sedimentary samples could be to make one or two radial cuts, then push the sample or let it slide down while keeping its stratigraphy in place (possibly with high inclination to minimize shear along tube surface, with a sliding stopper against the sample to control the sliding rate) into another tube with a closed longitudinal aperture that allows longitudinal opening later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' The physical state of each core (consolidated or friable) will not be known for certain until the samples are bought back to Earth, where CT-scanning will reveal the fine structure of the samples and guide the strategy that adopted for tube opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Future Work: The team suggests areas which require more work prior to sample return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' These include: • Investigate how/whether analogue sedimentary samples and aqueously altered cumulate rocks can be removed in a manner that preserves their structural integrity with only one radial cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' • Investigate ways to efficiently remove the fines left behind after core extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' • Impurities in all tube materials, coatings, and opening contraption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', materials used in the saw) must be characterized with appropriate techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=', ICP-MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' We suggest that a task group be established to undertake an in-depth contaminant characterization campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' • Investigate if it is possible to remove the alumina coating without compromising the sample, and without causing damage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' by vibration) to the martian sample inside the core tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' • Investigate the degree to which the different cutting protocols can introduce contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' • Integrate these studies with CT and related scanning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' • Investigate how the cutting and related techniques can be performed in a Biological Hazard Level BSL4 environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' A concept that is not discussed in this report, but that has been considered elsewhere, is that the opportunity exists to do penetrative imaging/mineralogical characterization of the sample-bearing Mars sample tubes once they make it to Earth, so that we can obtain data on the mechanical state of each sample as received prior to tube opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' This eliminates the need to make guesses based on pre-sampling field data, or accelerations measured by the return spacecraft, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' That imaging data will give us the opportunity to help make decisions on how to open each tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' We know that for samples with different kinds of mechanical integrity, different tube-opening strategies may be required to avoid the risk of damage that unnecessarily affects the scientific usefulness of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' A component of the technology program is needed to develop the datasets for what happens when tubes containing samples with different degrees of mechanical integrity are opened by each of the three methods described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' This will become the basis for future decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' We also need data 8 on the real contamination implications of making the horizontal cuts, and what kind of science is affected by such contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' References.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content=' Astrobiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='Jun 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='S-1-S- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='http://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='1089/ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'} +page_content='2021.' 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b/MNE3T4oBgHgl3EQfBAmd/content/tmp_files/2301.04263v1.pdf.txt @@ -0,0 +1,1668 @@ +arXiv:2301.04263v1 [math.AP] 11 Jan 2023 +Existence of solutions to fractional semilinear +parabolic equations in Besov-Morrey spaces +Erbol Zhanpeisov∗ +Okinawa Institute of Science and Technology +1919-1 Tancha, Onna-son, Kunigami-gun +Okinawa, Japan 904-0495 +Abstract +In this paper, we establish the existence of solutions to fractional semilinear +parabolic equations in Besov-Morrey spaces for a large class of initial data +including distributions other than Radon measures. We also obtain sufficient +conditions for the existence of solutions to viscous Hamilton-Jacobi equations. +1 +Introduction and main results +Consider a semilinear parabolic equation +� +∂tu + (−∆) +θ +2 u = |u|γ−1u, +x ∈ RN, t ∈ (0, T), +u(x, 0) = ϕ(x), +x ∈ RN +(1.1) +and a viscous Hamilton-Jacobi equation +� +∂tu + (−∆) +θ +2u = |∇u|γ, +x ∈ RN, t ∈ (0, T), +u(x, 0) = ϕ(x), +x ∈ RN, +(1.2) +where γ > 1, N ≥ 1, T > 0 and θ > 0 (resp. θ > 1) for problem (1.1) (resp. +problem (1.2)). The purpose of this paper is to obtain sufficient conditions for the +existence of solutions to the Cauchy problem (1.1) and (1.2) for a large class of +initial data by introducing inhomogeneous Besov-Morrey spaces. This enables us to +take distributions other than Radon measures as initial data. +Let us consider the Cauchy problem for the semilinear parabolic equation (1.1) +with θ > 0 and γ > 1. The solvability of problem (1.1) has been studied in many +papers, see e.g., [3, 7, 9–17, 19–21, 23–27]. (See also the monograph [22].) Among +2010 AMS subject classification. 35K58,35K25 +∗E-mail: erbol.zhanpeisov@oist.jp +1 + +others, Ishige, Kawakami, and Okabe [17] developed the arguments in [16] and ob- +tained sufficient conditions for the existence of solutions to problem (1.1) for general +θ > 0. As corollaries of their main results, they proved the following properties: +(a) Let 1 < γ < 1 + θ/N. Then problem (1.1) possesses a local-in-time solution if +sup +x∈RN ∥ϕ∥L1(B(x,1)) < ∞; +(b) Let γ = 1 + θ/N. Then there exists c > 0 such that, if +|ϕ(x)| ≤ c|x|−N +����log +� +e + 1 +|x| +����� +− N +θ −1 +, +x ∈ RN, +then probolem (1.1) possesses a local-in-time solution; +(c) Let γ > 1 + θ/N. Then there exists c > 0 such that, if +|ϕ(x)| ≤ c|x|− +θ +γ−1 , +x ∈ RN, +then probolem (1.1) possesses a local-in-time solution. +In the case of either 0 < θ ≤ 2 or θ ∈ {4, 6, . . . }, it is shown in [13] and [16] that +sufficient conditions in (b) and (c) are sharp. More precisely, there exists c′ > 0 +such that, if +ϕ(x) ≥ + + + + + + + +c′|x|−N +����log +� +e + 1 +|x| +����� +− N +θ −1 +if +γ = 1 + θ +N , +c′|x|− +θ +γ−1 +if +γ > 1 + θ +N , +x ∈ B(0, 1), +then problem (1.1) possesses no local-in-time nonnegative solutions. +On the other hand, in the case of (a), distributions other than Radon measures +such as the derivative of the Dirac distribution can be considered as the initial data +to problem (1.1) with θ = 2. For instance, problem (1.1) with θ = 2 is well-posed +in certain negative order inhomogeneous Besov-Morrey spaces Ns +p,q,r(RN), see [19] +and Remark 1.1. The arguments in [19] are based on delicate decay estimates of +the heat kernel in inhomogeneous Besov-Morrey spaces and the power nonlinearity +of the semilinear parabolic equation. It seems difficult to apply their arguments +directly to the Cauchy problem (1.1) and problem (1.2), in particular, the case +of fractional diffusion θ ̸= 2 and the case of the nonlinearity depending on ∇u. +In this paper, we develop the arguments in [19] and prove the unique existence +of the solution to problem (1.1) (resp. +problem (1.2)) in inhomogeneous Besov- +Morrey spaces Ns +p,q,r(RN) for general θ > 0 (resp. θ > 1). This enables us to take +2 + +distributions other than Radon measures as initial data and the results in the case +(a) is extended for more general initial data. +For viscous Hamilton-Jacobi equations (1.2), the solvability has been studied +in [1, 2, 4, 8, 18]. +Using the majorant kernel, Ishige, Kawakami, and Okabe [17] +obtained the same results for problem (1.2) as for problem (1.1). That is, when +1 < γ < 1 + (θ + 1)/(N + 1), there exists a solution to problem (1.2) if the initial +measure satisfies +sup +x∈RN ∥ϕ∥L1(B(x,1)) < ∞. +We extend these results to more general initial data. See Remark 1.2 for more details +on the relation to previous studies. +We recall the definition of local Morrey spaces and introduce inhomogeneous +Besov-Morrey spaces. +Definition 1.1 (local Morrey spaces) Let 1 ≤ q ≤ p < ∞. The local Morrey +space Mp +q (RN) is defined to be the set of measurable functions u in RN such that +∥u |Mp +q ∥ := +sup +x∈RN, 0<ρ≤1 +ρ +N +p − N +q ∥u |Lq(B(x, ρ))∥ < ∞. +The local measure space of the Morrey type Mp(RN) is defined as the sets of the +Radon measures µ on RN such that +∥µ|Mp∥ := +sup +x∈RN,0<ρ≤1 +ρ +N +p −N|µ|(B(x, ρ)) < ∞, +where |µ| denotes the total variation of the measure µ. +Let ζ(t) be a smooth function on [0, ∞) such that 0 ≤ ζ(t) ≤ 1, ζ(t) ≡ 1 for +t ≤ +3 +2 and supp ζ ⊂ [0, 5 +3). For j ∈ Z, put ϕj(ξ) := ζ(2−j|ξ|) − ζ(21−j|ξ|) and +ϕ(0)(ξ) := ζ(|ξ|). Then we have ϕj(ξ), ϕ(0)(ξ) ∈ C∞ +0 (RN) and +ϕ(0)(ξ) + +∞ +� +j=1 +ϕj(ξ) = 1 +for any +ξ ∈ RN. +Definition 1.2 (inhomogeneous Besov-Morrey space) Let 1 ≤ q ≤ p < ∞, +1 ≤ r ≤ ∞ and s ∈ R. The local Besov-Morrey space is defined as the sets of +distributions u ∈ S′(RN) such that F −1ϕ(0)(ξ)Fu ∈ Mp +q and F −1ϕj(ξ)Fu ∈ Mp +q for +every positive integer j, and that +∥u|Ns +p,q,r∥ := ∥F −1ϕ(0)(ξ)Fu|Mp +q ∥ + ∥{2sj∥F −1ϕj(ξ)Fu|Mp +q ∥}∞ +j=1|ℓr∥ < ∞, +where F denotes the Fourier transform on RN. +3 + +For every t > 0 and every u ∈ S′(RN), put S(t)u := F −1 exp(−t|ξ|θ)Fu. +We +formulate a solution to problem (1.1) and (1.2) . +Definition 1.3 Let T > 0 and ϕ ∈ Ns +p,q,r for some s ∈ R, 1 ≤ q ≤ p < ∞ and +1 ≤ r ≤ ∞. We say that u is a solution to problem (1.1) in RN × [0, T) if +u ∈ BC(RN × (τ, T)) +for τ ∈ (0, T), and u satisfies +u(x, t) = [S(t)ϕ](x) + +� t +0 +[S(t − τ)|u(·, τ)|γ−1u(·, τ)](x) dτ +for (x, t) ∈ RN × (0, T). +Definition 1.4 Let T > 0 and ϕ ∈ Ns +p,q,r for some s ∈ R, 1 ≤ q ≤ p < ∞ and +1 ≤ r ≤ ∞. We say that u is a solution to problem (1.2) in RN × [0, T) if +u , ∇u ∈ BC(RN × (τ, T)) +for τ ∈ (0, T), and u satisfies +u(x, t) = [S(t)ϕ](x) + +� t +0 +[S(t − τ)|∇u(·, τ)|γ](x) dτ +for (x, t) ∈ RN × (0, T). +We are ready to state the main results of this paper. +Theorem 1.1 Let γ > 1, γ ≤ q ≤ p < ∞, −θ/γ < s < 0 and s ≥ N/p − θ/(γ − 1). +Then there exist δ > 0 and M > 0 such that for every ϕ(x) ∈ Ns +p,q,∞ satisfying +lim sup +j→∞ +2sj∥F −1ϕjFϕ|Mp +q ∥ < δ, +(1.3) +problem (1.1) possesses the unique solution u(x, t) on RN × [0, T) for some T > 0 +with a bound sup0 1+θ/N and take p as max{γ, p0} < p < p0γ, then by Proposition 2.1 +and Proposition 2.2, we have +|x|− +θ +γ−1 ∈ Mp0 +p0γ/p,∞ ⊂ N0 +p0,p0γ/p,∞ ⊂ NN/p−θ/(γ−1) +p,γ,∞ +. +4 + +Since the assumption of Theorem 1.1 is satisfied with NN/p−θ/(γ−1) +p,γ,∞ +for above +p, we see by (1.3) that there exists c > 0 such that, if +|ϕ(x)| ≤ c|x|− +θ +γ−1, +x ∈ RN, +then probolem (1.1) possesses a local-in-time solution. This result is consistent +with that of [17] and thus the condition (1.3) is necessary. +• Let γ = 1 + θ/N. Then by Proposition 2.1 and Proposition 2.2, for any p > 1 +we have +Lp = Mp +p ⊂ N0 +p,p,∞ ⊂ N−N/p+N/γ +γ,γ,∞ +. +Since the assumption of Theorem 1.1 is satisfied with N−N/p+N/γ +γ,γ,∞ +, we see that +if ϕ ∈ Lp with p > 1, then probolem (1.1) possesses a local-in-time solution. +Note that in the case of θ = 2, problem (1.1) is not well-posed in L1 (See for +example, [5,6]). +• Let 1 < γ < 1 + θ/N. Then by Proposition 2.1 and Proposition 2.2, we have +δ(x) ∈ M1 ⊂ N0 +1,1,∞ ⊂ N−N+N/γ +γ,γ,∞ +. +Since the assumption of Theorem 1.1 is satisfied with N−N+N/γ +γ,γ,∞ +, we see that if +ϕ is a Radon measure, then probolem (1.1) possesses a local-in-time solution, +which is consistent with the result of [17]. Furthermore, since +∂|α|δ(x) ∈ N−N+N/γ−|α| +γ,γ,∞ +, +we see that probolem (1.1) possesses a local-in-time solution for ϕ = ∂[θ]δ and +γ < +N+θ +N+[θ] if θ is not an integer, and for ϕ = ∂θ−1δ and γ < +N+θ +N+θ−1 if θ is an +integer. +Theorem 1.2 Let 1 < γ < θ, γ ≤ q ≤ p < ∞, p > N(γ−1)/(θ−1), 1−θ/γ < s < 0 +and s ≥ N/p + (γ − θ)/(γ − 1). Then there exist δ > 0 and M > 0 such that for +every ϕ(x) ∈ Ns +p,q,∞ satisfying +lim sup +j→∞ +2sj∥F −1ϕjFϕ|Mp +q ∥ < δ, +problem (1.2) possesses the unique solution u(x, t) on RN × [0, T) for some T > 0 +with a bound sup0 0 such that, if +|ϕ(x)| ≤ c|x|− θ−γ +γ−1 , +x ∈ RN, +then probolem (1.2) possesses a local-in-time solution. This result is consistent +with that of [17]. +• Let γ = (N + θ)/(N + 1). Then by Proposition 2.1 and Proposition 2.2, for +any p > 1 we have +Lp = Mp +p ⊂ N0 +p,p,∞ ⊂ N−N/p+N/γ +γ,γ,∞ +. +Since the assumption of Theorem 1.2 is satisfied with N−N/p+N/γ +γ,γ,∞ +, we see that +if ϕ ∈ Lp with p > 1, then probolem (1.2) possesses a local-in-time solution. +• Let 1 < γ < (N + θ)/(N + 1). Then by Proposition 2.1 and Proposition 2.2, +we have +δ(x) ∈ M1 ⊂ N0 +1,1,∞ ⊂ N−N+N/γ +γ,γ,∞ +. +Since the assumption of Theorem 1.2 is satisfied with N−N+N/γ +γ,γ,∞ +, we see that if +ϕ is a Radon measure, then probolem (1.2) possesses a local-in-time solution. +This result is consistent with that of [17]. Furthermore, since +∂|α|δ(x) ∈ N−N+N/γ−|α| +γ,γ,∞ +, +we see that probolem (1.2) possesses a local-in-time solution for ϕ = ∂[θ]−1δ +and γ < +N+θ +N+[θ] if θ is not an integer, and for ϕ = ∂θ−2δ and γ < +N+θ +N+θ−1 if θ is +an integer. +We explain the idea of the proof of Theorem 1.1 and Theorem 1.2. Let S(t)u := +F −1 exp(−t|ξ|θ)Fu. By modifying the arguments in [19], we first prove the heat +kernel estimates of the fractional Laplacian in inhomogeneous Besov-Morrey spaces +and obtain the estimate +∥S(t)u|Nσ +p,q,1∥ ≤ C(1 + t(s−σ)/θ)∥u|Ns +p,q,∞∥, +∥∇S(t)u|Nσ +p,q,1∥ ≤ C(1 + t(s−σ−1)/θ)∥u|Ns +p,q,∞∥, +for t > 0 and σ > s. +Here, one of the main difficulties comes from the non- +smoothness of the function exp(−t|ξ|θ), see Lemma 2.2 and Remark 2.1. +6 + +Then we show that the approximate solutions converge in some Banach space +based on the local Morrey spaces with a bound near t = 0. +The rest of this paper is organized as follows. In Sections 2, we obtain the heat +kernel estimates of the fractional Laplacian in inhomogeneous Besov-Morrey spaces. +In section 3, we prove Theorem 1.1. In section 4, we prove Theorem 1.2. +2 +Preliminaries +In this section, we recall some preliminary facts about Besov-Morrey spaces and give +estimates of heat kernel of fractional Laplacian in these function spaces. +The following two propositions collect basic facts about Morrey spaces and Besov- +Morrey spaces. +Proposition 2.1 ( [19, Theorem 2.5]) Let 1 ≤ q ≤ p < ∞, r ∈ [1, ∞] and +s ∈ R. Then the following embeddings are continuous: +Ns +p,q,r ⊂ Bs−N/p +∞,r +, +(2.1) +Ns +p,q,r ⊂ Ns−N(1−l)/p +p/l,q/l,r +for any +l ∈ (0, 1). +(2.2) +Proposition 2.2 ( [19, Proposition 2.11]) Let 1 ≤ q ≤ p < ∞. Then the fol- +lowing embeddings are continuous: +N0 +p,q,1 ⊂ Mp +q ⊂ N0 +p,q,∞, +(2.3) +Mp ⊂ N0 +p,1,∞. +We modify the arguments in [19, Theorem 2.9 (2)] and prepare the following two +lemmas for the estimates of heat kernel of fractional Laplacian in inhomogeneous +Besov-Morrey spaces. Here, we denote by ⌊x⌋ the greatest integer less than or equal +to x ∈ R. +Lemma 2.1 Let m ∈ R, 1 ≤ q ≤ p < ∞ and P(ξ) ∈ C⌊N/2⌋+1(RN \ {0}). Assume +that there is A > 0 such that +���� +∂αP +∂ξα (ξ) +���� ≤ A|ξ|m−|α| +for all α ∈ (N ∪ {0})N with |α| ≤ ⌊N/2⌋ + 1 and for all ξ ̸= 0. Then the multiplier +operator P(D)u := F −1P(ξ)Fu satisfies the estimate +��F −1ϕjF(P(D)u)|Mp +q +�� ≤ CA2mj ��F −1ϕjFu|Mp +q +�� +for every positive integer j and u ∈ S′(RN) such that F −1ϕj(ξ)Fu ∈ Mp +q , where +C > 0 is a constant independent of j, A, and u. +7 + +Proof. Put Φj := ϕj−1 + ϕj + ϕj+1 and K(x) := F −1Φj(ξ)P(ξ) for j ∈ Z. Note +that supp ϕj(ξ) ⊂ {ξ ∈ RN; 2j+1/3 ≤ |ξ| ≤ 2j+1} and Φj ≡ 1 on supp ϕj(ξ). +Putting also N0 := ⌊N/2⌋ + 1, we have +∥K|L1(RN)∥ = +� +|x|≤2−j |K(x)| dx + +� +|x|≥2−j |K(x)| dx +≤ +�� +|x|≤2−j dx +�1/2 �� +|x|≤2−j |K(x)|2 dx +�1/2 ++ +�� +|x|≥2−j |x|−2N0 dx +�1/2 �� +|x|≥2−j |x|2N0|K(x)|2 dx +�1/2 +≤ C + +2−Nj/2∥K(x)|L2(RN)∥ + 2(N0−N/2)j � +|α|=N0 +∥xαK(x)|L2(RN)∥ + + += C + +2−Nj/2∥Φj(ξ)P(ξ)|L2(RN)∥ + 2(N0−N/2)j � +|α|=N0 +���� +∂|α| +∂ξα(Φj(ξ)P(ξ))|L2(RN) +���� + + +≤ C(2−Nj/22(m+N/2)jA + 2(N0−N/2)j2(m−N0+N/2)jA) = C2mjA +for some constant C > 0, depending on N, m, ∥ζ|BCN0(R)∥, but not on j and A. +Since F −1ϕjF(P(D)u) = K ∗ (F −1ϕjFu), we see by [19, Lemma 1.8] that +∥F −1ϕjF(P(D)u)|Mp +q ∥ ≤ CA2mj∥F −1ϕjFu|Mp +q ∥ +for every positive integer j, and the proof is complete. ✷ +Lemma 2.2 Let m > 0, 1 ≤ q ≤ p < ∞ and P(ξ) ∈ C⌊N/2⌋+1(RN \ {0}). Assume +that there is A > 0 such that +���� +∂αP +∂ξα (ξ) +���� ≤ A|ξ|m−|α| +for all α ∈ (N ∪ {0})N with |α| ≤ ⌊N/2⌋ + 1 and for all ξ ∈ B(0, 4) \ {0}. Then the +multiplier operator P(D)u := F −1P(ξ)Fu satisfies the estimate +∥F −1ϕ(0)F(P(D)u)|Mp +q ∥ ≤ CA∥F −1ϕ(0)Fu|Mp +q ∥ +for every u ∈ S′(RN) such that F −1ϕ(0)(ξ)Fu ∈ Mp +q , where C > 0 is a constant +independent of A and u. +Proof. Put Kj(x) := F −1ϕj(ξ)P(ξ) and Φ(0) := ϕ(0) + ϕ1. In the same way as in +Lemma 2.1, we have +∥F −1Φ(0)(ξ)P(ξ)|L1(RN)∥ ≤ +1 +� +j=−∞ +∥Kj|L1(RN)∥ +≤ +1 +� +j=−∞ +C2mjA ≤ CA +8 + +with some constant C > 0 independent of A. This implies in the same way as in +Lemma 2.1 +∥F −1ϕ(0)F(P(D)u)|Mp +q ∥ ≤ CA∥F −1ϕ(0)Fu|Mp +q ∥, +and the proof is complete. ✷ +Remark 2.1 Note that we do not assume the smoothness of P(ξ) at ξ = 0, which +is useful for the estimates of the derivative of heat kernel of fractional Laplacian +since P(ξ) = exp(−t|ξ|θ) is not smooth at ξ = 0 in general. In this respect, we +improved [19, Theorem 2.9 (2)] where the smoothness at ξ = 0 is needed. +In the following theorem, we obtain estimates of heat kernel of fractional Laplacian +in inhomogeneous Besov-Morrey spaces. +Theorem 2.1 Let s ≤ σ, 1 ≤ q ≤ p < ∞ and r ∈ [1, ∞]. Then there exists C > 0 +such that the estimate +∥S(t)u|Nσ +p,q,r∥ ≤ C(1 + t(s−σ)/θ)∥u|Ns +p,q,r∥ +for +t > 0 +(2.4) +holds. Furthermore, if s < σ, the estimate +∥S(t)u|Nσ +p,q,1∥ ≤ C(1 + t(s−σ)/θ)∥u|Ns +p,q,∞∥ +for +t > 0 +(2.5) +holds. +Proof. By induction we see that for every α ∈ NN there exist homogeneous poly- +nomials Pα,k(ξ) of degree |α| for k = 1, 2, . . . , |α| such that for ξ ̸= 0 +∂|α| exp(−t|ξ|θ) +∂ξα += exp(−t|ξ|θ)|ξ|−2|α| +|α| +� +k=1 +Pα,k(ξ)tk|ξ|kθ. +(2.6) +We have for m = s − σ +|ξ|−m+|α|∂|α| exp(−t|ξ|θ) +∂ξα +≤ Ct +m +θ exp(−t|ξ|θ) +|α| +� +k=1 +(t +1 +θ |ξ|)kθ−m +≤ Cαt +m +θ . +This together with Lemma 2.1 implies +∥F −1ϕj(ξ)F(S(t)u)|Mp +q ∥ ≤ Ct +m +θ 2mj∥F −1ϕj(ξ)Fu|Mp +q ∥ +(2.7) +for every positive integer and every t > 0. On the other hand, since +∥F −1ϕ(0)(ξ)F(S(t)u)|Mp +q ∥ +≤ ∥F −1Φ(0) ∗ F −1 exp(−t|ξ|θ)|L1(RN)|∥∥F −1ϕ(0)(ξ)Fu|Mp +q ∥ +≤ C∥F −1ϕ(0)(ξ)Fu|Mp +q ∥, +9 + +where Φ(0) is as in Lemma 2.2. This together with (2.7) implies the inequality (2.4). +The inequality (2.5) follows exactly in the same way as in [19, Theorem 3.1] from +the inequality (2.4), and the proof is complete. ✷ +In the following lemma, we obtain another estimate of the heat kernel of frac- +tional Laplacian by using the smallness condition on the initial data. +Lemma 2.3 Let 1 ≤ q ≤ p < ∞ and s < σ. Then there exists A > 0 such that, for +every u ∈ Ns +p,q,∞ and every B > 0, satisfying +A lim sup +j→∞ +2sj∥F −1ϕjFu|Mp +q ∥ < B, +there exists T > 0 such that +sup +0 0 such that +lim sup +j→∞ +2sj∥F −1ϕjFu|Mp +q ∥ < δ < B/A, +then for some m ∈ N, the estimate +2sj∥F −1ϕjFu|Mp +q ∥ ≤ δ < B/A +holds for every j ≥ m. Put u1 = F −1ϕ(0)(2−m·)Fu and u2 = u − u1. Since +supp ϕ(0)(2−mξ) ⊂ +� +ξ ∈ RN; |ξ| ≤ 5 +32m +� +, +supp ϕj(ξ) ⊂ +� +ξ ∈ RN; 2j+1 +3 +≤ |ξ| ≤ 2j+1 +� +, +ϕ(0)(2−mξ) ≡ 1 +on +{ξ ∈ RN; |ξ| ≤ 3 · 2m−1}, +we have +F −1ϕjFu1 = + + + + + +F −1ϕjFu +for +j ≤ m − 1, +F −1(ϕm−1 + ϕm)ϕjFu +for +j = m, m + 1, +0 +for +j ≥ m + 2, +and +F −1ϕjFu2 = + + + + + +0 +for +j ≤ m − 1, +F −1(ϕm+1 + ϕm+2)ϕjFu +for +j = m, m + 1, +F −1ϕjFu +for +j ≥ m + 2. +10 + +It follows from the the definition of the constant C1 and the fact ∥F −1ϕj|L1∥ = +∥F −1ϕ0|L1∥ that ∥u2|Ns +p,q,∞∥ ≤ C1δ. Therefore, we have +t(σ−s)/θ∥S(t)u2|Nσ +p,q,1∥ ≤ C0(1 + t(σ−s)/θ)∥u2|Ns +p,q,∞∥ +≤ C0C1δ(1 + T (σ−s)/θ) = Aδ(1 + T (σ−s)/θ) < Aδ + B +2 +(2.8) +for every t ∈ (0, T], by taking T > 0 sufficiently small. On the other hand, since +u1 ∈ N(σ+s)/2 +p,q,∞ +, we have the estimate +t(σ−s)/θ∥S(t)u1|Nσ +p,q,1∥ ≤ C0(t(σ−s)/2θ) + t(σ−s)/θ))∥u1|N(s+σ)/2 +p,q,∞ +∥ +≤ C0T (σ−s)/2θ(1 + T (σ−s)/2θ)∥u1|N(s+σ)/2 +p,q,∞ +∥ < B − Aδ +2 +(2.9) +for every t ∈ (0, T], by taking T > 0 sufficiently small. We obtain the conclusion +from (2.8) and (2.9), and the proof is complete. ✷ +3 +Proof of Theorem 1.1. +In this section, we prove Theorem 1.1 by using Theorem 2.1. Let XT denote the set +of Lebesgue measurable functions u(x, t) on RN × (0, T) such that +∥u|XT∥ := sup +0 1, T ≤ 1, γ ≤ q ≤ p < ∞, −θ/γ < s < 0 and s ≥ +N/p − θ/(γ − 1). Then there exists C2 > 0 independent of T such that +∥un+1|XT∥ ≤ ∥u0|XT∥ + C2∥un|XT∥γ +for n = 0, 1, . . .. +11 + +Proof. By (2.2), (2.3), (2.5) and (3.1), we see that +∥un+1(·, t) − u0(·, t)|Mp +q ∥ ≤ C∥un+1(·, t) − u0(·, t)|N0 +p,q,1∥ +≤ C +� t +0 +∥S(t − τ)|un(·, τ)|γ−1un(·, τ)|N0 +p,q,1∥ dτ +≤ C +� t +0 +∥S(t − τ)|un(·, τ)|γ−1un(·, τ)|NN(γ−1)/p +p/γ,q/γ,1 ∥ dτ +≤ C +� t +0 +{1 + (t − τ)−N(γ−1)/pθ}∥|un(·, τ)|γ|N0 +p/γ,q/γ,∞∥ dτ +≤ C +� t +0 +(t − τ)−N(γ−1)/pθ∥|un(·, τ)|γ|Mp/γ +q/γ ∥ dτ +≤ C +� t +0 +(t − τ)−N(γ−1)/pθ∥un(·, τ)|Mp +q ∥γ dτ +≤ C∥un|XT∥γ +� t +0 +(t − τ)−N(γ−1)/pθτ sγ/θ dτ +≤ Ct−N(γ−1)/pθ+sγ/θ+1∥un|XT∥γ. +Therefore, we have +∥un+1 − u0|XT∥ ≤ Ct1+(γ−1)(s/θ−N/pθ)∥un|XT∥γ +≤ C∥un|XT∥γ +for T ≤ 1, and the proof is complete. ✷ +Lemma 3.2 Let γ > 1, γ ≤ q ≤ p < ∞, −θ/γ < s < 0 and s ≥ N/p − +θ/(γ − 1). Then there exists C3 > 0 such that for every ϕ(x) ∈ Ns +p,q,∞ satisfy- +ing lim supj→∞ 2sj∥F −1ϕjFϕ|Mp +q ∥ < δ for some δ > 0, we can choose a positive +number T ≤ 1 so small that the inequality ∥u0|XT∥ < C3δ holds. Furthermore, we +can choose δ so small that supn ∥un|XT∥ ≤ M for some M > 0. +Proof. By Lemma 2.3 with B = Aδ, we can take T ≤ 1 such that the estimate +sup +0 0. +For δ > 0 satisfying +2γC2Cγ +3 δγ−1 < 1, +we see by induction that +sup +n ∥un|XT∥ ≤ 2C3δ =: M, +and the proof is complete. ✷ +12 + +Lemma 3.3 Let γ > 1, γ ≤ q ≤ p < ∞, −θ/γ < s < 0 and s ≥ N/p − θ/(γ − 1). +Suppose that δ and T ≤ 1 are small enough so that the assertion of Lemma 3.2 +holds. Then there exists a positive constant C independent of T such that +∥un+2 − un+1|XT∥ ≤ CMγ−1∥un+1 − un|XT∥ +for n = 0, 1, . . .. +Proof. By (2.2), (2.3), (2.5) and (3.1), we see that +∥un+2(·, t) − un+1(·, t)|Mp +q ∥ ≤ C∥un+2(·, t) − un+1(·, t)|N0 +p,q,1∥ +≤ C +� t +0 +∥S(t − τ)(|un+1(·, τ)|γ−1un+1(·, τ) − |un(·, τ)|γ−1un(·, τ))|N0 +p,q,1∥ dτ +≤ C +� t +0 +∥S(t − τ)(|un+1(·, τ)|γ−1un+1(·, τ) − |un(·, τ)|γ−1un(·, τ))|NN(γ−1)/p +p/γ,q/γ,1 ∥ dτ +≤ C +� t +0 +(t − τ)− N(γ−1) +pθ +∥|un+1(·, τ)|γ−1un+1(·, τ) − |un(·, τ)|γ−1un(·, τ)|N0 +p/γ,q/γ,∞∥ dτ +≤ C +� t +0 +(t − τ)− N(γ−1) +pθ +∥|un+1(·, τ)|γ−1un+1(·, τ) − |un(·, τ)|γ−1un(·, τ)|Mp/γ +q/γ ∥ dτ +≤ C +� t +0 +(t − τ)− N(γ−1) +pθ +∥|un+1(·, τ) − un(·, τ)|(|un+1(·, τ)|γ−1 + |un(·, τ)|γ−1)|Mp/γ +q/γ ∥ dτ +≤ CMγ−1 +� t +0 +(t − τ)− N(γ−1) +pθ +∥un+1(·, τ) − un(·, τ)|Mp +q ∥ dτ +≤ CMγ−1∥un+1 − un|XT∥ +� t +0 +(t − τ)−N(γ−1)/pθτ sγ/θ dτ +≤ CMγ−1t−N(γ−1)/pθ+sγ/θ+1∥un+1 − un|XT∥. +We used here [19, Lemma 1.4]. Therefore, we have +∥un+2 − un+1|XT∥ ≤ CMγ−1t1+(γ−1)(s/θ−N/pθ)∥un+1 − un|XT∥ +≤ CMγ−1∥un+1 − un|XT∥, +and the proof is complete. ✷ +Proof of Theorem 1.1. +Take δ and T so small that +∥un+2 − un+1|XT∥ ≤ 1 +2∥un+1 − un|XT∥ +for n = 0, 1, . . ., and we see that un(x, t) converges in XT. Set u(x, t) as a limit of +un(x, t) in XT and we see that +u(x, t) := [S(t)ϕ](x) + +� t +0 +[S(t − τ)|u(·, τ)|γ−1u(·, τ)](x) dτ. +(3.2) +13 + +We next prove that u(x, t) ∈ L∞([ε, T] × RN) for every ε > 0. Let n be the +smallest integer greater than Nγ/θp. +Then we can take an increasing sequence +of positive numbers {pj}n +j=1 such that p1 = p, N/pj+1 > N/pj − θ/γ for every +j = 1, 2, · · · , n − 1 and N/pn < θ/γ. We also define {qj}n +j=1 and {sj}n +j=1 as q1 = q, +qj+1 = pj+1qj/pj, s1 = s and sj+1 = N/pj+1 − N/pj. +By the obtained result, we see that the solution u(x, t) belongs to the spaces +L∞ �� ε +2n, T +� +, Mp +q +� +⊂ L∞ �� ε +2n, T +� +, N0 +p1,q1,∞ +� +⊂ L∞ �� ε +2n, T +� +, Ns2 +p2,q2,∞ +� +. +Since γ ≤ q2 ≤ p2, −γ/θ < s2 < 0 and s2 ≥ N/p2 − θ/(γ − 1), we can apply the +obtained result to see u(x, t) ∈ L∞ �� 2ε +2n, T +� +, Mp2 +q2 +� +. In the same way, since +L∞ +�� jε +2n, T +� +, Mpj +qj +� +⊂ L∞ +�� jε +2n, T +� +, N0 +pj,qj,∞ +� +⊂ L∞ +�� jε +2n, T +� +, Nsj +pj+1,qj+1,∞ +� +, +where γ ≤ qj+1 ≤ pj+1, −γ/θ < sj+1 < 0 and sj+1 ≥ N/pj+1 − θ/(γ − 1), we +have u(x, t) ∈ L∞ �� +(j+1)ε +2n , T +� +, M +pj+1 +qj+1 +� +for j = 1, 2, · · · , n − 1. Therefore, we have +u(x, t) ∈ L∞ �� ε +2, T +� +, Mpn +qn +� +, where pn > Nγ/θ. It follows from (2.1) that +���� +� t +ε/2 +S(t − τ)|u(·, τ)|γ−1u(·, τ) dτ|L∞ +���� +≤ C +� t +ε/2 +��S(t − τ)|u(·, τ)|γ−1u(·, τ)|B0 +∞,1 +�� dτ +≤ C +� t +ε/2 +���S(t − τ)|u(·, τ)|γ−1u(·, τ)|NNγ/pn +pn/γ,qn/γ,1 +��� dτ +≤ C +� t +ε/2 +� +1 + (t − τ)−Nγ/θpn� ��|u(·, τ)|γ−1u(·, τ)|N0 +pn/γ,qn/γ,∞ +�� dτ +≤ C +� t +ε/2 +(t − τ)−Nγ/θpn +���|u(·, τ)|γ|Mpn/γ +qn/γ +��� dτ +≤ C +� t +ε/2 +(t − τ)−Nγ/θpn ��u(·, τ)|Mpn +qn +��γ dτ +≤ C +� +t − ε +2 +�1−Nγ/θpn +sup +ε/2≤τ≤t +��u(·, τ)|Mpn +qn +��γ dτ +≤ CT 1−Nγ/θpn +sup +ε/2≤τ≤t +��u(·, τ)|Mpn +qn +��γ dτ < ∞ +(3.3) +for ε/2 ≤ t ≤ T ≤ 1. On the other hand, we have +∥S(t − ε/2)u(·, ε/2)|L∞∥ ≤ C∥S(t − ε/2)u(·, ε/2)|B0 +∞,1∥ +≤ C∥S(t − ε/2)u(·, ε/2)|NN/p +p,q,1∥ +≤ C +� +1 + (t − ε/2)−N/θp� ��|u(·, ε/2)||Mp +q +�� +≤ C(ε/2)−N/θp ��|u(·, ε/2)||Mp +q +�� < ∞ +(3.4) +14 + +for ε ≤ t ≤ T ≤ 1. Since +u(x, t) = +� +S +� +t − ε +2 +� +u +� +·, ε +2 +�� +(x) + +� t +ε/2 +� +S(t − s)|u(·, τ)|γ−1u(·, τ) +� +(x) dτ, +this together with (3.3) and (3.4) implies that u(x, t) ∈ L∞([ε, T] × RN) for every +ε > 0. +Finally, we prove the uniqueness of the solution. +Assume that u(1)(x, t) and +u(2)(x, t) are solutions to (3.2) satisfying sup0≤t≤T t−s/θ∥u(j)(·, t)|Mp +q ∥ < ∞. +Let +u = u(1) − u(2) and h(t) = ∥u(·, t)|Mp +q ∥. Then exactly in the same way as in the +proof of Lemma 3.3, we have +sup +0 0 be small and consider the Banach +space +YT := {u(x, t) on (0, T) × RN : ∥u |YT∥ < ∞}, +where +∥u |YT∥ := sup +0 0 and every u ∈ S′, put Sj(t)u := F −1(iξj) exp(−t|ξ|θ)Fu for +1 ≤ j ≤ N. +As in Section 2, we prove the derivative estimate for S(t) in the +following theorem. +Theorem 4.1 Let s ≤ σ, 1 ≤ q ≤ p < ∞ and r ∈ [1, ∞]. Then there exists C > 0 +such that the estimate +∥Sj(t)u|Nσ +p,q,r∥ ≤ C(1 + t(s−σ−1)/θ)∥u|Ns +p,q,r∥ +for +t > 0 +(4.2) +holds. Furthermore, if s < σ, the estimate +∥Sj(t)u|Nσ +p,q,1∥ ≤ C(1 + t(s−σ−1)/θ)∥u|Ns +p,q,∞∥ +for +t > 0 +(4.3) +holds. +15 + +Proof. By (2.6) we see that for every α ∈ NN there exists a homogeneous poly- +nomial Pα,k(ξ) of degree |α| for k = 1, 2, . . . , |α| and Pα−ej,k(ξ) of degree |α| − 1 for +k = 1, 2, . . . , |α| − 1 such that for ξ ̸= 0 +∂|α|(iξj) exp(−t|ξ|θ) +∂ξα += iξj exp(−t|ξ|θ)|ξ|−2|α| +|α| +� +k=1 +Pα,k(ξ)tk|ξ|kθ ++ iαj exp(−t|ξ|θ)|ξ|−2|α|+2 +|α|−1 +� +k=1 +Pα−ej,k(ξ)tk|ξ|kθ. +We have for m = s − σ +|ξ|−m+|α|∂|α|(iξj) exp(−t|ξ|θ) +∂ξα +≤ Ct +m−1 +θ +exp(−t|ξ|θ) +|α| +� +k=1 +(t +1 +θ |ξ|)kθ−m+1 +≤ Cαt +m−1 +θ . +This together with Lemma 2.1 implies +∥F −1ϕj(ξ)F(Sj(t)u)|Mp +q ∥ ≤ Ct +m +θ 2mj∥F −1ϕj(ξ)Fu|Mp +q ∥ +(4.4) +for every positive integer and every t > 0. On the other hand, by (2.6) we have +���� +∂|α|(iξj) exp(−t|ξ|θ) +∂ξα +���� ≤ Cα|ξ|1−|α|. +for every ξ ∈ B(0, 4) \ {0}. This together with Lemma 2.2 implies +∥F −1ϕ(0)(ξ)F(S(t)u)|Mp +q ∥ ≤ C∥F −1ϕ(0)(ξ)Fu|Mp +q ∥. +(4.5) +The inequality (4.2) follows from (4.4) and (4.5). The inequality (4.3) follows exactly +in the same way as in [19, Theorem 3.1] from the inequality 4.2, and the proof is +complete. ✷ +Lemma 4.1 Let 1 ≤ q ≤ p < ∞ and s < σ. Then there exists A > 0 such that, for +every u ∈ Ns +p,q,∞ and every B > 0, satisfying +A lim sup +j→∞ +2sj∥F −1ϕjFu|Mp +q ∥ < B, +there exists T > 0 such that +sup +0 0, m ∈ N, u1 and u2 as in the +proof of Lemma 2.3. Then we have +t(σ−s+1)/θ∥Sj(t)u2|Nσ +p,q,1∥ ≤ C0(1 + t(σ−s+1)/θ)∥u2|Ns +p,q,∞∥ +≤ C0C1δ(1 + T (σ−s+1)/θ) = Aδ(1 + T (σ−s+1)/θ) < Aδ + B +2 +(4.6) +for every t ∈ (0, T], by taking T > 0 sufficiently small. On the other hand, since +u1 ∈ N(σ+s)/2 +p,q,∞ +, we have the estimate +t(σ−s+1)/θ∥Sj(t)u1|Nσ +p,q,1∥ ≤ C(t(σ−s)/2θ) + t(σ−s)/θ))∥u1|N(s+σ)/2 +p,q,∞ +∥ +≤ CT (σ−s)/2θ(1 + T (σ−s)/2θ)∥u1|N(s+σ)/2 +p,q,∞ +∥ < B − Aδ +2 +(4.7) +for every t ∈ (0, T], by taking T > 0 sufficiently small. We obtain the conclusion +from (4.6) and (4.7). The proof is complete. ✷ +We prepare the following three lemmas for the proof of Theorem 1.2. +Lemma 4.2 Let 1 < γ < θ, T ≤ 1, γ ≤ q ≤ p < ∞, p > N(γ − 1)/(θ − 1), +1 − θ/γ < s < 0 and s ≥ N/p + (γ − θ)/(γ − 1). +Then there exists C2 > 0 +independent of T such that +∥un+1|YT∥ ≤ ∥u0|YT∥ + C2∥un|YT∥γ +for n = 0, 1, . . .. +17 + +Proof. By (2.2), (2.3), (2.5) and (4.1) we see that +∥un+1(·, t) − u0(·, t)|Mp +q ∥ ≤ C∥un+1(·, t) − u0(·, t)|N0 +p,q,1∥ +≤ C +� t +0 +∥S(t − τ)|∇un(·, τ)|γ|N0 +p,q,1∥ dτ +≤ C +� t +0 +∥S(t − τ)|∇un(·, τ)|γ|NN(γ−1)/p +p/γ,q/γ,1 ∥ dτ +≤ C +� t +0 +{1 + (t − τ)−N(γ−1)/pθ}∥|∇un(·, τ)|γ|N0 +p/γ,q/γ,∞∥ dτ +≤ C +� t +0 +(t − τ)−N(γ−1)/pθ∥|∇un(·, τ)|γ|Mp/γ +q/γ ∥ dτ +≤ C +� t +0 +(t − τ)−N(γ−1)/pθ∥∇un(·, τ)|Mp +q ∥γ dτ +≤ C∥un|YT∥γ +� t +0 +(t − τ)−N(γ−1)/pθτ (s−1)γ/θ dτ +≤ Ct−N(γ−1)/pθ+(s−1)γ/θ+1∥un|YT∥γ. +In the same way, by (2.2), (2.3), (4.1) and (4.3) we see that +∥∂jun+1(·, t) − ∂ju0(·, t)|Mp +q ∥ ≤ C +� t +0 +∥Sj(t − τ)|∇un(·, τ)|γ|N0 +p,q,1∥ dτ +≤ C +� t +0 +∥Sj(t − τ)|∇un(·, τ)|γ|NN(γ−1)/p +p/γ,q/γ,1 ∥ dτ +≤ C +� t +0 +{1 + (t − τ)−N(γ−1)/pθ−1/θ}∥|∇un(·, τ)|γ|N0 +p/γ,q/γ,∞∥ dτ +≤ C +� t +0 +(t − τ)−N(γ−1)/pθ−1/θ∥|∇un(·, τ)|γ|Mp/γ +q/γ ∥ dτ +≤ C +� t +0 +(t − τ)−N(γ−1)/pθ−1/θ∥∇un(·, τ)|Mp +q ∥γ dτ +≤ C∥un|YT∥γ +� t +0 +(t − τ)−N(γ−1)/pθ−1/θτ (s−1)γ/θ dτ +≤ Ct−N(γ−1)/pθ−1/θ+(s−1)γ/θ+1∥un|YT∥γ. +Therefore, we have +∥un+1 − u0|YT∥ ≤ Ct1+(γ−1)(s/θ−N/pθ)−γ/θ∥un|YT∥γ +≤ C∥un|YT∥γ +for T ≤ 1, and the proof is complete. ✷ +18 + +Lemma 4.3 Let 1 < γ < θ, T ≤ 1, γ ≤ q ≤ p < ∞, p > N(γ − 1)/(θ − 1), +1 − θ/γ < s < 0 and s ≥ N/p + (γ − θ)/(γ − 1). +Then there exists C3 > 0 +such that for every ϕ(x) ∈ Ns +p,q,∞ satisfying lim supj→∞ 2sj∥F −1ϕjFϕ|Mp +q ∥ < δ for +some δ > 0 we can choose a positive number T ≤ 1 so small that the inequality +∥u0|YT∥ < C0δ holds. Furthermore, we can choose δ so small that ∥un|YT∥ ≤ M for +some M > 0. +Proof. By Lemma 4.1 with B = Aδ, we can take a positive number T ≤ 1 such +that the estimate +sup +0 0. +For δ > 0 satisfying +2γC2Cγ +3 δγ−1 < 1, +we see by induction that +sup +n ∥un|XT∥ ≤ 2C3δ =: M, +and the proof is complete. ✷ +Lemma 4.4 Let 1 < γ < θ, T ≤ 1, γ ≤ q ≤ p < ∞, p > N(γ − 1)/(θ − 1), +1 − θ/γ < s < 0 and s ≥ N/p + (γ − θ)/(γ − 1). Suppose that δ and T ≤ 1 are +small enough so that the assertion of Lemma 4.3 holds. Then there exists a positive +constant C independent of T such that +∥un+2 − un+1|YT∥ ≤ CMγ−1∥un+1 − un|YT∥ +for n = 0, 1, . . .. +Proof. By (2.2), (2.3), (2.5) and (4.1) we see that +∥un+2(·, t) − un+1(·, t)|Mp +q ∥ ≤ C∥un+2(·, t) − un+1(·, t)|N0 +p,q,1∥ +≤ C +� t +0 +∥S(t − τ)(|∇un+1(·, τ)|γ − |∇un(·, τ)|γ)|N0 +p,q,1∥ dτ +≤ C +� t +0 +∥S(t − τ)(|∇un+1(·, τ)|γ − |∇un(·, τ)|γ)|NN(γ−1)/p +p/γ,q/γ,1 ∥ dτ +≤ C +� t +0 +{1 + (t − τ)−N(γ−1)/pθ}∥|∇un+1(·, τ)|γ − |∇un(·, τ)|γ|N0 +p/γ,q/γ,∞∥ dτ +≤ C +� t +0 +(t − τ)−N(γ−1)/pθ∥|∇un+1(·, τ)|γ − |∇un(·, τ)|γ|Mp/γ +q/γ ∥ dτ +≤ CMγ−1∥un+1 − un|YT∥ +� t +0 +(t − τ)−N(γ−1)/pθτ (s−1)γ/θ dτ +≤ CMγ−1t−N(γ−1)/pθ+(s−1)γ/θ+1∥un+1 − un|YT∥. +19 + +In the same way, by (2.2), (2.3), (4.1) and (4.3) we see that +∥∂j(un+2(·, t) − un+1(·, t))|Mp +q ∥ ≤ C∥∂j(un+2(·, t) − un+1(·, t))|N0 +p,q,1∥ +≤ C +� t +0 +∥Sj(t − τ)(|∇un+1(·, τ)|γ − |∇un(·, τ)|γ)|N0 +p,q,1∥ dτ +≤ C +� t +0 +∥Sj(t − τ)(|∇un+1(·, τ)|γ − |∇un(·, τ)|γ)|NN(γ−1)/p +p/γ,q/γ,1 ∥ dτ +≤ C +� t +0 +{1 + (t − τ)−N(γ−1)/pθ−1/θ}∥|∇un+1(·, τ)|γ − |∇un(·, τ)|γ|N0 +p/γ,q/γ,∞∥ dτ +≤ C +� t +0 +(t − τ)−N(γ−1)/pθ−1/θ∥|∇un+1(·, τ)|γ − |∇un(·, τ)|γ|Mp/γ +q/γ ∥ dτ +≤ CMγ−1∥un+1 − un|YT∥ +� t +0 +(t − τ)−N(γ−1)/pθ−1/θτ (s−1)γ/θ dτ +≤ CMγ−1t−N(γ−1)/pθ−1/θ+(s−1)γ/θ+1∥un+1 − un|YT∥. +We used here [19, Lemma 1.4]. Therefore, we have +∥un+2 − un+1|YT∥ ≤ CMγ−1t1+(γ−1)(s/θ−N/pθ)−γ/θ∥un+1 − un|YT∥ +≤ CMγ−1∥un+1 − un|YT∥, +and the proof is complete. ✷ +Proof of Theorem 1.2. Take δ and T so small that +∥un+2 − un+1|YT∥ ≤ 1 +2∥un+1 − un|YT∥ +for n = 0, 1, . . ., and we see that un(x, t) converges in YT. Set u(x, t) as a limit of +un(x, t) in YT and we see that +u(x, t) := u0(x, t) + +� t +0 +[S(t − τ)|∇u(·, τ)|γ](x) dτ. +(4.8) +We next prove that u(x, t) ∈ L∞([ε, T] × RN) and ∇u(x, t) ∈ L∞([ε, T] × RN) +for every ε > 0. Let n be the smallest integer greater than Nγ/(θ − γ)p. Then +we can take an increasing sequence of positive numbers {pj}n +j=1 such that p1 = p, +N/pj+1 > N/pj − (θ − γ)/γ for every j = 1, 2, · · · , n − 1 and N/pn < (θ − γ)/γ. +We also define {qj}n +j=1 and {sj}n +j=1 as q1 = q, qj+1 = pj+1qj/pj, s1 = s and sj+1 = +N/pj+1 − N/pj. +By the obtained result, we see that the solution u(x, t) and ∇u(x, t) belong to +the spaces +L∞ �� ε +2n, T +� +, Mp +q +� +⊂ L∞ �� ε +2n, T +� +, N0 +p1,q1,∞ +� +⊂ L∞ �� ε +2n, T +� +, Ns2 +p2,q2,∞ +� +. +20 + +Since γ ≤ q2 ≤ p2, p2 > N(γ − 1)/(θ − 1), 1 − γ/θ < s2 < 0 and s2 ≥ N/p2 − (θ − +γ)/(γ − 1), we can apply the obtained result to see u(x, t) ∈ L∞ �� 2ε +2n, T +� +, Mp2 +q2 +� +and +∇u(x, t) ∈ L∞ �� 2ε +2n, T +� +, Mp2 +q2 +� +. In the same way, since +L∞ +�� jε +2n, T +� +, Mpj +qj +� +⊂ L∞ +�� jε +2n, T +� +, N0 +pj,qj,∞ +� +⊂ L∞ +�� jε +2n, T +� +, Nsj +pj+1,qj+1,∞ +� +, +where γ ≤ qj+1 ≤ pj+1, pj+1 > N(γ − 1)/(θ − 1), 1 − γ/θ < sj+1 < 0 and +sj+1 ≥ N/pj+1 − (θ − γ)/(γ − 1), we have u(x, t) ∈ L∞ �� +(j+1)ε +2n , T +� +, M +pj+1 +qj+1 +� +for +j = 1, 2, · · · , n − 1. Therefore, we have u(x, t) ∈ L∞ �� ε +2, T +� +, Mpn +qn +� +and ∇u(x, t) ∈ +L∞ �� ε +2, T +� +, Mpn +qn +� +, where pn > Nγ/(θ − γ). It follows that +���� +� t +ε/2 +S(t − τ)|∇u(·, τ)|γ dτ|L∞ +���� +≤ C +� t +ε/2 +��S(t − τ)|∇u(·, τ)|γ|B0 +∞,1 +�� dτ +≤ C +� t +ε/2 +���S(t − τ)|∇u(·, τ)|γ|NNγ/pn +pn/γ,qn/γ,1 +��� dτ +≤ C +� t +ε/2 +� +1 + (t − τ)−Nγ/θpn� ��|∇u(·, τ)|γ|N0 +pn/γ,qn/γ,∞ +�� dτ +≤ C +� t +ε/2 +(t − τ)−Nγ/θpn +���|∇u(·, τ)|γ|Mpn/γ +qn/γ +��� dτ +≤ C +� t +ε/2 +(t − τ)−Nγ/θpn ��∇u(·, τ)|Mpn +qn +��γ dτ +≤ C +� +t − ε +2 +�1−Nγ/θpn +sup +ε/2≤τ≤t +��∇u(·, τ)|Mpn +qn +��γ dτ +≤ CT 1−Nγ/θpn +sup +ε/2≤τ≤t +��∇u(·, τ)|Mpn +qn +��γ dτ < ∞ +(4.9) +for ε/2 ≤ t ≤ T ≤ 1. On the other hand, we have +∥S(t − ε/2)u(·, ε/2)|L∞∥ ≤ C∥S(t − ε/2)u(·, ε/2)|B0 +∞,1∥ +≤ C∥S(t − ε/2)u(·, ε/2)|NN/p +p,q,1∥ +≤ C +� +1 + (t − ε/2)−N/θp� ��|u(·, ε/2)||Mp +q +�� +≤ C(ε/2)−N/θp ��|u(·, ε/2)||Mp +q +�� < ∞ +(4.10) +for ε ≤ t ≤ T ≤ 1. Since +u(x, t) = +� +S +� +t − ε +2 +� +u +� +·, ε +2 +�� +(x) + +� t +ε/2 +[S(t − τ)|∇u(·, τ)|γ] (x) dτ, +21 + +this together with (4.9) and (4.10) implies that u(x, t) ∈ L∞([ε, T] × RN) for every +ε > 0. We next prove that ∇u(x, t) ∈ L∞([ε, T] × RN)for every ε > 0. It follows +that +���� +� t +ε/2 +Sj(t − τ)|∇u(·, τ)|γ dτ|L∞ +���� +≤ C +� t +ε/2 +��Sj(t − τ)|∇u(·, τ)|γ|B0 +∞,1 +�� dτ +≤ C +� t +ε/2 +���Sj(t − τ)|∇u(·, τ)|γ|NNγ/pn +pn/γ,qn/γ,1 +��� dτ +≤ C +� t +ε/2 +� +1 + (t − τ)−Nγ/θpn−1/θ� ��|∇u(·, τ)|γ|N0 +pn/γ,qn/γ,∞ +�� dτ +≤ C +� t +ε/2 +(t − τ)−Nγ/θpn−1/θ ���|∇u(·, τ)|γ|Mpn/γ +qn/γ +��� dτ +≤ C +� t +ε/2 +(t − τ)−Nγ/θpn−1/θ ��∇u(·, τ)|Mpn +qn +��γ dτ +≤ C +� +t − ε +2 +�1−Nγ/θpn−1/θ +sup +ε/2≤τ≤t +��∇u(·, τ)|Mpn +qn +��γ dτ +≤ CT 1−Nγ/θpn−1/θ +sup +ε/2≤τ≤t +��∇u(·, τ)|Mpn +qn +��γ dτ < ∞ +(4.11) +for ε/2 ≤ t ≤ T ≤ 1. On the other hand, we have +∥Sj(t − ε/2)u(·, ε/2)|L∞∥ ≤ C∥Sj(t − ε/2)u(·, ε/2)|B0 +∞,1∥ +≤ C∥Sj(t − ε/2)u(·, ε/2)|NN/p +p,q,1∥ +≤ C +� +1 + (t − ε/2)−N/θp−1/θ� ��|u(·, ε/2)||Mp +q +�� +≤ C(ε/2)−N/θp−1/θ ��|u(·, ε/2)||Mp +q +�� < ∞ +(4.12) +for ε ≤ t ≤ T ≤ 1. Since +∇u(x, t) = +� +Sj +� +t − ε +2 +� +u +� +·, ε +2 +�� +(x) + +� t +ε/2 +[Sj(t − τ)|∇u(·, τ)|γ] (x) dτ, +this together with (4.11) and (4.12) implies that ∇u(x, t) ∈ L∞([ε, T] × RN) for +every ε > 0. +Finally, we prove the uniqueness of the solution. +Assume that u(1)(x, t) and +u(2)(x, t) are solutions to (4.8) satisfying +sup +0≤t≤T +t−s/θ∥u(j)(·, t)|Mp +q ∥ + t(−s+1)/θ∥∇u(j)(·, t)|Mp +q ∥ < ∞. +22 + +Let u = u(1) − u(2) and h(t) = ∥u(·, t)|Mp +q ∥. Then exactly in the same way as in the +proof of Lemma 4.4, we have +sup +0 0, doubly occupied momentum states +pay an energy penalty U. Hence, states close to the orig- +inal Fermi energy avoid double occupancy giving rise to +states with a single up or down electron. As a result, a +single occupied region S1 forms which includes all states +with energy µ − U < ϵk < µ, whereas in the region S2 +with states fulfilling ϵk < µ − U momenta remain double +occupied, see inset of Fig. 1. At half filling and for large + +3 +repulsion U a Mott insulating state emerges with a fully +singly occupied band. +In the doped Mott insulator regime the occupation re- +gions Si can be understood as pseudo Fermi seas. We +refer to the occupation edges as pseudo Fermi surfaces +(pFS) associated with the effective pseudo Fermi ener- +gies µi(0), where µ1(0) = µ and µ2(0) = µ−U. While at +first glance, the metallic regime of the HK model seems +to be analogous to a two-band metal, the interacting na- +ture is manifest in the unconventional excitations [40] +and thermodynamic properties [39] as detailed below. +Remarkably, we find that the application of an orbital +magnetic field, which introduces the new length scale +ℓB = +1 +√ +eB , conserves the full LL degeneracy with in- +teresting implications. First, it simplifies the many-body +problem enormously by simplifying the degrees of free- +dom, e.g. +only the LL index of the wave functions is +relevant, which offers the opportunity to study solely the +effects of LL mixing/repulsion. Second, we can directly +work in the thermodynamic limit which allows us to de- +rive the interaction vertex analytically. +The resulting +many-body problem can be efficiently solved numerically. +A direct application of Onsager’s semiclassical theory +to the HK model would lead to two distinct QO frequen- +cies for each of the two pseudo Fermi surfaces (pFSs) µi +with conventional LK behaviour [50]. As one of our main +results, we show that Onsager’s relation is only correct in +the semiclassical regime at small magnetic fields where +the size of the semiclassical orbit, i.e. the characteris- +tic size of the LLs at the Fermi energy +√ +2l⋆ℓB, with the +highest occupied LL l⋆ ≈ µ/ωc, is the dominant length- +scale of the system. +The reason for the appearance of a “semiclassical” +regime in interacting metals is very generic. For low mag- +netic field, i.e. large ℓB, multiple LLs are occupied. In- +side the region ℓB +� +l/5 which can be of macroscopic size, +they resemble plane waves. Hence, any interaction has +the same influence on high LLs at small magnetic fields +as on momentum eigenstates. +Therefore, the assump- +tions of Onsager’s and LK theory, where the properties +of the oscillations can be connected to electronic prop- +erties of the metal in zero magnetic field, remain true. +However, we show that even in the semiclassical regime +of the HK model QOs can have a temperature dependent +frequency drift because of the non-Fermi–Dirac distribu- +tion of excitations, see sec. IV B. +Beyond the semiclassical regime LL repulsion becomes +important. Surprisingly, we observe numerically that a +simple scenario of individual LLs persists. +Concretely, +the ground state (GS) remains close to a state with an +integer occupation of each LL, see Fig. 3 (b). +Quali- +tatively similar to the B = 0 case, a double occupied +region forms at low energies and single occupied one for +higher energies. However, as our main result we find that +the size of the regions now depend on the magnetic field +µi = µi(B), see Fig. 1 which leads to a breakdown of On- +sager’s relation with aperiodic QOs. A detailed study of +the QOs in the strongly correlated non-Onsager regime, +see Fig. 4,5 and 6 shows that non-trivial sum and combi- +nation frequencies appear in the QO spectrum. Finally, +while all frequencies show a LK temperature dependence, +they feature unusual effective mass renormalization at +odds with the canonical LK theory. +A. +A word of caution +As any fine-tuned exactly soluble Hamiltonian the +HK model should not be considered a microscopic de- +scription of (doped) Mott insulating materials. Never- +theless, it can show generic physics which needs to be +separated from artificial behavior originating from the +infinite-ranged interactions. Concretely, the strength of +the interaction between LLs is governed by two differ- +ent effects. First, the deviation of the LL wavefunctions +compared to plane waves leads to a very natural change +of the repulsion between LLs with opposite spin. It re- +duces the double occupied region S2 for multiple occupied +LLs stronger than for higher magnetic fields where less +LLs are occupied. Secondly, there is an artificial reduc- +tion of the effective interaction U ′ = ℓB +L U between LLs: +With increasing magnetic field LLs become more local- +ized, eventually decreasing the possibility for centre of +mass conserving scattering events and, hence, the effec- +tive interaction approaches an artificial non-interacting +limit in the high field regime. +In order to discuss the effect of LL repulsion beyond the +HK limit, we note that the HK interaction is essentially +the q = 0, k = k′ part of the standard Hubbard interac- +tion in momentum space ˜U � +k,k′,q c† +k−q,↑ck,↑c† +k′+q,↓ck′,↓. +One can then study the effect of LL repulsion by pro- +jecting the Hubbard term into the LL basis and keep +only inter-LL interactions but ignore LL degeneracy lift- +ing contributions. Remarkably, in sec. V we show that +we find similar aperiodic QO beyond the Onsager and +LK paradigm, see Fig. 7. +Overall, we argue that breaking Onsager’s relation is a +generic effect of strongly interacting metals with strong +LL repulsion. In practice this might occur as an addi- +tional effect on top of LL degeneracy lifting effects. Our +work focuses solely on the influence of interactions on LL +mixing, which can be studied in a controlled way in the +HK limit. +It should therefore be seen as the opposite +limit to standard treatments of interactions in quantum +hall physics where LL mixing is only treated perturba- +tivley and interactions are projected into individual LLs. + +4 +III. +RECAP OF THE HATSUGAI–KOHMOTO +MODEL +The HK model [39] is described by the Hamiltonian +H = − t +� +⟨r,r′⟩,σ +c† +r,σcr′,σ ++ U +L2 +� +r1,r2,r3,r4 +δr1+r3,r2+r4c† +r1,↑cr2,↑c† +r3,↓cr4,↓ (2) +where L is the linear length of the system. We measure all +lengthscales in terms of the dimensionless lattice constant +a = 1. The interaction is of infinite range and may be +interpreted as centre of mass scattering: A pair of a spin- +up and down electrons is scattered to a different location +but their centre of mass coordinate is conserved. +The HK model can be block-diagonalized to Eq. (1) by +simple Fourier transformation of the creation and anni- +hilation operators +ck = 1 +L +� +r +e−ikrcr, +(3) +see appendix A. +Initially, Hatsugai and Kohmoto [39] introduced the +model as a simplified yet soluble version for an interaction +driven metal insulator transition at half-filling. +Away +from the ’Mott-insulating’ half-filling limit the model is +metallic. However, it is not a simple Fermi liquid but +features for a non-zero interaction U singly S1, doubly S2 +and non-occupied S0 regions in the Brillouin zone with +pFSs separating them. It is then a natural question to +ask, whether these pFS give rise to QOs similar to an +ordinary metal? +The GS of the HK model is highly degenerate. Each +momentum state in S1 can be either occupied by a spin- +up or down electron. However, this degeneracy is artifi- +cial, i.e. it is unstable against perturbations. Projecting +a local Hubbard term ˜Unr↑nr↓ into the GS manifold re- +sults in an effective ferromagnetic interaction implying +that the spins of the electrons inside S1 point all in the +same direction [50]. Henceforth, we take +|GSσ⟩ = +� +k1∈S1 +c† +k1σ +� +k2∈S2 +c† +k2↑c† +k2↓ |0⟩ . +(4) +as the robust GS. +All finite temperature thermodynamic properties of +the HK model can be calculated exactly [39]. Here we +only show the distribution function because it already +offers a glimpse into the interacting nature of the doped +Mott insulator. The partition function +Z = Tr e−β(H−µN) +(5) += +� +k +� +1 + 2e−β(ϵk−µ) + e−2β(ϵk−µ)−βU� +(6) +FIG. 2. The HK model features at T = 0 regions Si in the +Brillouin zone in which ⟨nk⟩ = i. +S2 (S1) is bound by its +pseudo Fermi energies µ2 (µ1) in blue (red). At finite tem- +perature the occupation steps broaden asymmetrically, see +zoom-in, with the distribution function fHK (black, solid) +due to excitations which can be excited from S2 directly to +S0, see above the plot. +At high temperatures T ≳ U the +details of the interaction are washed out (dashed). +leads +to +the +non-Fermi–Dirac +distribution +function +fHK(ϵ−µ, T) for the occupation number ⟨n↑+n↓⟩ where +fHK(ϵ, T) = 2 +e−βϵ + e−2βϵ−βU +1 + 2e−βϵ + e−2βϵ−βU , +(7) +see Fig. 2. For T ≳ U all details of the interaction are +essentially washed out by temperature and the thermo- +dynamic properties resemble those of an ordinary metal. +The interesting limiting case is T ≪ U, where +fHK(ϵ, T) → [f(ϵ + U + T log 2, T) + 1] f(ϵ − T log 2, T) +(8) +is the combination of two Fermi–Dirac distribution func- +tions f. Each occupation edge in the HK-model broadens +in a Fermi–Dirac fashion with temperature, however an +asymmetry of the excitations leads to a slight tempera- +ture shift of the pseudo Fermi energies, see Fig. 2. +Finally, note that the dispersion of the band ϵk can be +of any type, depending on the form of the non-interacting +part of the Hamiltonian. Throughout this manuscript we +fix ϵk = +k2 +2m corresponding to the continuous real-space +term −c†(r) ∇2 +2mc(r) in order to calculate the exact LL +spectrum. +Formally the continuous approximation ap- +plies only for low fillings of a typical band, but we expect +our findings to be generic for doped Mott insulators be- +cause the qualitative feature of two pFSs with singly and +doubly occupied states persist. Note that by introducing +an unbounded band structure, we loose the concept of +bandwidth which is responsible for the Mott transition. +This could be artificially restored by introducing a UV +cutoff. + +5 +IV. +LANDAU LEVEL INTERACTIONS +A. +Transformation to LL eigenstates +We apply a magnetic field in z-direction which is per- +pendicular to the HK model lying in the x-y-plane, and +use standard minimal coupling −i∇ → −i∇ − eA in the +Landau gauge A = (−By, 0, 0)T. Note that the interac- +tion does not couple to the magnetic field. +We transform to the LL basis +cl,kx,σ = +� +x,y +Φl,kx(x, y)c(x,y),σ +(9) +with the LL wavefunction +Φl,kx(x, y) = e−ikxx +√LℓB +ψl +� y +ℓB ++ kxℓB +� +(10) +where +ψl(ξ) = +1 +� +2ll!√π +e− 1 +2 ξ2Hl (ξ) +(11) +are the normalized wave functions of the quantum har- +monic oscillator and Hl are the (physicist’s) Hermite +polynomials. The above transformation diagonalizes the +non-interacting part of the Hamiltonian and gives the +well known LL Hamiltonian where each LL state labeled +by l is NΦ = 2πL2 +ℓ2 +B -fold degenerate. +One of the key simplifications of the HK interaction +is that the LL transformation makes it block diagonal: +The interaction only couples states with different LLs +li but same momenta, giving rise to an interaction ver- +tex VL/(2ℓB) +l1l2l3l4 (kx). In general, the vertex VL/(2ℓB) +l1l2l3l4 (kx) for +a finite sized system is a difficult 3-dimensional integral +which needs to be carefully solved numerically as detailed +in the appendix and benchmarked in Fig. 8. Remarkably, +we find that in the thermodynamic limit L → ∞ all in- +tegrals of the vertex V∞ +l1l2l3l4(kx) = Vl1l2l3l4 can be solved +analytically, see appendix C. The full interacting Hamil- +tonian then reads +H = +� +l,kx,σ +ωc +� +l + 1 +2 +� +c† +l,kx,σcl,kx,σ ++ U ℓB +L +� +kx,l1,l2,l3,l4 +Vl1l2l3l4c† +l1,kx,↑cl2,kx,↑c† +l3,kx,↓cl4,kx,↓ +(12) +and is diagonal in kx. Note that, the prefactor ℓB/L = +� +2π/NΦ normalizes the multiple sums of the interaction +and hence the interaction can not be treated perturba- +tivley in the thermodynamic limit. +We have simulated the above Hamiltonian for up to +10 LLs with exact diagonalization (ED). We emphasize +that the required lattice size for a real-space calculation +would be beyond any numerical capabilities. The reason +why the HK model can be efficiently simulated in an or- +bital magnetic field has its origin in the center of mass +preserving interaction which does not mix different mo- +menta, thus, retains the full LL degeneracy. Note that +this is the opposite limit of most studies of the FQHE, +which usually ignore LL mixing and only treat interac- +tions projected to individual LLs. +B. +The semiclassical regime +Before studying generic field strengths, we discuss the +limit of small orbital magnetic fields. The application of +a magnetic field introduces a new lengthscale, the mag- +netic length ℓB which may be interpreted as the size of +a flux quantum Φ0 = (2πe)−1. +The cyclotron orbits, +i.e. +characteristic size of the highest occupied LL, are +much larger with a radius of ℓB +√ +2l [3]. For small mag- +netic fields only few fluxes are inserted into the system, +and the semiclassical cyclotron orbits are of macroscopic +size approaching L. In this limit the semiclassical the- +ory always remains valid, independent of the form of the +interaction. +A quantum mechanical argument for the validity of the +semiclassical theory is that inside the real space region +|y| < ℓB +� +l/5 LLs with index l resemble plane waves +ψ∞ +l (ξ) = +� 2 +π2l +� 1 +4 +cos +�√ +2lξ − lπ +2 +� +, +(13) +see appendix B 1. For low magnetic fields, leading to LLs +with a large LL index l at the Fermi energy, this region +is of macroscopic size. Hence, high LLs interact with ex- +actly the same interaction as momentum states interact +at zero magnetic field. Our semiclassical intuition carries +over and Onsager’s theorem remains valid. +The above statement applies for any metal and we now +focus on the specific case of the HK model. Using the +asymptotic form of the wavefunctions ψ∞ +l +we evaluate +the vertex VL/ℓB +l1l2l3l4, see appendix B 2. Remarkably, we +find that for sufficiently high LLs the vertex becomes +diagonal in each LL leading to a ‘LL-HK’ Hamiltonian +Hsc = +� +l,kx +ωc +� +l + 1 +2 +� +(nl,kx,↑ + nl,kx,↓) + U ′nl,kx,↑nl,kx,↓ +(14) +which is exactly the same as in zero magnetic field, but +for quantum numbers l, kx. +All known concepts from +B = 0 carry over exactly: LLs with ϵl < µ−U ′ are double +occupied, LLs with ϵl < µ are single occupied and higher +energetic LLs are not occupied. The occupation edges at +µ and µ − U ′ lead at T = 0 to QO with frequencies +µ +ωc +and µ−U ′ +ωc +which are indeed the areas of the pFSs. +Nevertheless the non-Fermi–Dirac distribution func- +tion of the HK model leads to unconventional behav- +ior at non-zero temperature T > 0. +We focus on the +limit T ≪ U ′ otherwise the effects of the interaction are +washed out by temperature. Hence, we can make use of +the approximate representation of fHK in terms of the + +6 +Fermi–Dirac distribution function Eq. (8) and follow ear- +lier work e. g. Ref. [51], to derive the characteristic form +of the QOs of an observable X (i.e. the magnetization or +resistance) +X ∝ +� +k>0 +cos +� +2πk µ + T log 2 +ωc +� +RT (m) ++ cos +� +2πk µ − U ′ − T log 2 +ωc +� +RT (m) +(15) +where RT (m) = +2π2mℓ2 +BT +sinh(2π2mℓ2 +BT) is the usual LK temper- +ature dependence. +Remarkably, the only effect of the +non-Fermi–Dirac distribution function in the HK model +is a temperature shift of the frequencies. +C. +The non-Onsager regime: Exact treatment +We now focus on the regime ℓB ≪ L such that all in- +tegration boundaries can be extended to infinity. In this +limit the vertex of the LL interaction can be computed +analytically with details relegated to appendix C. Due to +the degeneracy in kx, we drop the momentum index kx +from here on and work with completely filled LLs which +corresponds to working at fixed chemical potential. We +measure the filling of a LL nl in units of the LL degen- +eracy NΦ. +Although all matrix elements of the vertex Vijkl can be +found exactly, the resulting model remains far to complex +to be solved analytically. The vertex Vijkl is dense and +has off-diagonal and diagonal elements with no apparent +substructure. Nevertheless, the transformation to the LL +basis has simplified the problem enormously: First, it re- +duced the initial long-range interacting 2D model to a +1D long-range interacting model. Secondly, the transfor- +mation made use of the infinite system size, such that +we are actually working in the thermodynamic limit and +are only constrained by the number of LLs we can simu- +late. Overall, we can study interacting LLs with ED far +beyond any real space numerical calculation. +The first remarkable result of the ED study is that even +though the vertex Vijkl has a non-perturbative form, the +exact eigenstates of the system remain close to a Fock +state in the LL basis. This becomes apparent from the +fact that deviations to integer filling of each LL are small, +as well as from the fact that the many-body participation +ratio P −1(ψ) = dim(H) � +α |⟨α|ψ⟩|4 of the GS is small. +The many-body participation ratio measures how many +Fock states |α⟩ contribute to a many-body state |ψ⟩. At +the minimal value P = (dim(H))−1 only a single basis +state contributes, i.e. +the state is a single Fock state +whereas P takes its maximal value of 1 for a maximally +superpositioned state, e.g. � +α |α⟩. +The above results allow for a simple, perturbative un- +derstanding of the complicated vertex Vijkl: The density- +density interactions Viijj may be understood as ferro- +magnetic interactions between the LLs i and j, because +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +E/µ +ϵl⋆ +1 +ϵl⋆ +2 +ϵl +M +M = − ∂E +∂B +0 +1 +max. overlap +2 +4 +6 +8 +10 +µ/ωc +4−L +4−9 +P(GS) +(a) +(b) +FIG. 3. Panel (a): The occupation of different LLs (double +occupied: transparent blue; single occupied: transparent red) +is shown for inverse magnetic field µ/ωc ∝ 1/B. The dispersion +of the LLs ϵl = ωc +� +l + 1 +2 +� +are shown as gray dotted lines. The +red (blue) line shows the energy of the highest single (double) +occupied LL ϵl⋆ +1 (ϵl⋆ +2). Jumps occur when l⋆ +1 and l⋆ +2 change and +are also visible in the orbital magnetization (black). The data +is obtained from ED with L = 10 LLs and U′/µ = +� +µ/ωcL. +Panel (b) shows the many-body participation ratio P(GS) +of the GS on a log-scale (left axis, dark gray) as well as the +overlap with the closest Fock state maxα∈H +� +|⟨α|GS⟩|2� +(right +axis, light gray). +the state c† +i↑c† +j↓ |0⟩ has a density-density interaction en- +ergy > 0, whereas the state c† +i↑c† +j↑ |0⟩ has no interaction +energy. +Hence, the density-density interaction reduces +double occupancy and aligns the electron spins of differ- +ent LLs. On the other hand the off-diagonal elements of +the vertex, i.e. i ̸= j or k ̸= l, stabilize antiferromag- +netic LL occupations, hence also double occupancy. This +contribution diagonal in LL occupation states arises as +a perturbative effect via an enormous number of virtual +intermediate states coupled by the off-diagonal elements +of the vertex. +In summary, even though the repulsive +density-density interaction always wins, it is significantly +reduced by the latter effect, see sec. IV D. +The second important result is that the electrons keep +forming pseudo Fermi seas, i.e. energetic regions which +are for LL index l ≤ l⋆ +2 doubly occupied and for LL index +l⋆ +2 < l ≤ l⋆ +1 singly occupied. +A Fock state with these +properties, which is not the exact GS but close to it, is +|l⋆ +1, l⋆ +2⟩ = +� +l1≤l⋆ +1,l2≤l⋆ +2 +c† +l1↑c† +l2↓ |0⟩ . +(16) +We evaluate l⋆ +1,2 from the exact GS by calculating +l⋆ +2 = +� +l +min ({⟨nl↑⟩, ⟨nl↓⟩}) − 1 +(17) +l⋆ +1 = +� +l +max ({⟨nl↑⟩, ⟨nl↓⟩}) − 1. +(18) + +7 +non-inter- +acting +non-Onsager +semiclassical +10 +20 +30 +40 +50 +µ/ωc +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +E/µ +ϵl⋆ +1 +ϵl⋆ +2 +ϵl +M +M = − ∂E +∂B +FIG. 4. The occupation of different LLs (double occupied: transparent blue; single occupied: transparent red), obtained from +zeroth order Monte Carlo simulations at temperatures T ≪ ωc, shown for inverse magnetic field µ/ωc ∝ 1/B. The dispersion of +the LLs ϵl = ωc +� +l + 1 +2 +� +are shown as gray dotted lines. The plot should be compared to Fig. 3 but here we simulated L = 50 +LLs (U′/µ = +� +µ/ωcL). The semiclassical, non-Onsager and non-interacting regime are clearly distinguishable. The orbital +magnetization M experiences drops when the LL occupations change, consistent with the ED result. Additional noise in the +magnetization is due to a numerical derivative of the MC data. +As stated before, it is impossible to describe the semi- +classical low field regime correctly when extending the +system size to infinity, which is required to evaluate the +vertex analytically. However, for U ≥ µ no double oc- +cupied pseudo Fermi sea S2 exists and the semiclassical +regime and the low field behavior for infinite system size +coincide accidentally. We focus our numerical analysis +for simplicity on U = µ. +In Fig. 3 (a) the energy dispersion of the highest sin- +gle (double) occupied LL ϵl⋆ +1 (ϵl⋆ +2) is shown, as well as +the orbital magnetization obtained from the GS energy +as a function of µ/ωc ∝ 1/B. For small magnetic fields, +i.e. large µ/ωc, the number of occupied LLs drops periodi- +cally at µ/ωc = Z+1/2, i.e. when the energy of the highest +occupied LL becomes larger than the chemical potential +µ. These periodic QO appear in the magnetization in +accordance with Onsager’s seminal relation. However, at +a sufficiently strong magnetic fields the system can min- +imize its energy by occupying the lowest single occupied +LL with a spin down and a spin up electron, l⋆ +2 increases +by one. +Similarly, it might be energetically preferable +to keep the lowest double occupied LL and instead de- +populate the highest single occupied LL, l⋆ +1 decreases by +one. Both processes lead to jumps in the magnetization. +Importantly, these jumps are aperiodic and the critical +magnetic field values where they appear depend on the +details of the vertex and the interaction strength. The +main conclusion is that the resulting QOs become aperi- +odic breaking Onsager’s relation! +Henceforth, we can understand the effect of interac- +tions in terms of effective chemical potentials µi(B) for +the doubly and singly occupied states, which is analo- +gous to the B = 0 HK model where µ1(0) = µ and +µ2(0) = µ − U. +D. +Qualitative results for many LLs: A +Monte–Carlo study +1. +Zero temperature +The simple results from the ED simulations suggest +that a perturbative picture where LLs remain the exact +eigenstates might be sufficient to understand the under- +lying physics. In this picture the off-diagonal matrix el- +ements, i.e. +Vijkl for i ̸= j and k ̸= l are treated as +perturbations to the classical Hamiltonian +H0 = +� +l,σ +ωc +� +l + 1 +2 +� +nl,σ + U ′ � +l,l′ +Vlll′l′nl,↑nl′,↓. +(19) +The +eigenstates +of +H0 +are +known +exactly, +since +[nl,σ, H0] = 0. These are the Fock states in the LL ba- +sis |n0,↑, n0,↓; n1,↑, ...⟩. In principle, the energy of each +eigenstate can be computed efficiently, however finding +the GS by a direct calculation of all eigenstates is numer- +ically costly. In the following, we show that an efficient +way to find the GS and obtain the finite temperature +dynamics with respect to H0 is the use of Monte-Carlo +(MC) sampling employing the Metropolis algorithm. +In principle it is possible to include perturbations of +second or higher order (the first order vanishes) but in +practice the dense form of the off-diagonal vertex re- +quires to sum over a large fraction of states of the en- +tire Hilbert space such that the second order correction +of the eigenstate energy cannot be computed efficiently. +By a careful comparison between ED and MC results, +we have shown that even in the presence of off-diagonal +interactions states remain close to LL Fock states. Thus, +we can conclude that the zeroth order approximation is +sufficient for a correct qualitative picture. Higher order + +8 +non-inter- +acting +non-Onsager +semi- +classical +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +freq. F +10−2 +10−1 +100 +fft amplitude +25 +50 +75 +100 +125 +150 +175 +µ/ωc +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +freq. F +µ1 +µ2 +µ1 + µ2 +2µ1 + µ2 +µ1 + 2µ2 +2µ1 + 2µ2 +2µ1 + 3µ2 +3µ1 + 2µ2 +(a) +(b) +FIG. 5. Panel (a) shows a STFT of the particle number N for +a MC data set like the one shown in Fig. 4 but for 200 LLs +at effective low temperatures. Here, we show N because it is +numerically more stable than M which requires a derivative. +However, the oscillating properties of M and N are the same. +Inside the semiclassical regime the Fourier spectrum shows +peaks at multiples of the area of the FS. In the non-Onsager +regime a plethora of peaks which are dispersive in µ/ωc arise. +In panel (b) we extracted the peak positions of panel (a) (open +circles). We overlayed the data points with the expected peak +positions for frequencies associated with sum combinations of +the effective pseudo Fermi energies p1µ1 + p2µ2. Note that +in a STFT plot frequency peaks do not appear at the actual +frequencies, however the peak frequencies can be calculated +from the actual frequencies, see appendix D. Several higher +orders of (p1, p2) are visible, for clarity we focused only on the +ones indicated in the legend. +perturbations will decrease the strength of the diagonal +elements Vlll′l′ and, therefore, we observe that the zeroth +order MC simulation overestimates the strength of the +interaction U. +Fig. 4 shows the results of the MC simulation of +Eq. (19) in the same style as Fig. 3 for ED. The MC +simulation allows to access many more LLs and hence +more oscillations. We have subsequently decreased the +temperature to obtain the GS occupation. +Importantly, the MC simulations provide further nu- +merical evidence for the schematic image sketched in +Fig. 1: The number of doubly and singly occupied LLs set +the QO frequencies. For Fig. 5 we have collected data of +200 LLs at effectively zero temperature. Due to the fact +that the QO frequencies depend on the magnetic field, we +perform a short-time Fourier transformation (STFT) as +µ/ωc changes. In the STFT small, consecutive windows +of the complete data are Fourier transformed, allowing to +0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 +0.00 +0.25 +0.50 +0.75 +1.00 +RLK (arb. units) +(a): µ/¯ωc = 175 +µ1, m∗/m = 1.002 ± 0.005 +2µ1, m∗/m = 1.93 ± 0.06 +3µ1, m∗/m = 3.0 ± 0.2 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.00 +0.25 +0.50 +0.75 +1.00 +RLK (arb. units) +(b): µ/¯ωc = 70 +µ2, m∗/m = 2.66 ± 0.08 +µ1, m∗/m = 2.7 ± 0.1 +µ1 + µ2, m∗/m = 1.78 ± 0.01 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +T/¯ωc +0.00 +0.25 +0.50 +0.75 +1.00 +RLK (arb. units) +(c): µ/¯ωc = 40 +µ2, m∗/m = 1.985 ± 0.009 +2µ2, m∗/m = 1.6 ± 0.03 +3µ2, m∗/m = 3.26 ± 0.03 +FIG. 6. +Temperature dependence of the main peak fre- +quencies of Fig. 5 (open symbols). The temperature depen- +dence is extracted for 3 different windows, each ranging from +[µ/¯ωc − 10, µ/¯ωc + 10]. The data is fitted with the LK factor +RT (m∗) to obtain the effective mass (solid lines). The color +coding of the frequencies is in accordance with Fig. 5 (b). +Note that very low temperatures are not accessible due to +freezing of the MC simulation. +study the magnetic field dependence of the peak frequen- +cies (for details of the STFT method see appendix D). +Strikingly, Fig. 5 shows that the observed frequencies +match with the effective pseudo Fermi energies µ1(B) = +¯ϵl⋆ +1+1/2 (µ2(B) = ¯ϵl⋆ +2+1/2) of the singly (doubly) occupied +LLs, see the red (blue) solid line in Fig. 5 (b). Note that +in a STFT the frequencies F (µ/ωc) are not observed di- +rectly but due to the consecutive Fourier transformations +only F(t)+t dF +dt (t) where ·(t) denotes the average over the +window with midpoint t, see appendix D. +The most prominent feature in Fig. 5 (a) are not +the basis frequencies but the combination frequencies +p1ϵl⋆ +1+1/2 + p2ϵl⋆ +2+1/2 with integers p1, p2. Our two main +observations are: +(i) In the canonical theory of QOs +only multiples of the basis frequencies are allowed, i.e. +p1ϵl⋆ +1+1/2 and p2ϵl⋆ +2+1/2, whereas we observe sum com- + +9 +binations of these basis frequencies which is highly un- +usual. (ii) The higher orders come with anomalous ampli- +tudes. The sum frequency is clearly dominant in the non- +Onsager regime but the canonical higher orders (p1, 0) +and (0, p2) with p1, p2 > 1 are absent. +The observed QOs in Fig. 5 show a clear breakdown of +Onsager’s relation which would predict frequencies set by +µi(0) with i = 1, 2 and higher harmonics thereof. Nev- +ertheless, oscillations remain visible and they are set by +the effective pseudo Fermi energies ϵl⋆ +1,2+1/2 which is de- +termined from the interaction. The oscillatory part of a +thermodynamic quantity Xosc reads +Xosc ∝ +� +p1,p2>0 +A(p1,p2) cos +� +2π p1ϵl⋆ +1+1/2 + p2ϵl⋆ +2+1/2 +ωc +� +(20) +and some amplitudes A(p1,p2) are too small to be observed +in our numerics. +2. +Finite temperature +A further advantage of the MC simulation is that it +allows for an efficient computation of finite temperature +properties. +Fig. 6 shows the temperature dependence +of the amplitudes of the strongest peaks of Fig. 5 for +different windows centered around µ/¯ωc. We chose win- +dows in the semiclassical (Fig. 6 (a)) as well as in the +non-Onsager regime (Fig. 6 (b) and (c)). Strikingly, we +find for all frequencies and all windows a clear LK de- +pendence of the amplitudes which can be traced back +to the underlying Fermi–Dirac-like distribution of exci- +tation energies. However, fitting the amplitudes with the +LK factor RT (m∗) to obtain the effective mass m∗ of each +frequency shows a breakdown of the LK theory. In the +semiclassical regime the higher harmonics are damped +with an effective mass being integer multiples of the bare +charge carrier mass m, as expected, see Fig. 6 (a). Con- +trarily, in the non-Onsager regime the sum frequency +µ1 +µ2 has the lowest effective mass m∗ ≈ 1.7m whereas +the basis frequencies decay faster in temperature with +m∗ ≈ 2 to 3m. +Note that we do no find a clear indication for tem- +perature drifts of the frequencies as in the semiclassical +regime. +V. +LANDAU LEVEL REPULSION IN THE +HUBBARD MODEL +The HK model provides a good starting point to ex- +plore the LL spectrum of interacting metals because its +physics at zero magnetic field is well understood due to +its exact solubility. However, we argue that our findings +about LL repulsion leading to anomalous QOs are generic +and not a pure artifact of the infinitely ranged HK inter- +action. In this subsection we show that we obtain similar +results for the Hubbard model as summarized in Fig. 7. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +E/µ +ϵl⋆ +1 +ϵl⋆ +2 +ϵl +M +M = − ∂E +∂B +0 +1 +max. overlap +2 +4 +6 +8 +10 +µ/ωc +4−L +4−9 +4−8 +P(GS) +(a) +(b) +FIG. 7. +Panel (a): The occupation of different LLs in the +Hubbard model (double occupied: transparent blue; single +occupied: transparent red) is shown for inverse magnetic field +µ/ωc ∝ 1/B. +The dispersion of the LLs ϵl = ωc +� +l + 1 +2 +� +are +shown as gray dotted lines. +The red (blue) line shows the +energy of the highest single (double) occupied LL ϵl⋆ +1 (ϵl⋆ +2). +Jumps occur when l⋆ +1 and l⋆ +2 change and are also visible in +the orbital magnetization (black). The data is obtained from +ED with 10 LLs and ˜ +U′/µ = 5/ +� +µ/ωc. Panel (b) shows the +many-body participation ratio P(GS) of the GS on a log-scale +(left axis, dark gray) as well as the overlap with the closest +Fock state maxα∈H +� +|⟨α|GS⟩|2� +(right axis, light gray). +We +project +the +standard +Hubbard +interaction +˜U � +r nr,↑nr,↓ in the LL basis and ignore contributions +lifting the LL degeneracy, e.g. +its kx momentum +dependence. Analogously to the HK model we obtain +˜H = +� +l,kx,σ +ωc +� +l + 1 +2 +� +c† +l,kx,σcl,kx,σ ++ ˜U 1 +ℓB +� +kx,l1,l2,l3,l4 +˜Vl1l2l3l4c† +l1,kx,↑cl2,kx,↑c† +l3,kx,↓cl4,kx,↓ +(21) +where Hubbard quantities are marked by a tilde. The LL +vertex ˜Vijkl for the Hubbard model can also be computed +exactly, see appendix E, and is similarly dense and un- +structured. The main difference to the HK model is that +the effective interaction ˜U/ℓB increases for high magnetic +fields, causing the artifical non-interacting regime of the +HK model to disappear. +We have solved Eq. (21) for up to 10 LLs by ED, see +Fig. 7. Remarkably, the results for the Hubbard model +closely resemble the results of the HK model, Fig. 3. +Concretely, LLs with B-field dependent effective pseudo +Fermi energies remain a good description of the system +leading to a breakdown of the Onsager relation for QOs. + +10 +VI. +DISCUSSION +Our approach to study QOs in a doped Mott insula- +tor is based on the HK interaction and an exact trans- +formation to the LL basis in the thermodynamic limit. +We showed that the resulting LL vertex retains the LL +degeneracy even for strong interactions. However, the in- +teractions lead to magnetic field dependent pseudo Fermi +energies due to strong repulsion between different LLs. +As a result, we find QOs beyond Onsager’s relation with +unusual properties. The aperiodic QOs can be mainly un- +derstood on the basis of the magnetic field dependence of +the pseudo Fermi energies with three notable exceptions: +(i) The emergence of new QO frequencies which are the +sum of pFS µ1 and µ2. (ii) The anomalous amplitudes +of the different harmonics, e.g. the sum frequencies are +strong whereas ordinary second or higher harmonics are +absent. (iii) The unusual effective masses extracted from +the LK temperature dependence of the different harmon- +ics. +For canonical QOs different mechanisms are known +which could possibly explain the emergence of sum fre- +quencies. However, most of them are due to processes in +experimental setups, like magnetic interactions [3], and +can therefore be ruled out. Neither can oscillations of +the effective Fermi energies be the reason for observa- +tion (i), since they would lead to oscillation with sum +and difference frequencies. We suggest that in strongly +interacting systems the sum frequencies can be under- +stood as oscillations of the quasiparticle lifetime [51]. In +Ref. [51] interband scattering by impurities leads to a +coupling of LLs from different bands which gives rise to +QOs of the quasiparticle lifetime. New combination fre- +quencies of QOs appear in transport properties but no +difference (only sum) frequencies are observed in thermo- +dynamic quantities similar to the magnetization studied +here. The underlying mechanism in our case is qualita- +tively similar, e.g. the interaction driven feedback of the +different QO periods of the two occupation edges leads +to sum combination frequencies in thermodynamic quan- +tities. Consequently, we expect both sum and difference +frequencies p1µ1 + p2µ2 with p1, p2 ∈ Z to appear in +transport properties. +Observation (ii) and (iii) are beyond standard pertur- +bative effects of QOs in interacting system [3, 52, 53]. +Especially the small effective mass of the sum frequency +is in stark contrast with known theories of QOs, where +sum combinations µ1+µ2 have temperature dependencies +RT (m∗ +1 + m∗ +2) or RT (m∗ +1)RT (m∗ +2) and, hence, necessarily +decay faster in temperature then their basis frequencies. +So far we have concentrated on the effect of LL repul- +sion on QOs but it is interesting to speculate about other +non-perturbative effects. For example, LL repulsion can +also lead to an interesting interplay between the IQHE +effect and Mott physics. The Mott insulating state of the +HK model without a magnetic field appears at half filling +and U being larger than the bandwidth Λ such that the +entire Brillouin zone is singly occupied. In our continuum +model this can be artificially realized by introducing a UV +cut-off Λ = µ1(0). Applying a magnetic field leads to the +formation of double occupied LLs at Bc0 by deoccupying +the “highest” LL, analogous to Fig. 3. Then QOs, IQHE +or FQHE would only be visible for µ1(B) ̸= µ1(0) = Λ. +The transition between regimes with singly and doubly +occupied LLs would be accompanied by a reformation of +edge states, one associated to the particle pocket at the +lower Hubbard band and the other one associated to the +hole pocket at the upper Hubbard band. +As a result, +a magnetic field induced transition between a Mott in- +sulator and a Hall insulating state should occur with a +distinct Hall response. +VII. +CONCLUSION +We have studied the LL spectrum of the exactly soluble +HK model and the resulting QOs. The HK interaction +does not break the LL degeneracy but leads to a strong +repulsion between LLs. We found various exact results +for the interaction vertex between LLs which allowed the +efficient numerical simulation of up to ten LLs. Subse- +quently, we showed that the main qualitative effects can +already be understood from density-density interactions +between LLs, which allowed us to perform Monte Carlo +simulations for hundreds of LLs. +The most important +effect is the emergence of effective pseudo Fermi energies +µi(B) which depend on the magnetic field strength via +the interaction vertex. +The implications of the magnetic field dependent LL +repulsion are manifold: The resulting QOs and the criti- +cal magnetic fields of IQHE transitions become aperiodic. +Hence, QOs are not connected to the area of the pseudo +Fermi energies at zero field in contrast to Onsager’s sem- +inal relation. Furthermore, LL interactions give rise to +novel sum combination frequencies and LK temperature +decays of the QO amplitudes with unusual effective mass +renormalizations. +In the future it will be interesting to explore other +physical observables of the (partially) soluble HK model +in an orbital magnetic field. In addition, the fine-tuned +limit of infinite ranged interactions could be used as a +starting point for including generic perturbations, e.g. +those lifting the LL degeneracy. It would be very worth- +while to look for our aperodic QOs with numerical meth- +ods, e.g. recent extensions of DMFT to include orbital +magnetic fields [38]. Similarly, other exactly soluble mod- +els [54] could shed light on interaction effects and QOs in +doped Mott insulators and non-perturbative parton de- +scriptions can help to map out the possible phenomenolo- +gies [55]. +The canonical Onsager and LK theory of QOs, which is +essentially a semiclassical theory of non-interacting elec- +trons, has been unreasonably successful. Over the last +decades, it has been applied beyond its regime of valid- +ity to understand QO experiments of weakly as well as +strongly correlated systems. In that context our work ra- + +11 +tionalizes that even in the strongly interacting HK model +we recover canonical QOs in the semiclassical limit. How- +ever, there are by now several experimental examples of +strongly correlated materials showing QOs beyond the +canonical description [13–20]. Our study indeed provides +rigorous calculations for novel aperiodic QOs with un- +usual mass renormalizations, and we hope it can serve as +a stepping stone for exploring new theoretical scenarios +and generalizations of Onsager’s relation. +ACKNOWLEDGMENTS +We acknowledge helpful discussions with Inti Sode- +mann. +V.L. acknowledges support from the Studiens- +tiftung des deutschen Volkes. J.K. acknowledges support +from the Imperial- TUM flagship partnership, as well as +the Munich Quantum Valley, which is supported by the +Bavarian state government with funds from the Hightech +Agenda Bayern Plus. +DATA AVAILABILITY +Code and data related to this paper are available on +Zenodo [56] from the corresponding authors upon rea- +sonable request. + +12 +Appendix A: Transformation to the Landau level +basis +Here we show how the HK model becomes block diago- +nal by Fourier transformation, i.e. deriving Eq. (1) from +Eq. (2), and how the LL vertex arises, i.e. the derivation +of Eq. (12). +We start from the real space Hamiltonian Eq. (2) and +transform its interaction to the LL basis. For simplicity +we carry out the calculation separately for the x and y- +component. We begin with the x-component which is for +our gauge choice of the magnetic field analogous to the +HK model at zero magnetic field +1 +L +� +x1,x2,x3,x4 +δx1+x3,x2+x4c† +x1,↑cx2,↑c† +x3,↓cx4,↓ += 1 +L3 +� +k1,k2,k3,k4 +c† +k1,↑ck2,↑c† +k3,↓ck4,↓ +× +� +x1 +eix1(k1−k4) +� +�� +� +Lδk1,k4 +� +x2 +e−ix2(k2−k4) +� +�� +� +Lδk2,k4 +� +x3 +eix3(k3−k4) +� +�� +� +Lδk3,k4 += +� +k4 +c† +k4,↑ck4,↑c† +k4,↓ck4,↓ +(A1) +and for the y-component at a given momentum kx +1 +L +� +y1,y2,y3,y4 +δy1+y3,y2+y4c† +y1,↑cy2,↑c† +y3,↓cy4,↓ += 1 +Lℓ2 +B +� +l1,l2,l3,l4 +c† +l1,↑cl2,↑c† +l3,↓cl4,↓ +� L/2 +−L/2 +dy1dy2dy3 +× ψl1 +� y1 +ℓB ++ ℓBkx +� +ψl2 +� y2 +ℓB ++ ℓBkx +� +× ψl3 +� y3 +ℓB ++ ℓBkx +� +ψl4 +�y1 − y2 + y3 +ℓB ++ ℓBkx +� +=ℓB +L +� +l1,l2,l3,l4 +VL/(2ℓB) +l1l2l3l4 (kx)c† +l1,↑cl2,↑c† +l3,↓cl4,↓ +(A2) +where the general vertex is +Vν +l1l2l3l4(q) = +� ν +−ν +dξ1dξ2dξ3ψl1 (ξ1 + ℓBq) +× ψl2 (ξ2 + ℓBq) ψl3 (ξ3 + ℓBq) +× ψl4 (ξ1 + ξ3 − ξ2 + ℓBq) . +(A3) +Different matrix elements of the general vertex for q = 0 +are shown in Fig. 8. +Appendix B: Semiclassical limit +This section includes details of calculations in the semi- +classical limit. We derive the asymptotic wavefunction +for high LLs and derive the semiclassical vertex which is +diagonal in the LL index. +1. +Asymptotic wavefunction for high LLs +In this subsection we derive Eq. (13) from its definition +Eq. (11) by making use of the asymptotic form of the Her- +mite polynomials Hl(x) [57] inside the region |ξ| < +√ +2l +Hl (ξ) ≈ +� +� +� +� +2 +� +1 − ξ2 +2l +e +l +2 +� +log(2l)−1+ ξ2 +2l +� +× cos +�� +l +2ξ +� +1 − ξ2 +2l + +� +l + 1 +2 +� +arcsin +� ξ +√ +2l +� +− lπ +2 +� +. +(B1) +We expand the asymptotic form in ξ2/l, into harmonic +oscillations to obtain +Hl(ξ) = +√ +2e +l +2 (log(2l)−1)e +ξ2 +2 cos +�√ +2lξ − lπ +2 +� +(B2) +which holds true with a relative error η up to √4lη (esti- +mated from higher orders of the Taylor expansion). We +fix η = 5% such that the asymptotic form Eq. (13) is +valid for |ξ| < +� +l/5. +2. +Derivation of the semiclassical vertex +We derive the semiclassical vertex, leading to Eq. (14), +by assuming that the asymptotic form of the LLs ψl → +ψ∞ +l +holds inside the entire integration region of the ver- +tex. A basic calculation leads to + +13 +0 +1 +2 +3 +4 +5 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +ν +ν +ν +6 6 6 6 +0 +1 +2 +3 +4 +5 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +ν +ν +ν8 8 8 8 +0 +1 +2 +3 +4 +5 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ν +ν +ν +6 6 7 7 +0 +1 +2 +3 +4 +5 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ν +ν +ν1 1 8 8 +0 +1 +2 +3 +4 +-0.25 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +ν +ν +ν1 2 3 8 +0 +1 +2 +3 +4 +5 +-0.4 +-0.3 +-0.2 +-0.1 +0.0 +ν +ν +ν +3 4 5 6 +FIG. 8. Dependence of the integral Vν +ijkl(kx = 0) on the integration boundary ν for various indices i, j, k, l (blue dots with +error bars). +The numerical integration is done with the built-in “NIntegrate” function of Mathematica 12 which returns +the estimated error of the integration. For ν ≲ 1/2 +� +2¯l + 1 (gray, dashed) where ¯l is the mean of the 4 indices, the vertex +can be described by the semiclassical vertex Eq. (B3) (red line). +By extending the integration boundaries to infinity the +integral can be solved exactly, i.e. Vijkl = V∞ +ijkl (orange line, Eq. (C13)). This becomes approximately a good approximation +when ν ≳ ξ0(¯l) (gray, dashed), where ξ0(l) ≈ 2 +√ +l is the value above which the LL wavefunction is exponentially small, i.e. +ξ0(l) = min{ξ ∈ R: ∀x > ξ +|ψl(x)| ≤ 0.05}. +Vν +ijkl(0) = +� ν +−ν +dξ1dξ2dξ3ψ∞ +i (ξ1) ψ∞ +j (ξ2) ψ∞ +k (ξ3) ψ∞ +l (ξ1 − ξ2 + ξ3) += +ν3 +8π2(ijkl)1/4 +� +±(i,j,k,l) +e−i π +2 (±ii±jj±kk±ll) +� ν +−ν +dξ1dξ2dξ3eiξ1(±i +√ +2i±l +√ +2l)eiξ2(±j +√2j∓l +√ +2l)eiξ3(±k +√ +2k±l +√ +2l) += +π +(ijkl)1/4 +� +±(i,j,k,l) +e−i π +2 (±ii±jj±kk±ll)δ1/ν +� +±i +� +i/2 ±l +� +l/2 +� +δ1/ν +� +±j +� +j/2 ∓l +� +l/2 +� +δ1/ν +� +±k +√ +2k ±l +√ +2l +� +(B3) +i,j,k,l≫1 +≈ +2π +(ijkl)1/4 δ1/ν +�√ +2i − +√ +2l +� +δ1/ν +� +− +� +2j + +√ +2l +� +δ1/ν +�√ +2k − +√ +2l +� +cos +�π +2 (i − j + k − l) +� +(B4) +where +δ1/ν(x) = ν +π +sin(νx) +νx += 1 +2π +� ν +−ν +dξeixξ +(B5) +and the sum extends over the 16 terms arising from dif- +ferent combinations of the signs. We refer to Eq. (B3) as +the semiclassical vertex. +In +the +limit +where +i, j, k, l +≫ +1 +only +the +sum combinations (upper,lower,upper,lower)-sign and +(lower,upper,lower,upper)-sign remain relevant and δ1/ν +become effectively δ-functions. The vertex is hence di- +agonal in the LLs Vν +ijkl ∝ δi,j,k,l. When normalizing the +asymptotic wavefunction inside the integration interval +N 2 +l = +� ν +−ν |ψ∞ +l (ξ)|2dξ the entire prefactor for the interac- +tion ℓB +L VL/(2ℓB) +llll +(kx)/N 4 +l = 1 approaches 1. This leads to +an effective HK-Hamiltonian in the LL basis, i.e. Eq. (14) +in the semiclassical limit. +Appendix C: Calculation of the LL vertex Vijkl +1. +Introduction +In the limit L ≫ ℓB the integral VL/(2ℓB) +ijkl +(kx) can be +solved exactly for all indices. The only important approx- + +14 +imation for this limit is the extension of the integration +boundary to infinity L/(2ℓB) → ∞. All dependencies on +kx cancel out. +The vertex can be split up into two equivalent integrals +Vijkl = V∞ +ijkl(kx) += +� ∞ +−∞ +dzIij(z)Ikl(−z) +(C1) +where +Iij(z) = +� ∞ +−∞ +dxψi(x)ψj(x + z) +(C2) +2. +Properties of Iij +Here, we list some useful properties of Iij +Iij(0) = δij +(C3a) +Iij(z → ∞) = 0 +(C3b) +Iij(−z) = (−1)i+jIij(z) +(C3c) +Iji(z) = (−1)i+jIij(z) +(C3d) +which can be easily shown by using the properties of ψl. +Most importantly the integral Iij can be solved exactly, +the solution is +Iij(z) = e−z2/4zj−i +� +i! +j!2j−i +�j +i +� +1F1(−i, 1 + j − i, z2/2) +(C4) +for j ≥ i where 1F1 is Kummer’s (confluent hypergeo- +metric) function of the first kind [58] +1F1(α, β, z) = +∞ +� +n=0 +(α + n − 1)! +(α − 1)! +(β − 1)! +(β + n − 1)! +zn +n! . +(C5) +Whereas this is in general not a helpful representation, +we emphasize that in the above equation 1F1 is a sum +over i terms and hence a polynomial in z. +Eq. (C4) is derived below w.l.o.g for j +≥ i (see +Eq. (C3d)): +Iij(z) = +� +dyψi(y − z/2)ψj(y + z/2) +(C6) += +e−z2/4 +� +π2i+ji!j! +� +dye−y2Hi(y − z/2) +Hj(y + z/2) +� +�� +� +�j +k=0 ( +j +k)Hk(y)zj−k +(C7) += +e−z2/4 +� +π2i+ji!j! +i,j +� +k,k′=0 +�i +k +�� j +k′ +� +(−z)i−kzj−k′ � +dye−y2Hk(y)Hk′(y) +� +�� +� +2kk!√πδk,k′ +(C8) += +e−z2/4 +� +2i+ji!j! +2izj−ii! +i +� +k=0 +(−1)k +2kk! +� +j +i − k +� +z2k +(C9) +=e−z2/4zj−i +� +i! +j!2j−i +�j +i +� +1F1(−i, 1 + j − i, z2/2) +(C10) +3. +Properties of Vijkl +Here, we list some useful properties of Vijkl: First, half +of the integrals evaluate to 0 due to an odd integrand +Vijkl = 0 +for i + j + k + l odd. +(C11) +The permutative relations +Vjikl = (−1)i+jVijkl +(C12a) +Vkjil = Vijkl +(C12b) +Vljki = (−1)i+lVijkl +(C12c) +Vikjl = (−1)j+kVijkl +(C12d) +Vilkj = Vijkl +(C12e) +Vijlk = (−1)k+lVijkl +(C12f) + +15 +reduce the the number of independent tensor entries. +Most importantly, the integral can be evaluated exactly, the solution is for j ≥ i and l ≥ k +Vijkl =(−1)k+l +� +i!k! +j!l! +� +j +i +� � +l +k +� +1F1 +� +−i, 1 + j − i; − d +dc +� +1F1 +� +−k, 1 + l − k; − d +dc +� � +− d +dc +�(j−i+l−k)/2 � +2π +c +����� +c=1 += +√ +2π(−1)k+l +� +i!k! +j!l! +i,k +� +n,n′=0 +(−1)n+n′ +n!n′! +� +j +i − n +�� +l +k − n′ +�(2n + 2n′ + j − i + l − k − 1)!! +2n+n′+(j−i+l−k)/2 +. +(C13) +Note that 1F1 are finite polynomials and that j − i + l − k is even if and only if i + j + k + l is even (if odd Vijkl = 0). +By +� +− d +dc +�n we mean (−1)n dn +dcn (the entire differential operator needs to be calculated first) and for calculation we +may use that (−1)n dn +dcn +1 +√c = (2n−1)!! +2n +where the double factorial !! denotes a factorial over all numbers with the same +parity. +For some indices Eq. (C13) evaluates to simpler results. For equal indices Viikk = +√ +2πLi +� +− d +dc +� +Lk +� +− d +dc +� +1 +√c +��� +c=1 +where Lk(x) are the Laguerre polynomials. This form simplifies to V00ll = +√ +2 Γ(l+1/2) +Γ(l+1) +if one of the indices is 0. This +result can be used to obtain an estimate of the scaling of the long range interaction between the LLs, since V00ll → +� +2 +l +for l ≫ 1. +From Eq. (C13) each matrix element of Vijkl can be calculated exactly be evaluating the finite sums. The numerical +complexity increases for increasing indices. In practice one has to be careful when performing the sums. The summands +have different signs and each of them is larger (in terms of its absolute value) then the total sum. This renders all +summands relevant and requires arbitrary precision floating point operations from the numerical side. +Eq. (C13) is derived below w.l.o.g for j ≥ i and l ≥ k (see Eq. (C12a)-(C12f)): +Vijkl =(−1)k+l +� +dzIij(z)Ikl(z) +(C14) +=(−1)k+l +� +i!k! +j!l! +�j +i +��l +k +� � ∞ +−∞ +dze−z2/2 +�z2 +2 +� j−i+l−k +2 +1F1(−i, 1 + j − i, z2/2)1F1(−k, 1 + l − k, z2/2) +� +�� +� +polynomials +(C15) +=(−1)k+l +� +i!k! +j!l! +�j +i +��l +k +� � +− d +dc +� j−i+l−k +2 +1F1 +� +−i, 1 + j − i, − d +dc +� +1F1 +� +−k, 1 + l − k, − d +dc +� � ∞ +−∞ +dze−cz2/2 +����� +c=1 +(C16) +=(−1)k+l +� +i!k! +j!l! +�j +i +��l +k +� � +− d +dc +� j−i+l−k +2 +1F1 +� +−i, 1 + j − i, − d +dc +� +1F1 +� +−k, 1 + l − k, − d +dc +� � +2π +c +����� +c=1 +(C17) +(C5) += (−1)k+l +� +i!k! +j!l! +i,k +� +n,n′=0 +(−1)n+n′ +n!n′! +� +j +i − n +�� +l +k − n′ +� � +− d +dc +�n+n′+ j−i+l−k +2 +� +2π +c +����� +c=1 +(C18) += +√ +2π(−1)k+l +� +i!k! +j!l! +i,k +� +n,n′=0 +(−1)n+n′ +n!n′! +� +j +i − n +�� +l +k − n′ +�(2n + 2n′ + j − i + l − k − 1)!! +2n+n′+(j−i+l−k)/2 +(C19) +Appendix D: Short-time Fourier transformation +The STFT is a method from Fourier analysis to deter- +mine phase and frequency information for local sections +of a signal changing over time. The basic idea is to per- +form several fast Fourier transformations of consecutive +windows in the time domain to obtain the frequency for +a segment in time. +We will explain how typical STFT plots of oscillating +functions with time dependant frequencies look like by +considering a test function g(t) = exp (if(t)t) where f(t) +is a slowly varying function with respect to g. We wish +to evaluate the Fourier transform +I(t0, ω) = +� ∞ +−∞ +e−iωtg(t)w(t − t0) +(D1) +as function of frequency ω and time t0 and w(t) is any + +16 +windowing function which for proof of principle we choose +to be a gaussian wσ(t) = e−t2/2/σ2. +The windowing +function restricts the dominant part of integration re- +gion to |t − t0|/σ < 1. The vague statement of f be- +ing a slowly varying function can be formulated in more +rigorous terms: First, for |t − t0|/σ < 1 the Taylor ex- +pansion f(t) = f(t0) + f ′(t0)(t − t0) holds. +Secondly, +the oscillations are fast with respect to the width of the +window ω/σ ≫ 1. +Under these assumptions, which are met in our MC +data as well as in possible experimental data, the in- +tegral can be solved exactly by completing the square. +The main result is that I(t0, ω) is exponentially peaked +at ωmax = f(t0) + t0f ′(t0). Therefore the STFT does +not show f(t) directly but only its linear approximation +inside each segment. This can be used to efficiently re- +construct f(t). +Appendix E: LL interactions in the Hubbard model +Here, we provide details for the derivation of Eq. (21) +and derive the LL vertex for the Hubbard model ˜Vijkl. +1. +Obtaining the vertex +We project the local Hubbard interaction in the LL +basis ignoring its effects on the LL degeneracy. Hence, +the calculation is similar to appendix A Eq. (A2). We +take L/ℓB → ∞ directly because the semiclassical limit +is not of interest here. The interaction reads +˜U +� +y +c† +y,↑cy,↑c† +y,↓cy,↓ += +˜U +ℓ2 +B +� +l1,l2,l3,l4 +c† +l1,↑cl2,↑c† +l3,↓cl4,↓ +� ∞ +−∞ +dy +× ψl1 +� y +ℓB ++ ℓBkx +� +ψl2 +� y +ℓB ++ ℓBkx +� +× ψl3 +� y +ℓB ++ ℓBkx +� +ψl4 +� y +ℓB ++ ℓBkx +� += +˜U +ℓB +� +l1,l2,l3,l4 +˜Vl1l2l3l4c† +l1,↑cl2,↑c† +l3,↓cl4,↓ +(E1) +where the vertex is +˜Vl1l2l3l4 = +� ∞ +−∞ +dξψl1 (ξ) ψl2 (ξ) ψl3 (ξ) ψl4 (ξ) . +(E2) +2. +Calculation of vertex ˜Vijkl +We evaluate the LL vertex ˜Vijkl for the Hubbard model +Eq. (E2) exactly. From the properties of ψl it is obvious +that half of the entries are zero +˜Vijkl = 0 +for i + j + k + l odd. +(E3) +similar to the HK model. +Furthermore, the vertex is +symmetric in each index pair ˜Vijkl = ˜Vjikl = ˜Vkjil = ... +To solve the integral we use the series representation +of the Hermite polynomials [59] +Hl(x) = l! +⌊l/2⌋ +� +n=0 +(−1)n +n!(l − 2n)!(2x)l−2n +(E4) +where ⌊x⌋ is the largest integer ≤ x. For i + j + k + l +even the vertex is + +17 +˜Vijkl = +� ∞ +−∞ +dξ +1 +π +� +2i+j+k+li!j!k!l! +e−2ξ2Hi(ξ)Hj(ξ)Hk(ξ)Hl(ξ) +(E5) += 1 +π +� +i!j!k!l! +⌊i/2⌋,⌊j/2⌋,⌊k/2⌋,⌊l/2⌋ +� +n1,n2,n3,n4=0 +(−2)−n1−n2−n3−n4 +n1!n2!n3!n4!(i − 2n1)!(j − 2n2)!(k − 2n3)!(l − 2n4)! +× +� ∞ +−∞ +dξ(2ξ2)(i+j+k+l)/2−n1−n2−n3−n4e−2ξ2 +(E6) += +√i!j!k!l! +√ +2π +⌊i/2⌋,⌊j/2⌋,⌊k/2⌋,⌊l/2⌋ +� +n1,n2,n3,n4=0 +(−2)−n1−n2−n3−n4 +n1!n2!n3!n4!(i − 2n1)!(j − 2n2)!(k − 2n3)!(l − 2n4)! +× +� +− d +dc +�(i+j+k+l)/2−n1−n2−n3−n4 1 +√c +����� +c=1 +(E7) += +1 +√ +2π +� +i!j!k!l! +2i+j+k+l +⌊i/2⌋,⌊j/2⌋,⌊k/2⌋,⌊l/2⌋ +� +n1,n2,n3,n4=0 +(−1)n1+n2+n3+n4(i + j + k + l − 2[n1 + n2 + n3 + n4] − 1)!! +n1!n2!n3!n4!(i − 2n1)!(j − 2n2)!(k − 2n3)!(l − 2n4)! +(E8) +which is a series that can be computed exactly. +3. +QO in the HK model with fixed particle number +In the main text, we have concentrated on results for a +fixed chemical potential. In Fig.9 we show the schematic +effect of keeping the particle number fixed which will in- +troduce small quantiative changes. +non-inter- +acting +non-Onsager +semi- +classical +HK model +µ1(0) +µ2(0) +ℓB ∼ L +U′ ≪ ωc +ℓB ≪ L +ℓB +√ +l ∼ L +B = 0 +1/B +ϵF +E +U +µ2(B) +µ1(B) +E2 +E1 +ϵF (U = 0) +S2 +S1 +FIG. 9. +Schematic image of the DOS and the QO of e.g. +the GS energy E1 + E2 for fixed particle number in 2D, +whereas Fig. 1 is for fixed chemical potential. +In the HK +model at B = 0 momentum states are double occupied up to +µ2(0) = µ − U and single occupied from µ2(0) to µ1(0) = ϵF +where ϵF is the Fermi energy in the non-interacting limit. +Due to the constant DOS in 2D the interaction leads to a +symmetric single occupied region around the Fermi energy, +see inset. At finite magnetic field the pseudo Fermi energies +become magnetic field dependent, but such that the total par- +ticle number is conserved. 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(United States Departement of +Commerce, New York, 1972). + diff --git a/MNFAT4oBgHgl3EQfxB7a/content/tmp_files/load_file.txt b/MNFAT4oBgHgl3EQfxB7a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6391eeccfe0681473548020bf8c9f837c19b1805 --- /dev/null +++ b/MNFAT4oBgHgl3EQfxB7a/content/tmp_files/load_file.txt @@ -0,0 +1,1209 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf,len=1208 +page_content='Quantum oscillations in a doped Mott insulator beyond Onsager’s relation Valentin Leeb1, 2 and Johannes Knolle1, 2, 3 1Technical University of Munich, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' TUM School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' TQM 2Munich Center for Quantum Science and Technology (MCQST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 80799 Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Germany 3Blackett Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Imperial College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' London SW7 2AZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' United Kingdom (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2023) The kinetic energy of electrons in a magnetic field is quenched resulting in a discrete set of highly degenerate Landau levels (LL) which gives rise to fascinating phenomena like the de Haas–van Alphen effect (dHvAe) or the integer and fractional quantum Hall effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The latter is a result of interactions partially lifting the degeneracy within a given LL while inter-LL interactions are usually assumed to be unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Here, we study the LL spectrum of the Hatsugai–Kohmoto model, a Hubbard-like model which is exactly soluble on account of infinite range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For the doped Mott insulator phase in a magnetic field we find that the degeneracy of LLs is preserved but inter- LL interactions are important leading to a non-monotonous reconstruction of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' As a result, strong LL repulsion leads to aperiodic quantum oscillations of the dHvAe in contrast to Onsager’s famous relation connecting oscillation frequencies with the Fermi surface areas at zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In addition, we find unconventional temperature dependencies of quantum oscillations and interaction-induced effective mass renormalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We discuss the general importance of inter-LL interactions for understanding doped Mott insulators in magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' INTRODUCTION The most remarkable aspect of Landau level (LL) for- mation of electrons in a magnetic field is the quenching of kinetic energy from a continuous spectrum to a set of discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The resulting macroscopically large de- generacy lies at the heart of prominent effects like the in- teger quantum Hall effect (IQHE) [1], discovered 1980, as well as the dHvAe already measured 50 years earlier [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' There, the discreteness of the LL spectrum leads to quan- tum oscillations (QO) of thermodynamic and transport properties periodic in the inverse of the applied field [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' A natural and persistent research question then addresses the role of electron interactions on the stability of the LL degeneracies and on physical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The study of strong electron-electron correlations in orbital magnetic fields typically focuses on the single LL limit [4, 5] because in the high magnetic field regime the spacing between LLs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the cyclotron frequency ωc, is large compared to the energy scale of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Prominently, it is well known that interactions in low LLs lead to a partial lifting of the LL degeneracy giving rise to the fractional quantum Hall effect (FQHE) [6, 7] in two-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, the effect of LL formation is not constrained to two-dimensional systems nor to high magnetic fields where only very few LLs are occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For instance, QOs are routinely observed at much smaller fields in a huge variety of two- and three-dimensional ma- terials, from weakly [3] to strongly interacting ones [8], which calls for an investigation of strong correlation ef- fects beyond the few LL limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The effect of weak interactions on LLs is well under- stood within Fermi liquid theory and the semiclassical description of electron motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' At zero magnetic field effective single particle theories emerge as low-energy de- scriptions with renormalized parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In 1952 On- sager shaped our understanding of Fermi liquids in mag- netic fields by a semiclassical picture [9]: The electrons perform quantized orbital motion with the cyclotron fre- quency ωc, constrained by their energy-momentum dis- persion ϵk perpendicular to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This leads to Onsager’s famous relation: The area of the extremal orbits around the Fermi surface equal the QO frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that these also determine the critical fields of the IQHE transitions in two dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The standard the- ory of QO was then completed by Lifshitz and Kosevich who connected the cyclotron frequency, which is deter- mined by the effective mass ωc = eB/m, to the universal temperature decay of the QO amplitude [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' It is surprising that the canonical Onsager and Lifshitz–Kosevich (LK) theory, which is essentially a single particle theory, can be applied routinely even to strongly correlated systems like heavy fermion sys- tems [11] or cuprate high temperature superconductors [8, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Nevertheless, in recent years numerous exper- imental findings [13–20] have shown deviations to the standard theory of QOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, despite a number of effective theories available [21–28] a controlled calcula- tion including strong correlations is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Exactly soluble models have played an important role for understanding the physics of strongly correlated sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Many phenomena, for example LL physics or gapless quantum spin liquid phases only emerge for large system sizes, which are challenging for numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, in certain soluble limits rigorous progress can be made albeit with the trade-off of a fine- tuned set of parameters [29, 30] or unphysical interac- tions [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Important developments for understand- ing correlated electrons have been the dynamical mean field theory (DMFT), which is exact in infinite dimen- sion, or the strongly coupled Sachdev–Ye–Kitaev models, which achieve exact solubility by random all-to-all cou- plings [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Both limits have recently been extended to orbital magnetic field regimes and feature anomalous arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='08685v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='str-el] 20 Jan 2023 2 QOs [26, 36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Here, we concentrate on the Hatsugai–Kohmoto (HK) model, which is exactly soluble due to all-to-all scattering with a centre of mass constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' It was initially intro- duced as a soluble example of a correlated metal to Mott insulator transition at half filling [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Recently, it has received renewed interest shedding light on superconduc- tivity in doped Mott insulators [40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Furthermore, HK-type interactions have been used for studying inter- action effects in the Haldane model [44], the Kondo effect [45], the periodic Anderson model [46], the gapping of Weyl nodes [47] or non-equilibrium physics [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' It has been argued that the metal insulator transition in the tractable HK model and the intractable Hubbard model are controlled by the same fixpoint [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In this work we study the LL spectrum of the doped HK model and the resulting anomalous QOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' At finite magnetic field the solubility is only partially lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Re- markably, the LL degeneracy is retained exactly but dif- ferent LLs are strongly interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hence, we can study the little explored effect of LL mixing/repulsion on LL spectra and QOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Due to the HK interaction the effec- tive degrees of freedom are simplified enormously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We find an exact functional form of the interaction vertex which allows for an efficient numerical treatment in the thermodynamic limit as well as further approximation to a classical Hamiltonian amenable to Monte–Carlo sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' As a result, we find that strong LL repulsion leads to aperiodic QOs at odds with the Onsager relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In addition, we discover unconventional temperature de- pendencies of QO amplitudes and effective mass renor- malizations beyond LK theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Finally, we show that the inter-LL components of the standard Hubbard inter- action lead to a similar phenomenology, which highlights the general relevance of LL repulsion for interpreting QO spectra of strongly correlated quantum materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' II we summa- rizes our main findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' III introduces the HK model and the continuum version for calculating the exact LL spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' IV we show how to solve the model in the LL basis, discuss analytical results of the interaction vertex and use exact diagonalization and Monte–Carlo simulations to calculate QOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' V we show that the LL repulsion arising from the standard local Hubbard in- teraction gives rise to similar anomalous QOs as in the HK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We discuss our findings in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' VI and close with explaining the broader implications of our work in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' OVERVIEW The HK model is an exactly solvable Hubbard-like model in which integrability is achieved by an infinite ranged interaction [39] leading to a block-diagonalized FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Schematic image of the DOS and QO of the singly and doubly occupied GS energies E1 and E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the HK model at B = 0 momentum states are double occupied up to µ2(0) = µ − U and single occupied from µ2(0) to µ1(0) = µ where µ is the Fermi energy in the non-interacting limit, see inset and right part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the main panel we plot the entire en- ergetic region where the LLs are double (single, not) occupied in blue (red, white), neglecting the LL substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The ef- fective pseudo Fermi energies µi(B) (dashed) depend on the magnetic field and lead to QOs of the GS energy E1 + E2 whose frequencies are set by µi(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Different regimes emerge for increasing magnetic field going from right to left: In the semiclassical regime for sufficiently low B two QO frequen- cies can be observed, each associated with the pseudo Fermi seas Si at B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For higher magnetic fields the semiclassi- cal behavior breaks down: The LLs interact and transitions between them are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This interaction leads to a B-field dependence of the effective pseudo Fermi energies µi(B) which set the QO frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The QOs become aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For high magnetic fields the LLs are strongly localized at odds with the center of mass constraint, such that the effective interaction U ′ reduces to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hamiltonian H = � k ϵk(nk,↑ + nk,↓) + Unk,↑nk,↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (1) At each momentum, the local Hilbert space is 4- dimensional consisting of the states |0k⟩ , |↑k⟩ , |↓k⟩ and |↑↓k⟩ with energies 0, ϵk and 2ϵk + U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' One can then minimize the energy for each momentum and the ground state (GS) is a simple product state thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The GS for any interaction strength can be understood easily from the non-interacting limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For U = 0 all states below the Fermi energy µ are double occupied, leading to an ordinary Fermi sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' When turning on repulsive interactions U > 0, doubly occupied momentum states pay an energy penalty U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hence, states close to the orig- inal Fermi energy avoid double occupancy giving rise to states with a single up or down electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' As a result, a single occupied region S1 forms which includes all states with energy µ − U < ϵk < µ, whereas in the region S2 with states fulfilling ϵk < µ − U momenta remain double occupied, see inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' At half filling and for large 3 repulsion U a Mott insulating state emerges with a fully singly occupied band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the doped Mott insulator regime the occupation re- gions Si can be understood as pseudo Fermi seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We refer to the occupation edges as pseudo Fermi surfaces (pFS) associated with the effective pseudo Fermi ener- gies µi(0), where µ1(0) = µ and µ2(0) = µ−U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' While at first glance, the metallic regime of the HK model seems to be analogous to a two-band metal, the interacting na- ture is manifest in the unconventional excitations [40] and thermodynamic properties [39] as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Remarkably, we find that the application of an orbital magnetic field, which introduces the new length scale ℓB = 1 √ eB , conserves the full LL degeneracy with in- teresting implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' First, it simplifies the many-body problem enormously by simplifying the degrees of free- dom, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' only the LL index of the wave functions is relevant, which offers the opportunity to study solely the effects of LL mixing/repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Second, we can directly work in the thermodynamic limit which allows us to de- rive the interaction vertex analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The resulting many-body problem can be efficiently solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' A direct application of Onsager’s semiclassical theory to the HK model would lead to two distinct QO frequen- cies for each of the two pseudo Fermi surfaces (pFSs) µi with conventional LK behaviour [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' As one of our main results, we show that Onsager’s relation is only correct in the semiclassical regime at small magnetic fields where the size of the semiclassical orbit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the characteris- tic size of the LLs at the Fermi energy √ 2l⋆ℓB, with the highest occupied LL l⋆ ≈ µ/ωc, is the dominant length- scale of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The reason for the appearance of a “semiclassical” regime in interacting metals is very generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For low mag- netic field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' large ℓB, multiple LLs are occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In- side the region ℓB � l/5 which can be of macroscopic size, they resemble plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hence, any interaction has the same influence on high LLs at small magnetic fields as on momentum eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Therefore, the assump- tions of Onsager’s and LK theory, where the properties of the oscillations can be connected to electronic prop- erties of the metal in zero magnetic field, remain true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, we show that even in the semiclassical regime of the HK model QOs can have a temperature dependent frequency drift because of the non-Fermi–Dirac distribu- tion of excitations, see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Beyond the semiclassical regime LL repulsion becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Surprisingly, we observe numerically that a simple scenario of individual LLs persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Concretely, the ground state (GS) remains close to a state with an integer occupation of each LL, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Quali- tatively similar to the B = 0 case, a double occupied region forms at low energies and single occupied one for higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, as our main result we find that the size of the regions now depend on the magnetic field µi = µi(B), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 1 which leads to a breakdown of On- sager’s relation with aperiodic QOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' A detailed study of the QOs in the strongly correlated non-Onsager regime, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 4,5 and 6 shows that non-trivial sum and combi- nation frequencies appear in the QO spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Finally, while all frequencies show a LK temperature dependence, they feature unusual effective mass renormalization at odds with the canonical LK theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' A word of caution As any fine-tuned exactly soluble Hamiltonian the HK model should not be considered a microscopic de- scription of (doped) Mott insulating materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Never- theless, it can show generic physics which needs to be separated from artificial behavior originating from the infinite-ranged interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Concretely, the strength of the interaction between LLs is governed by two differ- ent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' First, the deviation of the LL wavefunctions compared to plane waves leads to a very natural change of the repulsion between LLs with opposite spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' It re- duces the double occupied region S2 for multiple occupied LLs stronger than for higher magnetic fields where less LLs are occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Secondly, there is an artificial reduc- tion of the effective interaction U ′ = ℓB L U between LLs: With increasing magnetic field LLs become more local- ized, eventually decreasing the possibility for centre of mass conserving scattering events and, hence, the effec- tive interaction approaches an artificial non-interacting limit in the high field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In order to discuss the effect of LL repulsion beyond the HK limit, we note that the HK interaction is essentially the q = 0, k = k′ part of the standard Hubbard interac- tion in momentum space ˜U � k,k′,q c† k−q,↑ck,↑c† k′+q,↓ck′,↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' One can then study the effect of LL repulsion by pro- jecting the Hubbard term into the LL basis and keep only inter-LL interactions but ignore LL degeneracy lift- ing contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Remarkably, in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' V we show that we find similar aperiodic QO beyond the Onsager and LK paradigm, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Overall, we argue that breaking Onsager’s relation is a generic effect of strongly interacting metals with strong LL repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In practice this might occur as an addi- tional effect on top of LL degeneracy lifting effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Our work focuses solely on the influence of interactions on LL mixing, which can be studied in a controlled way in the HK limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' It should therefore be seen as the opposite limit to standard treatments of interactions in quantum hall physics where LL mixing is only treated perturba- tivley and interactions are projected into individual LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' RECAP OF THE HATSUGAI–KOHMOTO MODEL The HK model [39] is described by the Hamiltonian H = − t � ⟨r,r′⟩,σ c† r,σcr′,σ + U L2 � r1,r2,r3,r4 δr1+r3,r2+r4c† r1,↑cr2,↑c† r3,↓cr4,↓ (2) where L is the linear length of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We measure all lengthscales in terms of the dimensionless lattice constant a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The interaction is of infinite range and may be interpreted as centre of mass scattering: A pair of a spin- up and down electrons is scattered to a different location but their centre of mass coordinate is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The HK model can be block-diagonalized to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (1) by simple Fourier transformation of the creation and anni- hilation operators ck = 1 L � r e−ikrcr, (3) see appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Initially, Hatsugai and Kohmoto [39] introduced the model as a simplified yet soluble version for an interaction driven metal insulator transition at half-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Away from the ’Mott-insulating’ half-filling limit the model is metallic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, it is not a simple Fermi liquid but features for a non-zero interaction U singly S1, doubly S2 and non-occupied S0 regions in the Brillouin zone with pFSs separating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' It is then a natural question to ask, whether these pFS give rise to QOs similar to an ordinary metal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The GS of the HK model is highly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Each momentum state in S1 can be either occupied by a spin- up or down electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, this degeneracy is artifi- cial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' it is unstable against perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Projecting a local Hubbard term ˜Unr↑nr↓ into the GS manifold re- sults in an effective ferromagnetic interaction implying that the spins of the electrons inside S1 point all in the same direction [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Henceforth, we take |GSσ⟩ = � k1∈S1 c† k1σ � k2∈S2 c† k2↑c† k2↓ |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (4) as the robust GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' All finite temperature thermodynamic properties of the HK model can be calculated exactly [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Here we only show the distribution function because it already offers a glimpse into the interacting nature of the doped Mott insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The partition function Z = Tr e−β(H−µN) (5) = � k � 1 + 2e−β(ϵk−µ) + e−2β(ϵk−µ)−βU� (6) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The HK model features at T = 0 regions Si in the Brillouin zone in which ⟨nk⟩ = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' S2 (S1) is bound by its pseudo Fermi energies µ2 (µ1) in blue (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' At finite tem- perature the occupation steps broaden asymmetrically, see zoom-in, with the distribution function fHK (black, solid) due to excitations which can be excited from S2 directly to S0, see above the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' At high temperatures T ≳ U the details of the interaction are washed out (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' leads to the non-Fermi–Dirac distribution function fHK(ϵ−µ, T) for the occupation number ⟨n↑+n↓⟩ where fHK(ϵ, T) = 2 e−βϵ + e−2βϵ−βU 1 + 2e−βϵ + e−2βϵ−βU , (7) see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For T ≳ U all details of the interaction are essentially washed out by temperature and the thermo- dynamic properties resemble those of an ordinary metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The interesting limiting case is T ≪ U, where fHK(ϵ, T) → [f(ϵ + U + T log 2, T) + 1] f(ϵ − T log 2, T) (8) is the combination of two Fermi–Dirac distribution func- tions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Each occupation edge in the HK-model broadens in a Fermi–Dirac fashion with temperature, however an asymmetry of the excitations leads to a slight tempera- ture shift of the pseudo Fermi energies, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Finally, note that the dispersion of the band ϵk can be of any type, depending on the form of the non-interacting part of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Throughout this manuscript we fix ϵk = k2 2m corresponding to the continuous real-space term −c†(r) ∇2 2mc(r) in order to calculate the exact LL spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Formally the continuous approximation ap- plies only for low fillings of a typical band, but we expect our findings to be generic for doped Mott insulators be- cause the qualitative feature of two pFSs with singly and doubly occupied states persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that by introducing an unbounded band structure, we loose the concept of bandwidth which is responsible for the Mott transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This could be artificially restored by introducing a UV cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' LANDAU LEVEL INTERACTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Transformation to LL eigenstates We apply a magnetic field in z-direction which is per- pendicular to the HK model lying in the x-y-plane, and use standard minimal coupling −i∇ → −i∇ − eA in the Landau gauge A = (−By, 0, 0)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that the interac- tion does not couple to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We transform to the LL basis cl,kx,σ = � x,y Φl,kx(x, y)c(x,y),σ (9) with the LL wavefunction Φl,kx(x, y) = e−ikxx √LℓB ψl � y ℓB + kxℓB � (10) where ψl(ξ) = 1 � 2ll!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='√π e− 1 2 ξ2Hl (ξ) (11) are the normalized wave functions of the quantum har- monic oscillator and Hl are the (physicist’s) Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The above transformation diagonalizes the non-interacting part of the Hamiltonian and gives the well known LL Hamiltonian where each LL state labeled by l is NΦ = 2πL2 ℓ2 B -fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' One of the key simplifications of the HK interaction is that the LL transformation makes it block diagonal: The interaction only couples states with different LLs li but same momenta, giving rise to an interaction ver- tex VL/(2ℓB) l1l2l3l4 (kx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In general, the vertex VL/(2ℓB) l1l2l3l4 (kx) for a finite sized system is a difficult 3-dimensional integral which needs to be carefully solved numerically as detailed in the appendix and benchmarked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Remarkably, we find that in the thermodynamic limit L → ∞ all in- tegrals of the vertex V∞ l1l2l3l4(kx) = Vl1l2l3l4 can be solved analytically, see appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The full interacting Hamil- tonian then reads H = � l,kx,σ ωc � l + 1 2 � c† l,kx,σcl,kx,σ + U ℓB L � kx,l1,l2,l3,l4 Vl1l2l3l4c† l1,kx,↑cl2,kx,↑c† l3,kx,↓cl4,kx,↓ (12) and is diagonal in kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that, the prefactor ℓB/L = � 2π/NΦ normalizes the multiple sums of the interaction and hence the interaction can not be treated perturba- tivley in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We have simulated the above Hamiltonian for up to 10 LLs with exact diagonalization (ED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We emphasize that the required lattice size for a real-space calculation would be beyond any numerical capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The reason why the HK model can be efficiently simulated in an or- bital magnetic field has its origin in the center of mass preserving interaction which does not mix different mo- menta, thus, retains the full LL degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that this is the opposite limit of most studies of the FQHE, which usually ignore LL mixing and only treat interac- tions projected to individual LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The semiclassical regime Before studying generic field strengths, we discuss the limit of small orbital magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The application of a magnetic field introduces a new lengthscale, the mag- netic length ℓB which may be interpreted as the size of a flux quantum Φ0 = (2πe)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The cyclotron orbits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' characteristic size of the highest occupied LL, are much larger with a radius of ℓB √ 2l [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For small mag- netic fields only few fluxes are inserted into the system, and the semiclassical cyclotron orbits are of macroscopic size approaching L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In this limit the semiclassical the- ory always remains valid, independent of the form of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' A quantum mechanical argument for the validity of the semiclassical theory is that inside the real space region |y| < ℓB � l/5 LLs with index l resemble plane waves ψ∞ l (ξ) = � 2 π2l � 1 4 cos �√ 2lξ − lπ 2 � , (13) see appendix B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For low magnetic fields, leading to LLs with a large LL index l at the Fermi energy, this region is of macroscopic size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hence, high LLs interact with ex- actly the same interaction as momentum states interact at zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Our semiclassical intuition carries over and Onsager’s theorem remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The above statement applies for any metal and we now focus on the specific case of the HK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Using the asymptotic form of the wavefunctions ψ∞ l we evaluate the vertex VL/ℓB l1l2l3l4, see appendix B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Remarkably, we find that for sufficiently high LLs the vertex becomes diagonal in each LL leading to a ‘LL-HK’ Hamiltonian Hsc = � l,kx ωc � l + 1 2 � (nl,kx,↑ + nl,kx,↓) + U ′nl,kx,↑nl,kx,↓ (14) which is exactly the same as in zero magnetic field, but for quantum numbers l, kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' All known concepts from B = 0 carry over exactly: LLs with ϵl < µ−U ′ are double occupied, LLs with ϵl < µ are single occupied and higher energetic LLs are not occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The occupation edges at µ and µ − U ′ lead at T = 0 to QO with frequencies µ ωc and µ−U ′ ωc which are indeed the areas of the pFSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Nevertheless the non-Fermi–Dirac distribution func- tion of the HK model leads to unconventional behav- ior at non-zero temperature T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We focus on the limit T ≪ U ′ otherwise the effects of the interaction are washed out by temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hence, we can make use of the approximate representation of fHK in terms of the 6 Fermi–Dirac distribution function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (8) and follow ear- lier work e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' [51], to derive the characteristic form of the QOs of an observable X (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the magnetization or resistance) X ∝ � k>0 cos � 2πk µ + T log 2 ωc � RT (m) + cos � 2πk µ − U ′ − T log 2 ωc � RT (m) (15) where RT (m) = 2π2mℓ2 BT sinh(2π2mℓ2 BT) is the usual LK temper- ature dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Remarkably, the only effect of the non-Fermi–Dirac distribution function in the HK model is a temperature shift of the frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The non-Onsager regime: Exact treatment We now focus on the regime ℓB ≪ L such that all in- tegration boundaries can be extended to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In this limit the vertex of the LL interaction can be computed analytically with details relegated to appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Due to the degeneracy in kx, we drop the momentum index kx from here on and work with completely filled LLs which corresponds to working at fixed chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We measure the filling of a LL nl in units of the LL degen- eracy NΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Although all matrix elements of the vertex Vijkl can be found exactly, the resulting model remains far to complex to be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The vertex Vijkl is dense and has off-diagonal and diagonal elements with no apparent substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Nevertheless, the transformation to the LL basis has simplified the problem enormously: First, it re- duced the initial long-range interacting 2D model to a 1D long-range interacting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Secondly, the transfor- mation made use of the infinite system size, such that we are actually working in the thermodynamic limit and are only constrained by the number of LLs we can simu- late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Overall, we can study interacting LLs with ED far beyond any real space numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The first remarkable result of the ED study is that even though the vertex Vijkl has a non-perturbative form, the exact eigenstates of the system remain close to a Fock state in the LL basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This becomes apparent from the fact that deviations to integer filling of each LL are small, as well as from the fact that the many-body participation ratio P −1(ψ) = dim(H) � α |⟨α|ψ⟩|4 of the GS is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The many-body participation ratio measures how many Fock states |α⟩ contribute to a many-body state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' At the minimal value P = (dim(H))−1 only a single basis state contributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the state is a single Fock state whereas P takes its maximal value of 1 for a maximally superpositioned state, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' � α |α⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The above results allow for a simple, perturbative un- derstanding of the complicated vertex Vijkl: The density- density interactions Viijj may be understood as ferro- magnetic interactions between the LLs i and j, because 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 E/µ ϵl⋆ 1 ϵl⋆ 2 ϵl M M = − ∂E ∂B 0 1 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' overlap 2 4 6 8 10 µ/ωc 4−L 4−9 P(GS) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Panel (a): The occupation of different LLs (double occupied: transparent blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' single occupied: transparent red) is shown for inverse magnetic field µ/ωc ∝ 1/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The dispersion of the LLs ϵl = ωc � l + 1 2 � are shown as gray dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The red (blue) line shows the energy of the highest single (double) occupied LL ϵl⋆ 1 (ϵl⋆ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Jumps occur when l⋆ 1 and l⋆ 2 change and are also visible in the orbital magnetization (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The data is obtained from ED with L = 10 LLs and U′/µ = � µ/ωcL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Panel (b) shows the many-body participation ratio P(GS) of the GS on a log-scale (left axis, dark gray) as well as the overlap with the closest Fock state maxα∈H � |⟨α|GS⟩|2� (right axis, light gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the state c† i↑c† j↓ |0⟩ has a density-density interaction en- ergy > 0, whereas the state c† i↑c† j↑ |0⟩ has no interaction energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hence, the density-density interaction reduces double occupancy and aligns the electron spins of differ- ent LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' On the other hand the off-diagonal elements of the vertex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' i ̸= j or k ̸= l, stabilize antiferromag- netic LL occupations, hence also double occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This contribution diagonal in LL occupation states arises as a perturbative effect via an enormous number of virtual intermediate states coupled by the off-diagonal elements of the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In summary, even though the repulsive density-density interaction always wins, it is significantly reduced by the latter effect, see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The second important result is that the electrons keep forming pseudo Fermi seas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' energetic regions which are for LL index l ≤ l⋆ 2 doubly occupied and for LL index l⋆ 2 < l ≤ l⋆ 1 singly occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' A Fock state with these properties, which is not the exact GS but close to it, is |l⋆ 1, l⋆ 2⟩ = � l1≤l⋆ 1,l2≤l⋆ 2 c† l1↑c† l2↓ |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (16) We evaluate l⋆ 1,2 from the exact GS by calculating l⋆ 2 = � l min ({⟨nl↑⟩, ⟨nl↓⟩}) − 1 (17) l⋆ 1 = � l max ({⟨nl↑⟩, ⟨nl↓⟩}) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (18) 7 non-inter- acting non-Onsager semiclassical 10 20 30 40 50 µ/ωc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 E/µ ϵl⋆ 1 ϵl⋆ 2 ϵl M M = − ∂E ∂B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The occupation of different LLs (double occupied: transparent blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' single occupied: transparent red), obtained from zeroth order Monte Carlo simulations at temperatures T ≪ ωc, shown for inverse magnetic field µ/ωc ∝ 1/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The dispersion of the LLs ϵl = ωc � l + 1 2 � are shown as gray dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The plot should be compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 3 but here we simulated L = 50 LLs (U′/µ = � µ/ωcL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The semiclassical, non-Onsager and non-interacting regime are clearly distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The orbital magnetization M experiences drops when the LL occupations change, consistent with the ED result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Additional noise in the magnetization is due to a numerical derivative of the MC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' As stated before, it is impossible to describe the semi- classical low field regime correctly when extending the system size to infinity, which is required to evaluate the vertex analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, for U ≥ µ no double oc- cupied pseudo Fermi sea S2 exists and the semiclassical regime and the low field behavior for infinite system size coincide accidentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We focus our numerical analysis for simplicity on U = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 3 (a) the energy dispersion of the highest sin- gle (double) occupied LL ϵl⋆ 1 (ϵl⋆ 2) is shown, as well as the orbital magnetization obtained from the GS energy as a function of µ/ωc ∝ 1/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For small magnetic fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' large µ/ωc, the number of occupied LLs drops periodi- cally at µ/ωc = Z+1/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' when the energy of the highest occupied LL becomes larger than the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' These periodic QO appear in the magnetization in accordance with Onsager’s seminal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, at a sufficiently strong magnetic fields the system can min- imize its energy by occupying the lowest single occupied LL with a spin down and a spin up electron, l⋆ 2 increases by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Similarly, it might be energetically preferable to keep the lowest double occupied LL and instead de- populate the highest single occupied LL, l⋆ 1 decreases by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Both processes lead to jumps in the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Importantly, these jumps are aperiodic and the critical magnetic field values where they appear depend on the details of the vertex and the interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The main conclusion is that the resulting QOs become aperi- odic breaking Onsager’s relation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Henceforth, we can understand the effect of interac- tions in terms of effective chemical potentials µi(B) for the doubly and singly occupied states, which is analo- gous to the B = 0 HK model where µ1(0) = µ and µ2(0) = µ − U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Qualitative results for many LLs: A Monte–Carlo study 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Zero temperature The simple results from the ED simulations suggest that a perturbative picture where LLs remain the exact eigenstates might be sufficient to understand the under- lying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In this picture the off-diagonal matrix el- ements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Vijkl for i ̸= j and k ̸= l are treated as perturbations to the classical Hamiltonian H0 = � l,σ ωc � l + 1 2 � nl,σ + U ′ � l,l′ Vlll′l′nl,↑nl′,↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (19) The eigenstates of H0 are known exactly, since [nl,σ, H0] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' These are the Fock states in the LL ba- sis |n0,↑, n0,↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' n1,↑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In principle, the energy of each eigenstate can be computed efficiently, however finding the GS by a direct calculation of all eigenstates is numer- ically costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the following, we show that an efficient way to find the GS and obtain the finite temperature dynamics with respect to H0 is the use of Monte-Carlo (MC) sampling employing the Metropolis algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In principle it is possible to include perturbations of second or higher order (the first order vanishes) but in practice the dense form of the off-diagonal vertex re- quires to sum over a large fraction of states of the en- tire Hilbert space such that the second order correction of the eigenstate energy cannot be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' By a careful comparison between ED and MC results, we have shown that even in the presence of off-diagonal interactions states remain close to LL Fock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Thus, we can conclude that the zeroth order approximation is sufficient for a correct qualitative picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Higher order 8 non-inter- acting non-Onsager semi- classical 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' F 10−2 10−1 100 fft amplitude 25 50 75 100 125 150 175 µ/ωc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' F µ1 µ2 µ1 + µ2 2µ1 + µ2 µ1 + 2µ2 2µ1 + 2µ2 2µ1 + 3µ2 3µ1 + 2µ2 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Panel (a) shows a STFT of the particle number N for a MC data set like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 4 but for 200 LLs at effective low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Here, we show N because it is numerically more stable than M which requires a derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, the oscillating properties of M and N are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Inside the semiclassical regime the Fourier spectrum shows peaks at multiples of the area of the FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the non-Onsager regime a plethora of peaks which are dispersive in µ/ωc arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In panel (b) we extracted the peak positions of panel (a) (open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We overlayed the data points with the expected peak positions for frequencies associated with sum combinations of the effective pseudo Fermi energies p1µ1 + p2µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that in a STFT plot frequency peaks do not appear at the actual frequencies, however the peak frequencies can be calculated from the actual frequencies, see appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Several higher orders of (p1, p2) are visible, for clarity we focused only on the ones indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' perturbations will decrease the strength of the diagonal elements Vlll′l′ and, therefore, we observe that the zeroth order MC simulation overestimates the strength of the interaction U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 4 shows the results of the MC simulation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (19) in the same style as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 3 for ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The MC simulation allows to access many more LLs and hence more oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We have subsequently decreased the temperature to obtain the GS occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Importantly, the MC simulations provide further nu- merical evidence for the schematic image sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 1: The number of doubly and singly occupied LLs set the QO frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 we have collected data of 200 LLs at effectively zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Due to the fact that the QO frequencies depend on the magnetic field, we perform a short-time Fourier transformation (STFT) as µ/ωc changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the STFT small, consecutive windows of the complete data are Fourier transformed, allowing to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 RLK (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' units) (a): µ/¯ωc = 175 µ1, m∗/m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='005 2µ1, m∗/m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='06 3µ1, m∗/m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 RLK (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' units) (b): µ/¯ωc = 70 µ2, m∗/m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='08 µ1, m∗/m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='1 µ1 + µ2, m∗/m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='14 T/¯ωc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 RLK (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' units) (c): µ/¯ωc = 40 µ2, m∗/m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='985 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='009 2µ2, m∗/m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='03 3µ2, m∗/m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='03 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Temperature dependence of the main peak fre- quencies of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 (open symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The temperature depen- dence is extracted for 3 different windows, each ranging from [µ/¯ωc − 10, µ/¯ωc + 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The data is fitted with the LK factor RT (m∗) to obtain the effective mass (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The color coding of the frequencies is in accordance with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that very low temperatures are not accessible due to freezing of the MC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' study the magnetic field dependence of the peak frequen- cies (for details of the STFT method see appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Strikingly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 shows that the observed frequencies match with the effective pseudo Fermi energies µ1(B) = ¯ϵl⋆ 1+1/2 (µ2(B) = ¯ϵl⋆ 2+1/2) of the singly (doubly) occupied LLs, see the red (blue) solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that in a STFT the frequencies F (µ/ωc) are not observed di- rectly but due to the consecutive Fourier transformations only F(t)+t dF dt (t) where ·(t) denotes the average over the window with midpoint t, see appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The most prominent feature in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 (a) are not the basis frequencies but the combination frequencies p1ϵl⋆ 1+1/2 + p2ϵl⋆ 2+1/2 with integers p1, p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Our two main observations are: (i) In the canonical theory of QOs only multiples of the basis frequencies are allowed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' p1ϵl⋆ 1+1/2 and p2ϵl⋆ 2+1/2, whereas we observe sum com- 9 binations of these basis frequencies which is highly un- usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (ii) The higher orders come with anomalous ampli- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The sum frequency is clearly dominant in the non- Onsager regime but the canonical higher orders (p1, 0) and (0, p2) with p1, p2 > 1 are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The observed QOs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 show a clear breakdown of Onsager’s relation which would predict frequencies set by µi(0) with i = 1, 2 and higher harmonics thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Nev- ertheless, oscillations remain visible and they are set by the effective pseudo Fermi energies ϵl⋆ 1,2+1/2 which is de- termined from the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The oscillatory part of a thermodynamic quantity Xosc reads Xosc ∝ � p1,p2>0 A(p1,p2) cos � 2π p1ϵl⋆ 1+1/2 + p2ϵl⋆ 2+1/2 ωc � (20) and some amplitudes A(p1,p2) are too small to be observed in our numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Finite temperature A further advantage of the MC simulation is that it allows for an efficient computation of finite temperature properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 6 shows the temperature dependence of the amplitudes of the strongest peaks of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 5 for different windows centered around µ/¯ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We chose win- dows in the semiclassical (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 6 (a)) as well as in the non-Onsager regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 6 (b) and (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Strikingly, we find for all frequencies and all windows a clear LK de- pendence of the amplitudes which can be traced back to the underlying Fermi–Dirac-like distribution of exci- tation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, fitting the amplitudes with the LK factor RT (m∗) to obtain the effective mass m∗ of each frequency shows a breakdown of the LK theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the semiclassical regime the higher harmonics are damped with an effective mass being integer multiples of the bare charge carrier mass m, as expected, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Con- trarily, in the non-Onsager regime the sum frequency µ1 +µ2 has the lowest effective mass m∗ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='7m whereas the basis frequencies decay faster in temperature with m∗ ≈ 2 to 3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Note that we do no find a clear indication for tem- perature drifts of the frequencies as in the semiclassical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' LANDAU LEVEL REPULSION IN THE HUBBARD MODEL The HK model provides a good starting point to ex- plore the LL spectrum of interacting metals because its physics at zero magnetic field is well understood due to its exact solubility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, we argue that our findings about LL repulsion leading to anomalous QOs are generic and not a pure artifact of the infinitely ranged HK inter- action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In this subsection we show that we obtain similar results for the Hubbard model as summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 E/µ ϵl⋆ 1 ϵl⋆ 2 ϵl M M = − ∂E ∂B 0 1 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' overlap 2 4 6 8 10 µ/ωc 4−L 4−9 4−8 P(GS) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Panel (a): The occupation of different LLs in the Hubbard model (double occupied: transparent blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' single occupied: transparent red) is shown for inverse magnetic field µ/ωc ∝ 1/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The dispersion of the LLs ϵl = ωc � l + 1 2 � are shown as gray dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The red (blue) line shows the energy of the highest single (double) occupied LL ϵl⋆ 1 (ϵl⋆ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Jumps occur when l⋆ 1 and l⋆ 2 change and are also visible in the orbital magnetization (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The data is obtained from ED with 10 LLs and ˜ U′/µ = 5/ � µ/ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Panel (b) shows the many-body participation ratio P(GS) of the GS on a log-scale (left axis, dark gray) as well as the overlap with the closest Fock state maxα∈H � |⟨α|GS⟩|2� (right axis, light gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We project the standard Hubbard interaction ˜U � r nr,↑nr,↓ in the LL basis and ignore contributions lifting the LL degeneracy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' its kx momentum dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Analogously to the HK model we obtain ˜H = � l,kx,σ ωc � l + 1 2 � c† l,kx,σcl,kx,σ + ˜U 1 ℓB � kx,l1,l2,l3,l4 ˜Vl1l2l3l4c† l1,kx,↑cl2,kx,↑c† l3,kx,↓cl4,kx,↓ (21) where Hubbard quantities are marked by a tilde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The LL vertex ˜Vijkl for the Hubbard model can also be computed exactly, see appendix E, and is similarly dense and un- structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The main difference to the HK model is that the effective interaction ˜U/ℓB increases for high magnetic fields, causing the artifical non-interacting regime of the HK model to disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We have solved Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (21) for up to 10 LLs by ED, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Remarkably, the results for the Hubbard model closely resemble the results of the HK model, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Concretely, LLs with B-field dependent effective pseudo Fermi energies remain a good description of the system leading to a breakdown of the Onsager relation for QOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 10 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' DISCUSSION Our approach to study QOs in a doped Mott insula- tor is based on the HK interaction and an exact trans- formation to the LL basis in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We showed that the resulting LL vertex retains the LL degeneracy even for strong interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, the in- teractions lead to magnetic field dependent pseudo Fermi energies due to strong repulsion between different LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' As a result, we find QOs beyond Onsager’s relation with unusual properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The aperiodic QOs can be mainly un- derstood on the basis of the magnetic field dependence of the pseudo Fermi energies with three notable exceptions: (i) The emergence of new QO frequencies which are the sum of pFS µ1 and µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (ii) The anomalous amplitudes of the different harmonics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the sum frequencies are strong whereas ordinary second or higher harmonics are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (iii) The unusual effective masses extracted from the LK temperature dependence of the different harmon- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For canonical QOs different mechanisms are known which could possibly explain the emergence of sum fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' However, most of them are due to processes in experimental setups, like magnetic interactions [3], and can therefore be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Neither can oscillations of the effective Fermi energies be the reason for observa- tion (i), since they would lead to oscillation with sum and difference frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We suggest that in strongly interacting systems the sum frequencies can be under- stood as oscillations of the quasiparticle lifetime [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' [51] interband scattering by impurities leads to a coupling of LLs from different bands which gives rise to QOs of the quasiparticle lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' New combination fre- quencies of QOs appear in transport properties but no difference (only sum) frequencies are observed in thermo- dynamic quantities similar to the magnetization studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The underlying mechanism in our case is qualita- tively similar, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the interaction driven feedback of the different QO periods of the two occupation edges leads to sum combination frequencies in thermodynamic quan- tities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Consequently, we expect both sum and difference frequencies p1µ1 + p2µ2 with p1, p2 ∈ Z to appear in transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Observation (ii) and (iii) are beyond standard pertur- bative effects of QOs in interacting system [3, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Especially the small effective mass of the sum frequency is in stark contrast with known theories of QOs, where sum combinations µ1+µ2 have temperature dependencies RT (m∗ 1 + m∗ 2) or RT (m∗ 1)RT (m∗ 2) and, hence, necessarily decay faster in temperature then their basis frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' So far we have concentrated on the effect of LL repul- sion on QOs but it is interesting to speculate about other non-perturbative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For example, LL repulsion can also lead to an interesting interplay between the IQHE effect and Mott physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The Mott insulating state of the HK model without a magnetic field appears at half filling and U being larger than the bandwidth Λ such that the entire Brillouin zone is singly occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In our continuum model this can be artificially realized by introducing a UV cut-off Λ = µ1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Applying a magnetic field leads to the formation of double occupied LLs at Bc0 by deoccupying the “highest” LL, analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Then QOs, IQHE or FQHE would only be visible for µ1(B) ̸= µ1(0) = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The transition between regimes with singly and doubly occupied LLs would be accompanied by a reformation of edge states, one associated to the particle pocket at the lower Hubbard band and the other one associated to the hole pocket at the upper Hubbard band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' As a result, a magnetic field induced transition between a Mott in- sulator and a Hall insulating state should occur with a distinct Hall response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' CONCLUSION We have studied the LL spectrum of the exactly soluble HK model and the resulting QOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The HK interaction does not break the LL degeneracy but leads to a strong repulsion between LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We found various exact results for the interaction vertex between LLs which allowed the efficient numerical simulation of up to ten LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Subse- quently, we showed that the main qualitative effects can already be understood from density-density interactions between LLs, which allowed us to perform Monte Carlo simulations for hundreds of LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The most important effect is the emergence of effective pseudo Fermi energies µi(B) which depend on the magnetic field strength via the interaction vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The implications of the magnetic field dependent LL repulsion are manifold: The resulting QOs and the criti- cal magnetic fields of IQHE transitions become aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hence, QOs are not connected to the area of the pseudo Fermi energies at zero field in contrast to Onsager’s sem- inal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Furthermore, LL interactions give rise to novel sum combination frequencies and LK temperature decays of the QO amplitudes with unusual effective mass renormalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the future it will be interesting to explore other physical observables of the (partially) soluble HK model in an orbital magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In addition, the fine-tuned limit of infinite ranged interactions could be used as a starting point for including generic perturbations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' those lifting the LL degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' It would be very worth- while to look for our aperodic QOs with numerical meth- ods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' recent extensions of DMFT to include orbital magnetic fields [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Similarly, other exactly soluble mod- els [54] could shed light on interaction effects and QOs in doped Mott insulators and non-perturbative parton de- scriptions can help to map out the possible phenomenolo- gies [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The canonical Onsager and LK theory of QOs, which is essentially a semiclassical theory of non-interacting elec- trons, has been unreasonably successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Over the last decades, it has been applied beyond its regime of valid- ity to understand QO experiments of weakly as well as strongly correlated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In that context our work ra- 11 tionalizes that even in the strongly interacting HK model we recover canonical QOs in the semiclassical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' How- ever, there are by now several experimental examples of strongly correlated materials showing QOs beyond the canonical description [13–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Our study indeed provides rigorous calculations for novel aperiodic QOs with un- usual mass renormalizations, and we hope it can serve as a stepping stone for exploring new theoretical scenarios and generalizations of Onsager’s relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge helpful discussions with Inti Sode- mann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' acknowledges support from the Studiens- tiftung des deutschen Volkes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' acknowledges support from the Imperial- TUM flagship partnership, as well as the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' DATA AVAILABILITY Code and data related to this paper are available on Zenodo [56] from the corresponding authors upon rea- sonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 12 Appendix A: Transformation to the Landau level basis Here we show how the HK model becomes block diago- nal by Fourier transformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' deriving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (1) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (2), and how the LL vertex arises, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We start from the real space Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (2) and transform its interaction to the LL basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For simplicity we carry out the calculation separately for the x and y- component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We begin with the x-component which is for our gauge choice of the magnetic field analogous to the HK model at zero magnetic field 1 L � x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='x4 δx1+x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='x2+x4c† x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑cx2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑c† x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓cx4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓ = 1 L3 � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k4 c† k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑ck2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑c† k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓ck4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓ × � x1 eix1(k1−k4) � �� � Lδk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k4 � x2 e−ix2(k2−k4) � �� � Lδk2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k4 � x3 eix3(k3−k4) � �� � Lδk3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k4 = � k4 c† k4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑ck4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑c† k4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓ck4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓ (A1) and for the y-component at a given momentum kx 1 L � y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='y3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='y4 δy1+y3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='y2+y4c† y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑cy2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑c† y3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓cy4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓ = 1 Lℓ2 B � l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l4 c† l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑cl2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑c† l3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓cl4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓ � L/2 −L/2 dy1dy2dy3 × ψl1 � y1 ℓB + ℓBkx � ψl2 � y2 ℓB + ℓBkx � × ψl3 � y3 ℓB + ℓBkx � ψl4 �y1 − y2 + y3 ℓB + ℓBkx � =ℓB L � l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l4 VL/(2ℓB) l1l2l3l4 (kx)c† l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑cl2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↑c† l3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓cl4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='↓ (A2) where the general vertex is Vν l1l2l3l4(q) = � ν −ν dξ1dξ2dξ3ψl1 (ξ1 + ℓBq) × ψl2 (ξ2 + ℓBq) ψl3 (ξ3 + ℓBq) × ψl4 (ξ1 + ξ3 − ξ2 + ℓBq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (A3) Different matrix elements of the general vertex for q = 0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Appendix B: Semiclassical limit This section includes details of calculations in the semi- classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We derive the asymptotic wavefunction for high LLs and derive the semiclassical vertex which is diagonal in the LL index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Asymptotic wavefunction for high LLs In this subsection we derive Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (13) from its definition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (11) by making use of the asymptotic form of the Her- mite polynomials Hl(x) [57] inside the region |ξ| < √ 2l Hl (ξ) ≈ � � � � 2 � 1 − ξ2 2l e l 2 � log(2l)−1+ ξ2 2l � × cos �� l 2ξ � 1 − ξ2 2l + � l + 1 2 � arcsin � ξ √ 2l � − lπ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (B1) We expand the asymptotic form in ξ2/l, into harmonic oscillations to obtain Hl(ξ) = √ 2e l 2 (log(2l)−1)e ξ2 2 cos �√ 2lξ − lπ 2 � (B2) which holds true with a relative error η up to √4lη (esti- mated from higher orders of the Taylor expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We fix η = 5% such that the asymptotic form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (13) is valid for |ξ| < � l/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Derivation of the semiclassical vertex We derive the semiclassical vertex, leading to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (14), by assuming that the asymptotic form of the LLs ψl → ψ∞ l holds inside the entire integration region of the ver- tex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' A basic calculation leads to 13 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 ν \uf785ν \uf785ν 6 6 6 6 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 ν \uf785ν \uf785ν8 8 8 8 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 ν \uf785ν \uf785ν 6 6 7 7 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='6 ν \uf785ν \uf785ν1 1 8 8 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='00 ν \uf785ν \uf785ν1 2 3 8 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='0 ν \uf785ν \uf785ν 3 4 5 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Dependence of the integral Vν ijkl(kx = 0) on the integration boundary ν for various indices i, j, k, l (blue dots with error bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The numerical integration is done with the built-in “NIntegrate” function of Mathematica 12 which returns the estimated error of the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For ν ≲ 1/2 � 2¯l + 1 (gray, dashed) where ¯l is the mean of the 4 indices, the vertex can be described by the semiclassical vertex Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (B3) (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' By extending the integration boundaries to infinity the integral can be solved exactly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Vijkl = V∞ ijkl (orange line, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This becomes approximately a good approximation when ν ≳ ξ0(¯l) (gray, dashed), where ξ0(l) ≈ 2 √ l is the value above which the LL wavefunction is exponentially small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' ξ0(l) = min{ξ ∈ R: ∀x > ξ |ψl(x)| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='05}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Vν ijkl(0) = � ν −ν dξ1dξ2dξ3ψ∞ i (ξ1) ψ∞ j (ξ2) ψ∞ k (ξ3) ψ∞ l (ξ1 − ξ2 + ξ3) = ν3 8π2(ijkl)1/4 � ±(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l) e−i π 2 (±ii±jj±kk±ll) � ν −ν dξ1dξ2dξ3eiξ1(±i √ 2i±l √ 2l)eiξ2(±j √2j∓l √ 2l)eiξ3(±k √ 2k±l √ 2l) = π (ijkl)1/4 � ±(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l) e−i π 2 (±ii±jj±kk±ll)δ1/ν � ±i � i/2 ±l � l/2 � δ1/ν � ±j � j/2 ∓l � l/2 � δ1/ν � ±k √ 2k ±l √ 2l � (B3) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l≫1 ≈ 2π (ijkl)1/4 δ1/ν �√ 2i − √ 2l � δ1/ν � − � 2j + √ 2l � δ1/ν �√ 2k − √ 2l � cos �π 2 (i − j + k − l) � (B4) where δ1/ν(x) = ν π sin(νx) νx = 1 2π � ν −ν dξeixξ (B5) and the sum extends over the 16 terms arising from dif- ferent combinations of the signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (B3) as the semiclassical vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the limit where i, j, k, l ≫ 1 only the sum combinations (upper,lower,upper,lower)-sign and (lower,upper,lower,upper)-sign remain relevant and δ1/ν become effectively δ-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The vertex is hence di- agonal in the LLs Vν ijkl ∝ δi,j,k,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' When normalizing the asymptotic wavefunction inside the integration interval N 2 l = � ν −ν |ψ∞ l (ξ)|2dξ the entire prefactor for the interac- tion ℓB L VL/(2ℓB) llll (kx)/N 4 l = 1 approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This leads to an effective HK-Hamiltonian in the LL basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (14) in the semiclassical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Appendix C: Calculation of the LL vertex Vijkl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Introduction In the limit L ≫ ℓB the integral VL/(2ℓB) ijkl (kx) can be solved exactly for all indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The only important approx- 14 imation for this limit is the extension of the integration boundary to infinity L/(2ℓB) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' All dependencies on kx cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The vertex can be split up into two equivalent integrals Vijkl = V∞ ijkl(kx) = � ∞ −∞ dzIij(z)Ikl(−z) (C1) where Iij(z) = � ∞ −∞ dxψi(x)ψj(x + z) (C2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Properties of Iij Here, we list some useful properties of Iij Iij(0) = δij (C3a) Iij(z → ∞) = 0 (C3b) Iij(−z) = (−1)i+jIij(z) (C3c) Iji(z) = (−1)i+jIij(z) (C3d) which can be easily shown by using the properties of ψl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Most importantly the integral Iij can be solved exactly, the solution is Iij(z) = e−z2/4zj−i � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2j−i �j i � 1F1(−i, 1 + j − i, z2/2) (C4) for j ≥ i where 1F1 is Kummer’s (confluent hypergeo- metric) function of the first kind [58] 1F1(α, β, z) = ∞ � n=0 (α + n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (α − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (β − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (β + n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C5) Whereas this is in general not a helpful representation, we emphasize that in the above equation 1F1 is a sum over i terms and hence a polynomial in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C4) is derived below w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g for j ≥ i (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C3d)): Iij(z) = � dyψi(y − z/2)ψj(y + z/2) (C6) = e−z2/4 � π2i+ji!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' � dye−y2Hi(y − z/2) Hj(y + z/2) � �� � �j k=0 ( j k)Hk(y)zj−k (C7) = e−z2/4 � π2i+ji!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' i,j � k,k′=0 �i k �� j k′ � (−z)i−kzj−k′ � dye−y2Hk(y)Hk′(y) � �� � 2kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='√πδk,k′ (C8) = e−z2/4 � 2i+ji!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2izj−ii!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' i � k=0 (−1)k 2kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' � j i − k � z2k (C9) =e−z2/4zj−i � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='2j−i �j i � 1F1(−i, 1 + j − i, z2/2) (C10) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Properties of Vijkl Here, we list some useful properties of Vijkl: First, half of the integrals evaluate to 0 due to an odd integrand Vijkl = 0 for i + j + k + l odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C11) The permutative relations Vjikl = (−1)i+jVijkl (C12a) Vkjil = Vijkl (C12b) Vljki = (−1)i+lVijkl (C12c) Vikjl = (−1)j+kVijkl (C12d) Vilkj = Vijkl (C12e) Vijlk = (−1)k+lVijkl (C12f) 15 reduce the the number of independent tensor entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Most importantly, the integral can be evaluated exactly, the solution is for j ≥ i and l ≥ k Vijkl =(−1)k+l � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' � j i � � l k � 1F1 � −i, 1 + j − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' − d dc � 1F1 � −k, 1 + l − k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' − d dc � � − d dc �(j−i+l−k)/2 � 2π c ����� c=1 = √ 2π(−1)k+l � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' i,k � n,n′=0 (−1)n+n′ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' � j i − n �� l k − n′ �(2n + 2n′ + j − i + l − k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2n+n′+(j−i+l−k)/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C13) Note that 1F1 are finite polynomials and that j − i + l − k is even if and only if i + j + k + l is even (if odd Vijkl = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' By � − d dc �n we mean (−1)n dn dcn (the entire differential operator needs to be calculated first) and for calculation we may use that (−1)n dn dcn 1 √c = (2n−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2n where the double factorial !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' denotes a factorial over all numbers with the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For some indices Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C13) evaluates to simpler results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For equal indices Viikk = √ 2πLi � − d dc � Lk � − d dc � 1 √c ��� c=1 where Lk(x) are the Laguerre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This form simplifies to V00ll = √ 2 Γ(l+1/2) Γ(l+1) if one of the indices is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This result can be used to obtain an estimate of the scaling of the long range interaction between the LLs, since V00ll → � 2 l for l ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C13) each matrix element of Vijkl can be calculated exactly be evaluating the finite sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The numerical complexity increases for increasing indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In practice one has to be careful when performing the sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The summands have different signs and each of them is larger (in terms of its absolute value) then the total sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This renders all summands relevant and requires arbitrary precision floating point operations from the numerical side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C13) is derived below w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g for j ≥ i and l ≥ k (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (C12a)-(C12f)): Vijkl =(−1)k+l � dzIij(z)Ikl(z) (C14) =(−1)k+l � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' �j i ��l k � � ∞ −∞ dze−z2/2 �z2 2 � j−i+l−k 2 1F1(−i, 1 + j − i, z2/2)1F1(−k, 1 + l − k, z2/2) � �� � polynomials (C15) =(−1)k+l � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' �j i ��l k � � − d dc � j−i+l−k 2 1F1 � −i, 1 + j − i, − d dc � 1F1 � −k, 1 + l − k, − d dc � � ∞ −∞ dze−cz2/2 ����� c=1 (C16) =(−1)k+l � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' �j i ��l k � � − d dc � j−i+l−k 2 1F1 � −i, 1 + j − i, − d dc � 1F1 � −k, 1 + l − k, − d dc � � 2π c ����� c=1 (C17) (C5) = (−1)k+l � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' i,k � n,n′=0 (−1)n+n′ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' � j i − n �� l k − n′ � � − d dc �n+n′+ j−i+l−k 2 � 2π c ����� c=1 (C18) = √ 2π(−1)k+l � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' i,k � n,n′=0 (−1)n+n′ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' � j i − n �� l k − n′ �(2n + 2n′ + j − i + l − k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2n+n′+(j−i+l−k)/2 (C19) Appendix D: Short-time Fourier transformation The STFT is a method from Fourier analysis to deter- mine phase and frequency information for local sections of a signal changing over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The basic idea is to per- form several fast Fourier transformations of consecutive windows in the time domain to obtain the frequency for a segment in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We will explain how typical STFT plots of oscillating functions with time dependant frequencies look like by considering a test function g(t) = exp (if(t)t) where f(t) is a slowly varying function with respect to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We wish to evaluate the Fourier transform I(t0, ω) = � ∞ −∞ e−iωtg(t)w(t − t0) (D1) as function of frequency ω and time t0 and w(t) is any 16 windowing function which for proof of principle we choose to be a gaussian wσ(t) = e−t2/2/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The windowing function restricts the dominant part of integration re- gion to |t − t0|/σ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The vague statement of f be- ing a slowly varying function can be formulated in more rigorous terms: First, for |t − t0|/σ < 1 the Taylor ex- pansion f(t) = f(t0) + f ′(t0)(t − t0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Secondly, the oscillations are fast with respect to the width of the window ω/σ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Under these assumptions, which are met in our MC data as well as in possible experimental data, the in- tegral can be solved exactly by completing the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The main result is that I(t0, ω) is exponentially peaked at ωmax = f(t0) + t0f ′(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Therefore the STFT does not show f(t) directly but only its linear approximation inside each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' This can be used to efficiently re- construct f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Appendix E: LL interactions in the Hubbard model Here, we provide details for the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (21) and derive the LL vertex for the Hubbard model ˜Vijkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Obtaining the vertex We project the local Hubbard interaction in the LL basis ignoring its effects on the LL degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Hence, the calculation is similar to appendix A Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' We take L/ℓB → ∞ directly because the semiclassical limit is not of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' The interaction reads ˜U � y c† y,↑cy,↑c† y,↓cy,↓ = ˜U ℓ2 B � l1,l2,l3,l4 c† l1,↑cl2,↑c† l3,↓cl4,↓ � ∞ −∞ dy × ψl1 � y ℓB + ℓBkx � ψl2 � y ℓB + ℓBkx � × ψl3 � y ℓB + ℓBkx � ψl4 � y ℓB + ℓBkx � = ˜U ℓB � l1,l2,l3,l4 ˜Vl1l2l3l4c† l1,↑cl2,↑c† l3,↓cl4,↓ (E1) where the vertex is ˜Vl1l2l3l4 = � ∞ −∞ dξψl1 (ξ) ψl2 (ξ) ψl3 (ξ) ψl4 (ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (E2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Calculation of vertex ˜Vijkl We evaluate the LL vertex ˜Vijkl for the Hubbard model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (E2) exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' From the properties of ψl it is obvious that half of the entries are zero ˜Vijkl = 0 for i + j + k + l odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (E3) similar to the HK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Furthermore, the vertex is symmetric in each index pair ˜Vijkl = ˜Vjikl = ˜Vkjil = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' To solve the integral we use the series representation of the Hermite polynomials [59] Hl(x) = l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' ⌊l/2⌋ � n=0 (−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (l − 2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (2x)l−2n (E4) where ⌊x⌋ is the largest integer ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' For i + j + k + l even the vertex is 17 ˜Vijkl = � ∞ −∞ dξ 1 π � 2i+j+k+li!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' e−2ξ2Hi(ξ)Hj(ξ)Hk(ξ)Hl(ξ) (E5) = 1 π � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' ⌊i/2⌋,⌊j/2⌋,⌊k/2⌋,⌊l/2⌋ � n1,n2,n3,n4=0 (−2)−n1−n2−n3−n4 n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (i − 2n1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (j − 2n2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (k − 2n3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (l − 2n4)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' × � ∞ −∞ dξ(2ξ2)(i+j+k+l)/2−n1−n2−n3−n4e−2ξ2 (E6) = √i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' √ 2π ⌊i/2⌋,⌊j/2⌋,⌊k/2⌋,⌊l/2⌋ � n1,n2,n3,n4=0 (−2)−n1−n2−n3−n4 n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (i − 2n1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (j − 2n2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (k − 2n3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (l − 2n4)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' × � − d dc �(i+j+k+l)/2−n1−n2−n3−n4 1 √c ����� c=1 (E7) = 1 √ 2π � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 2i+j+k+l ⌊i/2⌋,⌊j/2⌋,⌊k/2⌋,⌊l/2⌋ � n1,n2,n3,n4=0 (−1)n1+n2+n3+n4(i + j + k + l − 2[n1 + n2 + n3 + n4] − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='n4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (i − 2n1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (j − 2n2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (k − 2n3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (l − 2n4)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' (E8) which is a series that can be computed exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' QO in the HK model with fixed particle number In the main text, we have concentrated on results for a fixed chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='9 we show the schematic effect of keeping the particle number fixed which will in- troduce small quantiative changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' non-inter- acting non-Onsager semi- classical HK model µ1(0) µ2(0) ℓB ∼ L U′ ≪ ωc ℓB ≪ L ℓB √ l ∼ L B = 0 1/B ϵF E U µ2(B) µ1(B) E2 E1 ϵF (U = 0) S2 S1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Schematic image of the DOS and the QO of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' the GS energy E1 + E2 for fixed particle number in 2D, whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 1 is for fixed chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' In the HK model at B = 0 momentum states are double occupied up to µ2(0) = µ − U and single occupied from µ2(0) to µ1(0) = ϵF where ϵF is the Fermi energy in the non-interacting limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Due to the constant DOS in 2D the interaction leads to a symmetric single occupied region around the Fermi energy, see inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' At finite magnetic field the pseudo Fermi energies become magnetic field dependent, but such that the total par- ticle number is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' Due to energetic constraints the drop of µ2(0) in the non-Onsager regime will be less pro- nounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFAT4oBgHgl3EQfxB7a/content/2301.08685v1.pdf'} +page_content=' 18 [1] K.' metadata={'source': 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b/QtAyT4oBgHgl3EQf7vqM/content/tmp_files/2301.00844v1.pdf.txt @@ -0,0 +1,3779 @@ +Understanding the main failure scenarios of subsea blowout preventers +systems: An approach through Latent Semantic Analysis +Gustavo Jorge Martins de Aguiar, Ramon Baptista Narcizo, Rodolfo Cardoso, Iara Tammela, +Edwin Benito Mitacc Meza, Danilo Colombo, Luiz Antônio de Oliveira Chaves, Jamile Eleutério +Delesposte +Federal Fluminense University, Rua Recife, Rio das Ostras, 28895-532, Rio de Janeiro, Brazil +Abstract +The blowout preventer (BOP) system is one of the most important well safety barriers during the +drilling phase because it can prevent the development of blowout events. This paper investigates +BOP system’s main failures using an LSA-based methodology. +A total of 1312 failure records +from companies worldwide were collected from the International Association of Drilling Contrac- +tors’ RAPID-S53 database. +The database contains recordings of halted drilling operations due +to BOP system’s failures and component’s function deviations. The main failure scenarios of the +components annular preventer, shear rams preventer, compensated chamber solenoid valve, and hy- +draulic regulators were identified using the proposed methodology. The scenarios contained valuable +information about corrective maintenance procedures, such as frequently observed failure modes, +detection methods used, suspected causes, and corrective actions. The findings highlighted that the +major failures of the components under consideration were leakages caused by damaged elastomeric +seals. The majority of the failures were detected during function and pressure tests with the BOP +system in the rig. This study provides an alternative safety analysis that contributes to under- +standing blowout preventer system’s critical component failures by applying a methodology based +on a well-established text mining technique and analyzing failure records from an international +database. +Keywords: +Blowout preventer, Offshore drilling, Failure analysis, Latent semantic analysis, +Maintenance procedures, Operations safety +1. Introduction +When the pressure in the underground reser- +voir exceeds the pressure applied by the drilling +fluid column, an uncontrolled flow of gas, oil, +or other well fluids occurs through the wellbore +and into the atmosphere or the sea, depend- +ing on whether the operation is subsea or sur- +face [1]. +This event is known as a blowout. +Though rare, blowouts are among the most +feared and violent accidents in the oil and +gas industry, posing a significant threat to as- +sets, human lives, and the environment [2]. +Blowouts in deepwater oil and gas well explo- +ration and development cause not only massive +environmental disasters and property losses but +also fatalities [3]. It is one of the major events +that contribute to the risks of offshore drilling +operations [4, 5]. +Blowouts risk alone con- +tributes 40 percent to 50 percent of the total +risk of loss of life, environmental impact, and +economic value loss for fixed platforms in the +North Sea [6]. +January 4, 2023 +arXiv:2301.00844v1 [cs.IR] 2 Jan 2023 + +The blowout preventer (BOP) system is one +of the most important well safety barriers dur- +ing the drilling phase because it can prevent +the development of blowout events by closing +and sealing the well [7]. +Due to the risks of +blowout events, blowout preventers are criti- +cal to the monitoring and maintenance of well +integrity and the safety of the crew and the +environment [8]. +The Gulf of Mexico, the +North Sea, offshore West Africa, and offshore +Brazil are experiencing increasing oil and gas +exploration and production from deepwater lo- +cations with water depths exceeding 300 me- +ters [9]. +Deepwater drilling activities utilize +subsea blowout preventers, and the distance be- +tween the BOP system and the drilling rig is +approximately three kilometers [10]. +Drilling +operations face operational and technological +challenges that are becoming increasingly com- +plex due to the extreme variables of the deep- +water drilling seabed environment, such as low +temperature and high pressure [11]. +A blowout preventer is a group of valves re- +motely controlled from the rig and serves as +the main barrier in well control in the event of +a blowout [7, 12]. While the primary form of +control in the drilling phase is hydrostatic pres- +sure, which is counterbalanced by the weight of +the drilling mud, the BOP system is considered +one of the last resources capable of prevent- +ing blowouts [13]. Blowout preventers are de- +signed to seal the well during emergency control +events and test and training scenarios to guar- +antee the system’s capability to perform the re- +quired function [14]. Recognized industry reg- +ulations and standard requirements have been +implemented as guidelines for testing strategies +that can be used to detect potential failures and +reduce blowout preventer system unavailabil- +ity [15–18]. Despite their importance in per- +forming the required function, the components +are primarily monitored through testing, and +numerous functional and pressure tests are re- +quired to ensure the safety barriers during op- +erations [19]. +The world realized the critical importance +of the BOP system after the Macondo well +blowout accident at the Deepwater Horizon +semi-submersible drilling rig in 2010 [7, 8, 20]. +The blowout resulted in 11 deaths, 4.9 million +barrels of crude oil spilled, and significant pol- +lution and ecological damage [8, 21]. +Aside +from the estimated $42 billion in losses, the +event raised concerns about the BOP system +and compelled the oil and gas industry to es- +tablish new maintenance and operating stan- +dards to improve BOP reliability [20, 22, 23]. +Following the accident, BOP risk analysis be- +came more critical in oil exploration and pro- +duction projects, and the public’s perception +of the risk of offshore drilling increased due to +the event’s consequences [24, 25]. However, risk +management practices have not kept up with +the growing complexity of drilling operations +(due to the industry’s expansion into deepwa- +ter areas), leading to more severe disasters [3]. +Failures in equipment and components of +complex systems are problems that have long +demanded engineering attention for different +purposes [26, 27]. +Deviations from the oper- +ating condition of equipment or systems can +cause not only financial damage but also loss +of human life and environmental damage [27]. +In this context, offshore drilling also requires +a high level of safety. +Offshore drilling rig +blowouts and explosions pose significant finan- +cial, environmental, and human life risks [24, +28, 29]. Failures in the blowout preventer can +affect the availability of the entire production +system because of the high risks [30]. Making +decisions in the face of indications of failure of +one of the BOP system’s components is one of +the industry’s most significant challenges [31]. +In many instances, when a BOP’s failure is +detected, the equipment must be taken out of +operation for repair, and the return to opera- +tion may take up to a week or more depending +on availability on board [20]. Furthermore, the +withdrawal of the blowout preventer stack and +the marine drilling riser system due to failure +January 4, 2023 + +is one of the most expensive downtime (non- +productive time) events in offshore drilling, and +millions of dollars that are lost each year due +to downtime can be avoided through improve- +ments in BOP system’s reliability [32]. +This +event becomes even more costly than suspen- +sion due to scheduled maintenance [33]. In [34], +subsea BOP system’s failures in the Gulf of +Mexico between 2007 and 2009 were investi- +gated, and the study collected 156 failures that +resulted in 13448 hours (560 days) of downtime. +Given that any problem or failure that requires +the withdrawal of the blowout preventer costs +approximately $1.2 million per day [23], a total +of $672.4 million was lost as a result of the 156 +failures. +Conducting BOP system’s failure studies +is critical in assisting oil and gas companies +in identifying which systems and components +are prone to failure and, as a result, provid- +ing knowledge about failure’s scenarios for use +in loss prevention decision-making [35]. +The +study of blowout preventer system failures re- +quires a systematic analysis of the BOP sys- +tem and an understanding of the main critical +failures and their consequences if a blowout oc- +curs [36]. +More and more companies in the oil and gas +industry are striving to collect data and cre- +ate databases about their operations to extract +knowledge and support future management de- +cisions [37]. +Failure records are essential be- +cause they are used as input for data analyt- +ics processes to improve industry safety, per- +formance, and knowledge [38]. While text de- +scriptions of the interventions written by the +maintenance technicians can be found in a +database of maintenance records, maintenance +optimization is frequently limited to informa- +tion about the time of maintenance interven- +tions due to the difficulty of automatically an- +alyzing texts [39]. Natural language efforts in +maintenance data pose a significant challenge +due to its unstructured nature and technical +language with particular terms [40]. +Few studies have examined maintenance text +records to understand equipment maintenance +interventions better and improve future deci- +sions. +In [41], a semi-supervised conditional +random field (CRF)-based information extrac- +tion approach is proposed to extracting in- +formation entities from bridge inspection re- +ports identifying existing deficiencies, and per- +form maintenance actions. In [42], a text min- +ing method is used to extract the most com- +mon product failure stories and their causes +of failure and regression analyses to look into +the links between consumer repair experiences +and future purchasing behaviors. +In [39], a +method is proposed to using textual informa- +tion to identify equipment degradation states +by clustering maintenance records and develop- +ing a stochastic multi-state degradation model +using a Convolutional Neural Network (CNN). +In [43], a text mining method is developed to +understand defects of a secondary device in +an intelligent substation that combines global +vectors for word representation (GloVe) and +attention-based bidirectional long short-term +memory (BiLSTM-Attention). In [44], a text +mining method is used to extracting informa- +tion about failure patterns in building systems +and components from CMMS databases on in- +teresting parts of the database containing work +orders about component’s failures. +In [45], +a hybrid natural language processing (hybrid- +NLP) algorithm is used to extract entities that +represent electrical equipment. In [46], a sus- +tainable fault diagnosis model based on imbal- +anced text mining and natural language pro- +cessing technology is used to extract fault fea- +ture words from field fault data. +In [47], a +method is developed to identifying root cause +factors by extracting root cause text from un- +structured data using a keyword extraction +method. +In [48], a deep learning method is +proposed to processing a power grid malfunc- +tion report that combines text data mining ori- +ented recurrent neural networks (RNN) with +long short-term memory (LSTM). In [49], a +January 4, 2023 + +text mining method is used to extract addi- +tional information reports and used an inte- +grated spatial-temporal approach, namely the +Geographically and Temporally Weighted Or- +dered Logisgression (GTWOLR) to model the +natural gas pipeline incident reports. In [50], +a new methodology based on a series of steps +is proposed to preprocess and decompose the +service history to identify relevant words and +sentences that distinguish an unhealthy wind +turbine from a healthy one. +This study aims to identify and analyze the +most common failure scenarios of the follow- +ing components of the BOP system: +annu- +lar, shear rams, regulator, and compensated +chamber solenoid valve. Furthermore, as well, +to determine how the failures were detected +and which maintenance actions were taken. +In order to accomplish this, latent semantic +analysis (LSA) was used to generate concepts +that represent topics based on BOP corrective +maintenance text records from an international +database known as RAPID-S53, which is de- +rived from the BOP Reliability Joint Industry +Project managed by the International Associa- +tion of Drilling Contractors (IADC). The LSA +was used in this paper because it can retrieve +multiple conceptual topics based on documents +with similar contexts, resulting in topic group- +ings of documents and terms [51, 52]. Given +the records’ context, these concepts can be in- +terpreted as failure scenarios incorporating do- +main expert knowledge. +Understanding the +scenarios is critical for acting to improve in- +dustry safety by reducing accident risks and in- +creasing equipment availability. +LSA has proven to be a valuable tool +for +analyzing +reviews, +products +feedbacks, +maintenance interventions reports, and other +textual records capable of assisting decision- +making [53]. +Customers were segmented and +analyzed by [54] using a combination of group +RFM analysis and probabilistic latent semantic +analysis models, and the results indicated that +the developed approach provides insight and +captures a wide range of customer preferences. +In [55], Latent Semantic Analysis, Text2Vec, +and Doc2Vec techniques are used to analyze +data from the Parkinson’s Progression Markers +Initiative (PPMI). In [56], the main accident +topics in a database of railroad equipment +accident descriptions maintained by the Fed- +eral Railroad Administration in the United +States are identified using LSA and LDA. +In [51], groups of contextually similar terms +from future-oriented data sources, including +experts’ and the general public’s concerns +about drone technology, are extracted using +the LSA. In [57], the LSA is used to extract +essential topics in a large group of paper +abstracts from the field of Multiple Criteria +Decision Making (MCDM), and they were able +to identify principle categories and significant +themes contained in the text. +The proposed research’s main contribution +is that the information gathered could as- +sist in the development of maintenance plan- +ning actions, reliability studies, and support +in identifying critical conditions for the well +drilling system’s shutdown decision. Further- +more, text mining results provide a rich addi- +tional source of data for future predictive an- +alytics workflows, and leveraging unstructured +data resources allows for more accurate predic- +tive models [38]. +The paper is structured as follows. Section +2 discusses the theoretical background and rel- +evant literature on accidents in the oil and gas +industry and the components and functions of +blowout preventer systems. Section 3 describes +the research methodology, which includes data +collection and the use of LSA. Sections 4 and +5 present the research findings and the discus- +sion. Section 6 presents the findings’ conclu- +sions and implications. Finally, Section 7 dis- +cusses the research’s limitations as well as sug- +gestions for further studies. +January 4, 2023 + +2. Theoretical background +2.1. Blowouts in offshore drilling +The demand for oil and gas sources has +increased in recent decades due to economic +growth, and oil and gas remain the primary en- +ergy resources [24]. Drilling is a crucial part +of the oil and gas industry, and its success +will be determined by its ability to improve +its processes’ operational reliability and avail- +ability significantly [58]. Deepwater drilling life +cycle phases are well planning, drilling, and +completion, with the drilling phase encompass- +ing many activities such as drilling, running +casing, cementing, circulation, fluid displace- +ment, and clean-up [59]. Given that deepwater +drilling entails complex operations that must +be completed in short periods and that errors +can cost tens of millions of dollars, balanc- +ing risks, schedules, and budgets is a difficult +task [59, 60]. +Because of the significant financial invest- +ment required and the fact that drilling wells +are hazardous operations, safety is a top pri- +ority, and the activities are strictly regulated +through regulations and laws [61]. +Accidents +during the drilling phase generally halt pro- +duction and harm the image of companies in- +volved, so measures to reduce the frequency +and severity of accidents are critical [62]. Fur- +thermore, as the industry approaches reservoirs +at deeper water depths and in more complex ge- +ological formations, drilling operations become +more complex, resulting in increased risks [3]. +The well control operation aims to keep the +well integrity by addressing the procedures to +be followed when formation fluids begin to flow +into the wellbore [2, 63]. Blowout is the uncon- +trolled flow of formation fluids to the surface, +and it is one of the major events that contribute +to the risks associated with offshore drilling [5]. +A blowout occurs when a kick (the flow of +formation fluid into the wellbore) is not dis- +covered early enough, or when safety barriers +in the well, such as blowout preventers, fail to +seal [2]. +When the formation pressure over- +comes the pressure exerted by the fluid column, +such as drilling fluid, a blowout occurs [16]. +Blowouts are an expensive and feared opera- +tional hazard in offshore drilling, causing de- +lays as well as fires, explosions, casualties, asset +damage, and environmental damage [6]. +Because of the severe consequences of deep- +water blowouts, scholars have conducted exten- +sive research on them [25]. April 20, 2010, well +blowout, also known as the Deepwater Horizon +accident, prompted questions about the safety +of deepwater drilling [8]. +The gas exploded +and caught fire on the rig’s deck, killing eleven +workers, and the blowout caused oil to flow for +two months from the damaged well [8]. This +environmental disaster affects local economies, +sensitive coastlines, and wildlife throughout the +Gulf region and still [64–66]. On August 16, +1984, a blowout occurred on the Enchova Cen- +tral oil rig in Brazil. The majority of the work- +ers on the platform were safely rescued, but 42 +people died due to a failure in the lifeboat low- +ering system, and six of them died while jump- +ing from a height of 30 to 40 meters into the +water [67]. A second blowout occurred on April +24, 1988, when the BOP could not seal the +well and attempts to prevent the event failed. +The platform was destroyed after a drill pipe +was forced out of the well and struck one of +the platform’s legs, causing sparks to ignite gas +from the blowout [62]. +The operators of the +Montara H1 oil and gas well lost control of the +well on August 21, 2009, resulting in a blowout. +There were no serious injuries or deaths due +to the incident, but uncontrolled hydrocarbons +spilled into the atmosphere for 75 days [61]. On +October 21, 1982, on Eugene Island Block 361, +well-control was lost in the Gulf of Mexico, and +a significant blowout and fire occurred. One or +more shallow gas sands with slightly abnormal +pressures charged the gas flow, which flowed +outside the drill pipe and upward through the +annular preventer, resulting in a full-scale gas +blowout [68]. +January 4, 2023 + +According to [62] analysis from data col- +lected since 1956, blowouts are historically the +most common type of accident in all world +regions. +In North America, 38 accidents in- +volving blowouts were recorded, accounting for +38% of all accident types in the region. The +same pattern was observed in Europe and for +North Sea operations, with 9 (28%) blowouts +recorded. +Blowouts were also the most com- +mon accident type in South America, Asia, and +Africa, accounting, respectively, for 29 (82.8%), +12 (46.2%), and 8 (57.2%). +This data em- +phasizes the dominance of blowouts over other +types of accidents during well drilling opera- +tions. +2.2. Blowout preventers system +A BOP is an electro-hydraulic system used to +seal, control and monitor oil and gas wells [7, +16]. It was built to handle high pressures and +prevent blowouts by sealing the well with an an- +nular preventer and shear rams preventers [63]. +The two main subsystems of the subsea BOP +system are the BOP stack and the control sys- +tem [15]. A blowout preventer stack is consti- +tuted of one or two annular preventers, three to +six ram preventers, two connectors (one con- +necting the BOP to the wellhead connector +and the other connecting the lower marine riser +package to the BOP), and four to ten choke +and kill valves, according to [18]. Drilling com- +panies use several annular preventers and ram +preventers as system redundancies to improve +the reliability of the BOP stack [7]. Figure 1 +shows a typical configuration of a subsea BOP +system. +One of the most critical barriers capable of +preventing blowouts during deepwater drilling +operations is the annular blowout preven- +ter [70]. +An annular preventer is a blowout +preventer that seals between the tube and the +wellbore or an open hole using a shaped seal- +ing elastomeric component [16]. The annular +preventer effectively maintains a seal around +the drill pipe even as it rotates during drilling, +but it is not as effective as ram preventers +Figure 1: +Typical configuration of a subsea BOP +stack [69]. +at sealing on an open hole, and it must be +capable of closing the wellbore to comply with +regulations [7]. +The steadily increasing wall thickness, be- +cause drilling ultradeep wells places significant +demands on the drill string, required to absorb +high tensile loads, has exceeded the capacity of +some shear rams to shear drill pipe successfully +in some cases [71]. Pipe rams, blind shear rams +(BSR), and casing shear rams (CSR) are the +three main types of ram preventers. Pipe rams +close around a drill pipe, restricting flow be- +tween the drill pipe and the wellbore [7]. While +blind shear rams are designed to cut the drill +string as the rams close off the well, casing +shear rams should be able to cut the casing as +well, and they should be able to effectively seal +January 4, 2023 + +CentralControlUnit +Drilling Rig +UpperAnnularPreventer +LMRPConnector +Subsea BluePod +SubseaYellowPod +LowerAnnularPreventer +BlindShearRam +PipeRam +PipeRam +PipeRam +WellheadConnectorthe hole against the flow of oil and gas in emer- +gencies such as potential blowouts [15]. Among +the various devices on the BOP, blind shearing +rams serve as the last line of defense, and two +BSRs are used in some deepwater BOP to in- +crease the chances of cutting the drill pipe and +sealing the well [72, 73]. +Although there are two types of subsea BOP +control systems, hydraulic and multiplexed +(MUX), the MUX system is the more recent +and widely used because as exploration depth +increased, problems with the reaction time of +umbilicals used in the hydraulic control sys- +tem were observed, and hydraulic lines control- +ling the pilot valves were replaced with sepa- +rate electric cables operating the compensated +chamber solenoid valves (CCSV) [18, 35]. Most +subsea blowout preventers used for deepwater +drilling are similar to those used for shallow- +water drilling, but the BOP is controlled by +a multiplex control system, which reduces the +time it takes for the BOP functions to acti- +vate [73]. +The blowout preventer MUX control sys- +tem comprises two subsystems: an electrical +system and a hydraulic system that includes +components such as pumps, valves, accumu- +lators, fluid storage and mixing equipment, +and a manifold [7, 15]. The drilling rig’s sur- +face control components include three com- +puters in the central control unit (CCU), the +driller’s station, and the toolpusher’s station, +as well as three programmable logic controllers +(PLC) [74, 75]. The CCU is the brain behind +the subsea BOP control system, and it uses the +PLCs to send commands from surface control +stations and panels to the subsea control point +of distribution (POD) [74]. Six fiber optic re- +peaters complete the communications between +the CCU and the blue and yellow subsea elec- +tronics modules (SEM) via two fully indepen- +dent subsea umbilical cables made up of optical +fibers and electrical wires to transmit signals +and power [75]. The blue and yellow SEMs are +designed to decode surface signals, and their +complete independence allows for a fully redun- +dant system for controlling all subsea functions +and communicating with the CCU [76]. The +decoded signal activates an electrical solenoid +valve in the subsea POD, which sends a hy- +draulic pilot signal to the appropriate func- +tion hydraulic valve, also known as subplate- +mounted valves (SPM) [76]. +Control points of distribution (PODs) are +essential to the performance of a subsea con- +trol system. Each POD contains the compen- +sated chamber solenoid valves, pressure trans- +ducers, subplate-mounted valves, pressure reg- +ulators, flowmeters, and hydraulic accumula- +tors [74, 77]. +PODs are identical and can +be mounted in either the blue or yellow loca- +tion [16]. Decoded signals operate the PODs +from the SEMs, but only one of them receives +hydraulic fluid for performing BOP functions, +making the operation of the other POD inef- +fective [78]. The CCSV then sends a hydraulic +signal to the corresponding SPM, which is ac- +tuated and sends pressurized hydraulic fluid to +the BOP system’s function [76]. +Hydraulic regulators, which reduce the pres- +sure in the supplied fluid through the rigid hy- +draulic line before entering the POD, are an- +other important component of the BOP con- +trol system [16, 18]. +The regulator’s spools +control sizes of orifices of supply, output, and +vent ports according to the pressure down- +stream and upstream the valve, but there are +two types of pressure regulators valves present +in the BOP control system [79]. The first type +is a manual regulating valve known as manual +koomey regulators (MKR) and has the desired +outlet pressure set on the surface using the ad- +justment nut before it is put into operation in +the BOP system [80]. The other is remotely ad- +justable regulators known as hydraulic koomey +regulators (HKR), which can be regulated re- +motely from the surface via the BOP system’s +control panel [79]. +Each POD contains an accumulator used in +the MUX system to store hydraulic fluid to be +January 4, 2023 + +used in case of hydraulic flow demands and the +loss of power supply to the pumps [18, 80]. The +accumulators in the BOP stack should provide +enough volume and pressure of an available hy- +draulic fluid to actuate the specified well con- +trol equipment and enough residual pressure to +maintain sealing capability [16, 81]. +To bet- +ter understand how accumulators operate, con- +sider an order to close the BOP ram. As pre- +sented before, the signal will be transmitted +from the central control unit (CCU) to the SEM +for decoding and then to the POD. The spe- +cific compensated chamber solenoid valve will +then open to performing the function, trigger- +ing the SPM valve to change position and al- +lowing high-pressure fluid stored in the accu- +mulator to pass through, closing the ram [78]. +BOP +components +are +mainly +monitored +through tests, and two of the most critical +tests performed on the BOP are the pressure +test and the function test [19]. A function test +is an operation of a BOP system’s component +that aims to verify its intended operation (that +it can do what it is intended to do) and must +be performed at least once a week [16]. Only +the ability to perform the function of a BOP +is tested when it is function tested, which +is insufficient to ensure the safety of drilling +operations because the capability of sealing +the wellbore is not guaranteed [5]. +Pressure +testing the BOP entails evaluating both the +capability to act the BOP function and seal off +a pressure, it must also be performed regularly, +with no more than 21 days between tests, and +it consists of two tests: the low-pressure test +and the high-pressure test [16]. +3. Methodology +The method was divided into three major +phases: +initial procedures, LSA procedures, +and post-LSA procedures. Each major phase +was made up of stages and smaller steps, and +the first phase, initial procedures, consisted of +collecting failure records and preparing data. +The second phase, LSA procedures, consists of +preprocessing, which prepares the failure de- +scriptions for analysis, and LSA processing of +the preprocessed records. The third phase is +the post-LSA procedures, which begin with +creating failure scenarios using the concepts- +terms and document-concepts matrices, fol- +lowed by the final validation of the MFS matrix +by domain experts. The method was applied +to BOP corrective maintenance text records +from the RAPID-S53 international database, +managed by the International Association of +Drilling Contractors (IADC). Figure 2 shows an +overview of the methodology of this research. +Figure 2: Overview of proposed methodology. +3.1. Data collection +The RAPID-S53 database used in this re- +search results from the BOP Reliability Joint +Industry Project, in which oil and gas ex- +ploration companies, drilling contractors, and +original equipment manufacturers (OEM) from +all over the world participated. The database +January 4, 2023 + +RAPID-S53 +Preprocessing +Database +procedures +LSA procedures +Preprocessed +records +Failures records +Initial procedures +collection +LSA processing +Data preparation +Concepts-Terms +Documents-Concepts +Matrix +Matrix +Failure records +Concepts Interpretation +Post LSA procedures +Domain +Main Failure +Experts +Scenarios Matrix +Domain experts +validation +Final MFS Matrixincludes global records of deviations in equip- +ment functions and drilling operations inter- +ruptions associated with the BOP system, and +it is maintained by the International Associa- +tion of Drilling Contractors (IADC). The main +objective of this database is to provide a large +amount of data that individual companies can +use to improve the reliability and efficiency of +well control equipment. +There are guidelines for the register of events +in the RAPID-S53 database. +It is necessary +to generate a report for each event in which a +component of the well control equipment was +considered to be malfunctioning. If no immedi- +ate physical corrective action is required to be +applied directly to a component in order for it +to operate as designed, the event does not need +to be reported. However, events during main- +tenance and testing should continue to be re- +ported, but only defects (failures) and not any +event related to preventive maintenance, such +as a component preventive replacement [82]. +The data was collected from RAPID-S53 to +a worksheet. There are records of failure events +from January 2012 to November 2018 from 26 +different companies, 6380 registers. Although +the registers recorded information related to +the failure event, such as the amount of us- +age at the time of failure, hours of repair time, +hours of non-productive time, this research fo- +cused on the event description to identify and +understand the main failure scenarios of the +BOP system. The failure texts are 448 terms +long on average, corresponding to two or three +paragraphs explaining the failure observed, the +components and parts affected, detection meth- +ods, and corrective actions. +3.2. Data preparation +Given the complexity of the blowout preven- +ter system, which is made up of several sub- +systems with numerous components and parts, +it was necessary to work with subsets of the +registers to improve the ability of the concepts +generated by the LSA approach to describe +the main failure scenarios for some components +perceived more critical for the BOP system. +In [5], data on subsea BOPs on the Gulf of +Mexico’s outer continental shelf for ten months, +from July 1997 to May 1998, is gathered and +the 117 failures recorded resulted in 3638 hours +of downtime (non-productive time). The ram +preventers account for 41% of the downtime, +with a total time loss of 1505 hours, the pri- +mary control system accounts for 28% (1021 +hours), and the annular preventer accounts for +9% (337 hours) of the downtime caused by +BOP’s failures. In [34], a reliability study that +observed subsea BOP’s failures in the Gulf of +Mexico is presented, and the data comprises +156 failures registered and 13448 hours of non- +productive time from 2007 to 2009. +Failure +events caused by the BOP control system, such +as regulator’s failure, solenoid valve’s failure, +and control fluid leak, were the primary con- +tributors, accounting for 35% of the unplanned +non-productive time, with a total time loss of +4712 hours. Annular preventers and ram pre- +venters are the second and fourth most signif- +icant sources of downtime events in [34], ac- +counting for 17% (2345 hours) and 13% (1766 +hours). +Key components of the blowout preventer +system were chosen for analysis using the data +presented. The data was segmented based on +the components identified in the failure records. +The latent semantic analysis was performed +on textual descriptions of failures of the fol- +lowing components: annular preventers, shear +ram preventers, compensated chamber solenoid +valves, and hydraulic regulators. It is critical +to note that the database does not differentiate +between blind shear ram and casing shear ram. +However, it was expected that the LSA applied +to the failure event description field would al- +low for the distinction of the two components’ +failure scenarios. Similarly, the component in- +dication as regulator is used in the RAPID- +S53 database to record failures of both the hy- +draulic koomey regulator (HKR) and the man- +January 4, 2023 + +ual koomey regulator (MKR). +The +RAPID-S53 +was +cut +into +smaller +datasets according to the components, and +the data associated with the components se- +lected was gathered in specific worksheets. The +tokenization of terms (unigrams, bigrams, and +trigrams) was then performed with a minimum +frequency of 2.5 percent of the total records for +each component worksheet to form an initial +bag-of-words (BoW). These BoWs were ana- +lyzed and used to assist in creating a dictionary +of terms containing bigrams and trigrams that +are common in a database containing technical +texts such as the RAPID-S53. The purpose is +that terms such as annular element, soak test, +seal plate, vent port, and weep hole, which +used to appear separately, will appear in the +concepts with the character underline joining +the two or three words that make up the term, +facilitating concept interpretation. +This list +was also used to create a synonym dictionary. +Figure 3 illustrates the relationship between +this study’s latent semantic analysis and data +preparation. +Figure 3: Overview of the research initial procedures. +Python, a programming language increas- +ingly being used in academic research, was used +in this study [83], was used in this study. An- +other tool used that has become popular in +the data science area is the Jupyter Notebook. +The NumPy, Matplotlib, Seaborn, Scikit-learn, +Pandas, Pandas-Profiling, NLTK, and SciPy li- +braries were used in this research [84–90]. +The specific worksheets correspond to a total +of 1312 failure records and 6565 hours of down- +time. Considering the previously stated costs of +downtime, the failures documented in the work- +sheets resulted in a monetary loss of more than +$300 million. +While annular component fail- +ures account for 247 records, contributing to +2778 hours of non-productive time, shear rams’ +failures account for 310 records, contributing +to 1706 hours of downtime. +Failures of the +regulator and CCSV, both components of the +BOP control system, correspond to 421 failures +(1121 hours of non-productive time) and 334 +failures (960 hours of non-productive time), re- +spectively. +3.3. Latent Semantic Analysis +Many models have been developed in re- +cent decades to understand the use of words +found in textual documents, which is a sig- +nificant challenge due to the various contexts +in which words can be used [91]. +Because a +text is an unstructured form of information +with high complexity, and because it is com- +posed of many words or terms (such as bigrams +and trigrams) combined to form different ideas +throughout the text, reducing the number of +terms by removing words that are not relevant +to the ideas presented in the text is essential +to making the analysis possible [92–94]. Work- +ing with thousands of dimensions can have a +negative impact on textual mining results, and +reducing the number of dimensions can improve +accuracy and efficiency without sacrificing sig- +nificant aspects of the data [95]. +The latent +semantic analysis (LSA), first introduced as +an information retrieval technique [96, 97] and +later defined as knowledge acquisition, induc- +tion, and representation theory [98], is a model +that can retrieve topics and ideas from texts +records with similar contexts [51]. +LSA is a well-known mathematical approach +for extracting and presenting textual data in a +January 4, 2023 + +Generate +Term-Document Matrix +Generate TF-IDF Matrix +Text preprocessing +All Documents +All Documents +Terms +Terms +Frequency +Relative Frequency +Failure +Records +(Corpus) +Main Failure +Apply SVD +Scenarios Matrix +Analyse Concepts +(post-LSA) +Singular +Interpretation +Values +Concepts +TF-IDF +Failure +Vaiues +Concepts +Matrix +Scenarios +Matrices +Failure Scenarios +Concepts +Matrices +Document-Concepts +Concepts-Terms +Matrix +Matrix +Concepts +Terms + Documents +Concepts +Documents +Terms +Loadings +Loadingssemantic structure [51, 96]. The mathematical +model is based on the idea that words with sim- +ilar meanings are more likely to occur in similar +contexts [56]. It acquires semantic knowledge +by utilizing word associations found in training +documents [96]. As a result, the system’s inter- +pretation of a word is determined by how the +training corpus uses language and how words +are associated in the training corpus [91]. The +model generates semantic representations for +words by analyzing the statistical pattern of +joint occurrence of words across the training +corpus [56]. +The semantic space constructed +from corpus documents contains semantic vec- +tors, also known as concepts or topics. Each +of them has a specific value associated with +each word in the set of documents, and this +value can be interpreted as that word’s con- +tribution to the formation of the specific con- +cept [97]. Thus, latent semantic analysis can +examine the relationships and similarities be- +tween documents and the terms (the compo- +sition of one or more words) contained within +those documents and identify the ideas (topics) +presented in texts [98]. +LSA consists of three major steps, accord- +ing to [51] and [57]. The first step is to create +a term frequency (TF) matrix, in which each +line represents a word or term, and each col- +umn represents a document or context, and +individual cell entries contain the frequency +with which a term occurs in a document [99]. +Some studies consider using the preprocess- +ing procedures as the first step to getting bet- +ter results [57]. +The frequency of terms is +then transformed to form a matrix of terms +and documents with transformed values known +as term frequency-inverse document frequency +(TF-IDF), reflecting how important a word +is to a document in a collection [53]. +Fi- +nally, to reduce the matrix’s dimensionality was +used the singular-value decomposition (SVD), +which is an eigenvector decomposition and fac- +tor analysis technique [96]. This division into +three major steps is quite similar to the divi- +sion presented by [99], which divides into four +major steps, the first three of which are iden- +tical to those presented and the last of which +is associated with information retrieval through +vector similarity. +Preprocessing steps, in general, involve fil- +tering out documents of interest to the analyst +and eliminating words and terms that are ir- +relevant [100]. +Several studies emphasize the +importance of preprocessing procedures in text +mining research to improve techniques used for +information retrieval and topics modeling [92– +94]. Removal of stopwords and stemming are +the most commonly used techniques for prepro- +cessing textual data [94]. Other essential pre- +processing techniques include removing unnec- +essary punctuation, converting letters to lower- +case, and lemmatization [92]. +Tokenization, the first activity of the text +preprocessing stage in this study, transforms +these texts into vectors composed of all of the +words and terms contained within them [93]. It +was possible to identify the essential terms that +will be considered in the preprocessing stage +and to build a dictionary of terms and syn- +onyms by using initial experimental tokeniza- +tion. Tokenization was very important for the +current research because many technical terms +and expressions, such as component names and +maintenance procedures, were found frequently +in RAPID-S53 database failure records. In this +study, unnecessary punctuation was removed, +letters were converted to lowercase, and stop- +words were removed from the registers in this +order. Following that, the texts were lemma- +tized with WordNet PoS tagging and the Word- +Net lemmatizer [101, 102]. According to some +studies, lemmatization, despite having a higher +computational cost, is more precise than stem- +mization and produces better results [103, 104]. +Another reason for using lemmatization rather +than stemming in this study is that stemming is +more likely to produce incorrect or non-existent +terms, which is exacerbated by the highly tech- +nical nature of the texts describing the BOP +January 4, 2023 + +system’s failures. +Following the preprocessing procedures is the +stage of constructing the matrix of terms and +documents and calculating the frequency of +each term presented in the dictionary built +for each failure record. +The frequency value +is then transformed to construct the TF-IDF +matrix, which reduces the influence of words +proportionally to their occurrence, given that +these words do not significantly contribute to +the understanding of the various topics in the +records [105]. According to [99], there are var- +ious methods for calculating the entries of the +TF-IDF matrix, and in this study, it was cal- +culated as +wij = tfij · idfi +(1) +where, +idfi = log +�1 + N +1 + ni +� ++ 1 +(2) +N is the total number of records in the corpus, +and ni is the number of records containing the +term indicated by index i. The idfi indicates +the rarity of occurrence of the term indicated by +index i in document j, and the higher the value +obtained, the rarer it is. The euclidean norm +is then applied to the resulting TF-IDF vectors +for each document. As a result, the final TF- +IDF vectors are denoted as vj and calculated +as follows. +vj = +wj +� +w1j2 + w2j2 + · · · + wnj2 +(3) +The ability of the singular values decomposi- +tion technique to reduce the dimensions of large +volumes of data to a more manageable number +without losing a significant amount of informa- +tion from the original variables is why LSA uses +it [105]. The SVD satisfies the requirement of +working with the TF-IDF matrix, which con- +tains a large volume of data [96]. +Mathematically, SVD decomposes the terms +and documents matrix or the TF-IDF matrix, +At×d, into the product of three other matri- +ces: Gt×m, an orthogonal matrix with m repre- +senting the dimensionality of the data, Sm×m, +a diagonal matrix with single values ordered in +descending order, and Dd×m, a transpose of the +column orthogonal matrix [96–98]. That is to +say, +At×d = Gt×m · Sm×m · (Dd×m)T +(4) +Where t denotes the number of terms and d +denotes the number of documents in the cor- +pus. The matrices are truncated in an arbitrary +number of concepts, denoted as k , to remove +the noise in the original matrix and thus extract +the semantic relations of the documents [106]. +The SVD result is the best k-dimensional ap- +proximation to the original matrix in terms of +the least square error [99]. +In the same latent semantic space created +by singular values decomposition, each term +and document is represented as a k-dimensional +vector [99]. Each n latent semantic concept, in +the interval I = [1, 2, . . . , k], is associated with +a set of values, known as loadings, for terms and +documents. So, the SVD generates two matri- +ces of concepts loadings, one for the words and +one for the documents. These terms and docu- +ment loadings can be used to interpret or label +each concept [51]. Figure 4 illustrates the LSA +stages and outputs. +The optimal number of concepts is an open +problem in science, and analyzing the graph of +the singular values of the concepts produced +is one of the most common methods for deter- +mining this number [105]. The logic goes that +the higher the singular value associated with +the concept, the better it can explain the data +variance [107]. In the context of failure records, +concepts with higher singular values almost cer- +tainly define the components’ main failure sce- +narios (which appeared more frequently in the +records). Visualizing the elbow in the graph or +when there are accelerating decreasing returns +is one possible criterion for defining the con- +cepts under consideration [105, 108]. +Several +studies have investigated the terms that con- +tribute the most to concepts (via term load- +ings) in conjunction with the singular values +technique to uncover topics hidden in textual +January 4, 2023 + +Figure 4: Overview of LSA and post-LSA procedures and outputs. +records or documents [109, 110]. Given the im- +possibility of analyzing all of the concepts gen- +erated by the SVD, using singular values as a +cut-off rule to define the number of concepts +that should be analyzed is an interesting solu- +tion. +Constructing scenarios involves concept in- +terpretation using the term loadings of the +Concept-Term (CT) matrix and the documents +loadings of the Document-Concept (DC) ma- +trix [52]. +The high-loading terms and docu- +ments were analyzed to identify failure scenar- +ios, which are composed of information regard- +ing corrective maintenance operations. Defin- +ing the appropriate term and document loading +threshold value is critical and can impact the +interpretation process, but there are no estab- +lished methods for determining the value [51]. +According to [111], +researchers must empiri- +cally determine the appropriate loading thresh- +old values for each scenario based on coherence. +Each concept may have a different threshold +value for its terms or documents, but high load- +ing values are required to ensure the scenario is +reasonably concrete [51]. +Given that the corpus used in this study was +composed of failure records texts, the generated +concepts’ high loading terms were highly tech- +nical. The majority of these terms were compo- +nent names, detection methods, observed fail- +ures, and corrective actions. So, the domain ex- +perts responsible for interpreting the concepts +and constructing the failure scenarios did a sim- +ple classification of the terms, which can sup- +port the construction of failure scenarios stage +as an initial step of the interpretation process. +Figure 5 shows an example of a high loading +terms classification for a theoretical concept. +The classification is not necessary for interpret- +ing the concepts, but it does help with interpre- +tation. +4. Results +To extract the main failure scenarios, the +maximum number of concepts, k, was empir- +ically set as ten since this number of concepts +is higher than the number of main failure sce- +narios expected by the authors for each BOP +component. It was then used the singular val- +January 4, 2023 + +6.5 +6.5 +6 +6 +5.5 +5.5 +Value + Annular +Value +5. + Blind Shear Ram +5. +4.5 +4.5 + Singular +4 +4 +3.5 +3 - +3 . +2.5 +2.5 +2 +C1 +C2 +C3 +C4 +C5 +C6 +C7 +C8 +C9 +C10 +C1 +C2 +C3 +C4 +C6 +C7 +C8 +C9 +C5 +C10 +LSA Concepts +LSA Concepts +7.5 +6.5 - +7 +6.5 +6 +6 +5.5 - +Value +5.5 + Regulator ++ Compensated Chamber Solenoid Valve +57 +5 +4.5 +ular +4.5 +4 +4. +3.5 +3 . +2.5 +2.5 - +2 +2. +C1 +C2 +C3 +C4 +C5 +C6 +C7 +C8 +C9 +C10 +C1 +C2 +C3 +C4 +C5 +C6 +C7 +C8 +C9 +C10 +LSA Concepts +LSA ConceptsFigure 5: Example of terms classification of a theoreti- +cal concept. +ues elbow graph combined with the high load- +ing terms and documents contextual analysis +to determine the number of concepts to be an- +alyzed by domain experts. Figure 6 shows sin- +gular values graphs for each BOP component +to the ten concepts with higher singular values. +Following the analyses, eight concepts asso- +ciated with BOP stack components were cho- +sen, four annular preventer’s concepts and four +shear ram preventer’s concepts. +As with the +BOP stack’s components, eight concepts re- +lated to BOP control system’ components have +been selected, five regulator’s concepts, and +three compensated chamber solenoid valve’s +concepts. When analyzing the concepts, it was +evaluated that a maximum of twenty-five terms +was sufficient for understanding each concept’s +failure scenario. All concepts are given names +(denoted by the combination of the component +name abbreviation and the number of the order +in which it was generated, for example, AC1 for +the annular preventer’s concept one), and their +highest loading terms are reported in Table 1 +and Table 2. +The concepts were independently interpreted +by four experts of BOP system’s functions, fail- +ures, and maintenance operations based on the +CT matrix terms loadings of the terms that +contribute the most to each concept and the +DC matrix highly related failure records, which +have high loadings. +The results were then +compared, and the interpretation was similar +for the majority of the scenarios constructed. +Some minor disagreements were solved through +discussion. The following failure scenarios rep- +resent the concepts’ final interpretation. +Scenario AC1: The annular’s leakage fail- +ure is the subject of this scenario. Aside from +the leakage failure, the records highly associ- +ated with this scenario show that scratches on +the annular piston were observed during inspec- +tions. +Some records described them as light +scratches, while others were more specific, stat- +ing that they did not exceed an acceptable +depth defined by the original equipment man- +ufacturer (OEM). The most affected compo- +nents are the head seals, chamber seals, piston +seals, and annular sealing element (packer el- +ement). +The leaks were caused by seal cuts, +pinching, and wear, according to the records. +Because the seals are made of elastomeric ma- +terial (rubber), seal damage and wear can re- +sult in fluid leaks due to leakage paths. There +appears to be no difference in the likelihood +of failure occurrence for both lower and up- +per annular preventers. Failures were detected +through surface pressure tests (the term surface +refers to the fact that the tests were performed +with the BOP in the rig). +According to the +records highly connected to this scenario, after +the leakage was noticed, the annular was disas- +sembled at the surface to investigate and iden- +tify the cause of the incident. The corrective +action is the installation of a new component +during maintenance. The observed failure may +affect the capability of the BOP system to seal +the well in an emergency. +Scenario AC2: The scenario is about pro- +trusion failure of the annular’s sealing spher- +ical element due to the passage of drill pipes +and their joints. +The annular element pro- +January 4, 2023 + +BOP +Annular +System, Subsystem, +and Components +Element +Piston +Seal +Pressure Test +Inspect +Corrective +Replace +action +Detection +Method +Damage +Leak +Observed +FailureFigure 6: Singular values graphs +Table 1: BOP Stack components related concepts and their highest loading terms +Concept +Singular +Values +Terms that contribute the most to each concept +Annular’s failures concepts +AC1 +6.304 +seal; +annular; +test; +pressure; +leak; +pressure_test; +element; +mainte- +nance; +BOP; piston; +new; +description; +fail; +upper_annular; +inspect; +lower_annular; root_cause; open; replace; instal; observe; close; damage. +AC2 +3.333 +element; fail; test; annular_element; root_cause; pressure_test; main- +tenance; +description; +BOP; +rubber; +upper_annular_element; +tool; +drill_pipe; attempt; cycle; joint; pull; pass; protrude; hold; closure; packer. +AC3 +2.840 +piston; +element; +score; +maintenance; +new; +description; +replace; +adapter_ring; inspect; damage; pit; annular_piston; cause; root_cause; +change; surface; sent; rubber; disassembly; annular_element; area; bore; +replacement; seal_area; swarf. +AC4 +2.509 +seal; lower_annular; roll; wellbore; upper; cap; o-ring; mud; housing; +adapter; replace; instal; fluid; pull; rubber; weep_hole; packer; element; +anti_extrusion; remove; leak; polypak_seal; adapter_ring. +Shear ram’s failures concepts +SRC1 +6.485 +seal; pressure; test; bonnet; leak; blind_shear_ram; door; BOP; operator; +open; maintenance; inspect; pressure_test; fail; close; o-ring; ram; new; +description; damage; remove; hinge; replace. +SRC2 +3.717 +bolt; blade; inspect; upper; crack; block; ram_block; low; maintenance; +non_destructive_test; shear_ram; fail; pressure_test; blind_shear_ram; +rubber; description; lateral_seal; torque; shear; end_of_well. +SRC3 +3.186 +pressure; pressure_test; ram; fail; low; rubber; shear_ram; hold; test; BOP; +blind_shear_ram; prior; attempt; wellbore; complete; packer; high; sur- +face; change; initial; cavity; good; cycle. +SRC4 +2.927 +door; hinge; aft; assembly; piston; forward; damage; pressure; year; body; +pressure_test; close_chamber; remove; notice; crew; poslock; pit; cham- +ber_test; plug; shear_ram; cavity; fwd; lock. +trudes into the wellbore area. +The test drill +pipe joints can wedge it because it is made +of an elastomeric material (rubber). +Failures +were discovered due to the impossibility of pass- +ing drill pipes and their joints or testing tools +through the annular element and pressure tests +performed during operation (subsea). The de- +tection of the failure through the impossibil- +ity of passing drill pipe joints with the annular +in operation was more common than through +pressure tests in the records highly associated +with this scenario. +The corrective action for +this scenario is the replacement of the compo- +nent during maintenance. +Scenario AC3: This scenario involves an- +nular’s piston damage such as pitting and scor- +ing (the terms, scuffing, and scratches were +used as synonyms too). Pitting failure is a type +of corrosion that results in oxidation marks on +the component. The scoring is usually found on +January 4, 2023 + +the upper section of the piston, where it comes +into contact with the adapter ring seals. Ac- +cording to some highly associated records, the +scoring occurrence is related to piston damage +that resulted in a raised edge between the pis- +ton and adapter ring. Other records indicated +that foreign material was found on the cham- +ber head. The majority of the failures were ob- +served during routine maintenance operations +through surface inspection with annular disas- +sembly. +The corrective action is the replace- +ment of the component during maintenance. +The drill’s rotation generates acceptable metal- +lic debris during drilling operations, and there +is also ferrous debris leftover from milling or +cutting the casing. +When drilling mud exits +the well and returns to the surface, rock debris, +metallic and ferrous particles pass through the +BOP stack, potentially entering the cavities of +the annular and shear ram preventers and dam- +aging parts such as the piston. A few records +related to this scenario indicated that the cause +of failure was that there is nothing to protect +the piston from scratches and also that the ma- +terial removed from scratches was found at the +top when the piston was returning to the open +position. +Scenario AC4: This scenario is about an- +nular’s leakage failure caused by rolling seals. +A leak path is formed when the seal rolls out +of the seal profile. According to failure records, +this failure occurs more frequently in the an- +nular cap seal. While some failure records ar- +gue that this failure is a recurring issue with +the cap seal and a design flaw, others claim +that failures occurred due to maintenance er- +rors, such as installing the incorrect seal. The +seal rolling identification by disassembling and +inspecting the annular after one pressure test- +ing with leaks appears frequent in the highly +associated failure records. Leakage frequently +occurs through a weep hole, which is a hole +placed downstream of a seal to open a leakage +path and accuse the seal of losing integrity. The +component replacement is the corrective action +for this scenario. +Scenario +SRC1: +This scenario involves +leakage failures caused by damaged seals, such +as o-rings, on the blind shear ram. Seal damage +and wear can result in leakage paths and, as a +result, fluid leakage. Failures were detected us- +ing functional tests and pressure tests, the ma- +jority of which were performed at the surface +and the stump test of the blind shear ram. Af- +ter visually detecting the leak, a common pro- +cedure is to bleed off the pressure and remove +the BSR door from the body, and then dam- +aged seals, such as pinched, and leakage paths +are observed. Some highly related records show +that the problem occurred more frequently with +backup o-rings and polypak seals, but other +records show that the failure can affect other +seals. According to one of these maintenance +records, the bonnet studs were coated with grit +and dirt around the cylinder head, with even +more debris on the inner piston. The correc- +tive action for this scenario is the component +replacement. Another common procedure ap- +pears to be the execution of chamber tests to +verify the BSR’s capability to perform its re- +quired function. The leak was more common +in the blind shear ram door or door hinge than +in others parts. +Scenario SRC2: The fractures or cracks in +the blind shear ram blade’s bolts are the sub- +jects of this scenario. +Failures were detected +by non-destructive tests and inspection of the +component. According to the highly associated +records, this failure is an ongoing issue, and it +probably is a design or material issue. The cor- +rective action is the replacement of the cracked +blade bolts. The presence of the term end of +well indicates that this failure scenario is typ- +ically detected at the ending of drilling opera- +tions procedures, such as surface tests. +Scenario SRC3: This scenario is about the +failure of blind shear ram’s seals, which affects +the component’s capability to maintain the +pressure levels required. High-pressure tests ex- +ecuted with the BOP system in the rig detected +January 4, 2023 + +the failures. A common procedure after detect- +ing a failure is to bleed off the pressure and +disassemble the BSR to investigate the cause. +According to the records, the failure was de- +tected only during high-pressure tests, not dur- +ing low-pressure tests that yielded successful re- +sults. In this scenario, the corrective action is +to replace the seals in the BSR. According to +some records, this failure was caused by a de- +sign flaw or a manufacturing error, while others +report it was caused by seal wear and tear. +Scenario SRC4: The leakage failure in the +blind shear ram’s doors, bonnets, and hinges +is the subject of this scenario. +Some highly +related records indicated that these failures oc- +curred due to problems with parts such as bolts +and seals. The records reported that the oc- +currence was caused by wear in the compo- +nents and parts, but it is worth noting that a +few pointed to maintenance errors as the root +cause of the failure. The majority of the fail- +ures were identified during routine maintenance +operations by performing a chamber test on the +BSR. In this scenario, the corrective action is to +replace the damaged components or parts dur- +ing maintenance. Another common procedure +appears to be the execution of chamber tests +to determine whether or not the component’s +leakage has been resolved. +Scenario RC1: This scenario is about leak- +age failure in the HKR and MKR regulators +due to damaged seals, such as o-rings. The ma- +jority of the failures were detected during sur- +face soak tests. After visually detecting a leak +coming from the vent tube, a common proce- +dure is to bleed off the pressure and disassemble +the regulator, where damaged seals and leak- +age paths were discovered. The corrective ac- +tion for this scenario is to replace or rebuild the +damaged component during maintenance using +a repair kit. +Finally, the regulator is tested, +and if no leaks are found, the maintenance is +completed. According to the strongly related +records, the failures were caused by wear and +tear in the regulators and their parts. +Scenario +RC2: This scenario describes a +failure caused by the HKR regulators’ inability +to reach and (or) maintain the required pres- +sure, despite increase and decrease commands +executed on the control panels. Highly related +failure records indicated that pressure fluctua- +tions above the required set pressure occurred. +Failures were detected during surface functional +tests as well as with the BOP in subsea op- +eration. +According to the records, after the +pressure oscillations were detected, the regu- +lator was disassembled to investigate possible +causes of the failure. The corrective action for +this scenario is the replacement or the repair +(overhaul) of the regulator during maintenance. +Wear and tear in components, such as o-rings, +was suspected as the cause of the failures. +Scenario RC3: This scenario involves reg- +ulator’s leakage failure in both the HKR and +the MKR. Failures were detected during func- +tional tests at the surface and soak tests, with +fluid leaking through the vent tube. This sce- +nario is similar to scenario RC1. These leaks +were detected by passing fluid through the weep +hole, which is a hole placed downstream of a +seal to open a leak path and accuse the spring +housing seal of failing. The vent hole and vent +tube are connected to the spring housing com- +ponent, and also any damage to the seals, such +as o-rings, can result in fluid entering the spring +housing and being detected as a leak when flow- +ing through the spring house seal’s weep hole. +The corrective action for this scenario is the re- +placement of the damaged components, such as +o-rings and spring housing seals, during main- +tenance. +Scenario RC4: This scenario is similar to +the scenario RC3. The differences between the +scenarios are due to the characteristics of the +leakage. The vast majority of the cases involved +drips. +Soak tests and surface function tests +were used to detect the failures. The compo- +nent replacement during maintenance was the +corrective action for this scenario. +Scenario +RC5: +This scenario describes +January 4, 2023 + +Table 2: BOP Control System components related concepts and their highest loading terms +Concept Singular +Values +Terms that contribute the most to each concept +Regulator’s failures concepts +RC1 +7.379 +regulator; leak; pressure; vent; note; maintenance; bop; blue_POD; instal; +yellow_POD; soak_test; test; remove; fluid; o_ring; replace; description; +valve; POD; new; observe; rebuilt; seal; oem. +RC2 +3.358 +pressure; maintenance; description; POD; increase; function_test; regu- +late; decrease; overhaul; surface; swap; subsea; maintain; yellow_POD; +pilot_pressure; fail; root_cause; set; check; spare; new; supply_regulator; +circuit; rov. +RC3 +3.092 +soak_test; +leak; +root_cause; +replace_new; +blue_POD; oem; +mainte- +nance; note; description; assembly; manifold_regulator; yellow_POD; +supply_regulator; +function; +spring_housing; +BOP; manual_regulator; +vent_tube; function_test; crew; spare; detect; start; weep_hole; con- +trol_POD. +RC4 +2.945 +valve; +vent; +constantly; +vent_tube; +manifold_regulator; +pressure; +soak_test; time; replace_new; function; root_cause; drip; pilot_regulator; +stop; notice; blue_POD; cycle; increase; range. +RC5 +2.755 +note; +pressure; +yellow_POD; +damage; +manifold_regulator; +o_ring; +shear_seal; +vent_port; +seal_plate; +crew; +blue_POD; soak_test; +re- +place_new; come; annular_regulator; internal; BOP; inspect; subsea; low; +revision; seal; component; regulate. +Compensated chamber solenoid valve’s failures concepts +SVC1 +6.738 +CCSV; leak; valve; solenoid; POD; function; blue; maintenance; note; +soak_test; remove; BOP; test; yellow_POD; replace; fluid; instal; descrip- +tion; observe; open; inspect; vent; new; rebuilt; o_ring. +SVC2 +3.999 +valve; +test; +maintenance; +pass; +built; +built_incorrectly; +event; +re- +ceive_oem_unused; pressure; straight; sustain; receive_oem; dedicate; +end_of_well; test_bench; oem; apply; prior; warehouse; seal; vent; de- +scription; root_cause; vent; fluid; damage. +SVC3 +3.578 +solenoid; shear_seal; seal_plate; common; troubleshoot; leak; BOP; re- +place_oem; multiple; disassemble; bench_test; test; solenoid_bank; reacti- +vation; report; cameron; fail; currently; rig; run; investigation; drain; wear; +receive; surface. +January 4, 2023 + +HKR regulator leakage caused by damaged +components such as shear seals, o-rings, and +seal plates. The failures were detected during +surface soak tests, with the leaking occurring +through the vent port. +Following the detec- +tion, the regulator was disassembled in order to +determine the cause of the failure. According +to some highly associated records, they found +damages in the shear seals, such as cut, ex- +truding, and nibbling o-rings, shear seals, and +scratches in the seal plate. The corrective ac- +tion for this scenario is to replace damaged +components during maintenance. +Scenario +SVC1: The failure of compen- +sated chamber solenoid valves due to leakage +is the subject of this scenario. Soak tests and +functional tests, both performed at the surface, +were used to detect the failures. According to +highly related records, leakage frequently oc- +curs in the vent tube. After visually detecting a +leak from the vent tube, a common procedure is +to remove and disassemble the CCSV for inves- +tigation. The immediate corrective action for +this scenario is to replace the component, which +can be a rebuilt or new valve, and test to see +if the failure has been resolved. +The records +with a high score in this scenario indicated a +variety of failure causes. They report that the +leaks were caused by wear and tear on the seal +plates and damage to the seals, including the +seal assembly, o-rings, and shear seals. Some of +the seal plates had been worn and scuffed. +Scenario SVC2: This scenario is similar to +scenario SVC1. According to the highly related +records, when pressure was applied, the valve +would not seal and allow fluid to pass straight +through to the vent. The tests were conducted +at the surface, probably near the well’s end or +before the event. +Scenario +SVC3: This scenario is about +CCSV’s leakage. Leaks were discovered during +functional tests performed at the surface due to +a stream of fluid passing through the common +vent or common drain on the solenoid bank. +According to the highly associated records, a +common procedure is troubleshooting to iden- +tify the specific compensated chamber solenoid +valve leaking and then disassembling to inves- +tigate. +Some reported finding scratches and +wear signals on the seal plate and shear seal +and leaks caused by wear and tear. In this sce- +nario, the corrective action is to replace the en- +tire CCSV or just the seals during maintenance. +5. Discussion +The failure scenarios reported in this re- +search evidenced that critical failure modes in +BOP system’s components can occur for vari- +ous reasons and can be detected using a vari- +ety of methods. Some scenarios were similar to +others, but they differed in minor ways, such +as the failures observed and detection methods +or detection conditions. For example, some an- +nular leakage failures occurred due to different +component problems, such as damaged head +seals, chamber seals, or piston seals, and these +failures were sometimes detected through pres- +sure tests and other times through functional +tests. +Despite the use of the singular values +elbow graph, domain experts were essential to +solve the problem of analyzing failure scenarios +that were very similar to others and concepts +that lacked essential parameters, such as fail- +ure modes, by helping to determine the cut-off +number of concepts to be considered for analy- +sis in this study. It is worth noting that some +of the excluded concepts had similar terms and +high loading failure records to the selected con- +cepts for analysis. They may represent a varia- +tion of the failure scenario, taking into account +a less frequent dimension, such as one different +detection method. +LSA results highlighted the scenarios of fail- +ure that were more frequent in the RAPID-S53 +non-planned corrective maintenance records of +the BOP system. Three or more failure scenar- +ios reported more frequently in the database +were found for the components under consid- +eration, as expected by the authors. Also was +January 4, 2023 + +possible to identify the affected systems, sub- +systems, components, detection methods used, +observed failures, and immediate corrective ac- +tions taken to solve the problem for all scenar- +ios. Aside from the fact that the components +interact with each other, findings show that +failures and maintenance actions differ depend- +ing on the component and even within the same +component due to the type of failure observed. +This characteristic of the results is most likely +due to the complexity of the blowout preven- +ter system and its components, which contain +a large number of subcomponents and parts. +To fully comprehend the findings of this +study, it is essential to assess the similarities +and differences between the failure scenarios +and the findings of other studies related to +the selected BOP system’s components. +An- +nular failure scenarios reported leakage failures +caused by damaged or rolled seals, annular’s +piston damage, and protrusion of the annu- +lar’s sealing spherical element. In [5], was ob- +served that six of the 12 annular preventer’s +failures collected in the study were internal +leakage (leakage through a closed annular) fail- +ures, while the other six were failures to fully +open. In a more recent study, out of a total of +24 annular preventer’s failures collected, failed +to fully open and internal leakage were the most +common failure modes, accounting for 15 of +the failures [112]. These observed major fail- +ure modes are present in the failure scenarios. +The internal leakage failure mode is included +in the leakage failure, and the protrusion of +the annular’s sealing spherical element, which +is part of scenario AC2, is equivalent to the fail- +ure to fully open. One internal leakage failure +collected in [112] had a similar context to sce- +nario AC3 because after opening the upper an- +nular preventer, some metal swarf was observed +stuck between the annular piston and the hous- +ing, and the corrective action was to remove the +piston, adaptor ring, and packing element and +replace all with new parts. In [34], many inter- +nal leakage failures that occurred through the +weep holes due to damaged seals, as seen in sce- +narios AC1 and AC4, were reported. They also +reported one undetailed annular failure discov- +ered during testing before running the BOP, +which was corrected by replacing the piston. +Similar to the failures in scenario AC3, this fail- +ure was probably discovered during inspections +of the disassembled annular preventer. +The detection method varies depending on +the failure observed, but in general, in annu- +lar preventer’s failure scenarios, surface pres- +sure tests and subsea pressure tests are the +most frequent, followed by inspection and sur- +face functional tests. Several failure records as- +sociated with the concepts reported using hy- +draulic tests in conjunction with the disassem- +bled annular inspection to identify the causes +of leakage failures, a fact which [34] also ob- +served. Most fully open failures observed in re- +liability studies were due to the inability to pass +drill pipes or testing tools through the annu- +lar spherical element, necessitating an increase +in tool load (over-pull), and leakage failures +are primarily identified through subsea pres- +sure tests [34, 112]. +Furthermore, the most +common maintenance corrective actions for an- +nular preventer’s failure scenarios were compo- +nent replacement, but in some cases, the entire +annular preventer was replaced. These main- +tenance procedures were observed for failures +collected in other reliability studies from 1999 +to 2019 [5, 34, 112], suggesting that having +a standby annular preventer stored is less ex- +pensive than the non-productive time costs of +awaiting repair affected components in some +cases. +The shear rams’ failures included both blind +shear ram’s and casing shear ram’s failures, but +the high loading terms and records for each con- +cept only included BSR’s failures. CSR’s fail- +ures were almost certainly reported much less +frequently than BSR’s failures, and this could +be examined further in a more focused study. +When the pressure within the drilling system +becomes uncontrollable, the blind shear ram in +January 4, 2023 + +a BOP system is the critical last line of de- +fense, and if the BSR is available on-demand, +a blowout will not occur [15]. Thus, the BSR +is a critical component to the safety of drilling +operations and the blowout preventer system. +Failure modes of leakage due to damaged seals, +damaged components such as doors and hinges, +fractures and crackings in the bolts, and in- +ability to maintain required pressure levels due +to damaged seals were all present in the blind +shear rams’ scenarios. The two previous relia- +bility studies [34, 112] collected and analyzed +a few blind shear ram’s failures, and when +the two studies were combined, a total of six +blind shear ram’s failures were gathered. Five +of the six failures were caused by leakages in +the BSRs, four of which were internal leakages +(through a closed ram), and one was an exter- +nal leakage. One internal leakage was reported +without the affected component or the causes +specified, so this failure could be included in +scenarios SRC1, SRC3, or SRC4, depending on +the real unreported causes and detection meth- +ods. Another internal leakage occurred due to +lateral t-seal failure, which allows for the exis- +tence of leakage paths and is compatible with +scenario SRC1. A failure in the bonnet seals +caused one other internal leakage, and the ex- +ternal leakage was caused by a failure in the +bonnet door seals, both of which are similar +to the highly related failures observed in sce- +nario SRC4. The fifth reported leakage failure +was leakage through the BSR EVO lock motor +while it was in the unlock position. The sixth +failure occurred when a BSR failed to close due +to the hydraulic hose not being connected to +the manifold, and the failure occurred during +BOP’s maintenance. +None of the previously +reported failures are compatible with SRC2, +which involves fractures or cracks in the shear +ram blade’ bolts. +Surface pressure tests are the most frequently +reported method of detecting BSR’s failures in +the blind shear ram’s preventer failure scenar- +ios, followed by inspection and surface func- +tional tests. The use of hydraulic tests to iden- +tify leaks, followed by disassembling the blind +shear ram’s for inspection to determine the +causes of the failures, is a common procedure +in the scenarios’ highly related failures. Three +of the five leakage failures identified in the +previous reliability studies [34, 112] were de- +tected during the BOP installation testing, one +through pressure tests and one during the exe- +cution of a function test. In three of the leakage +failures, the blowout preventer was pulled to re- +pair and replace the affected components, one +did not specify the corrective actions, and one +reported that the failure was corrected thirteen +days later when the BOP was on the rig for +other reasons. +The scenarios indicated more +failures in tests performed with the blowout +preventer in the rig and did not address the +need to pull the BOP from subsea to the rig. +However, the failures reported in the reliability +studies, in which the BOP was pulled, and the +drilling company lost productive time, high- +light the importance of the blind shear ram pre- +venter to drilling operations safety. +This study’s regulator’s failure scenarios in- +dicated leakage due to damaged seals, compo- +nent’s failures such as seal plates and spring +housing, and failures due to the regulator’s in- +ability to reach and (or) maintain the required +pressure. The findings, such as scenarios RC1 +and RC2 (which suggested seal wear and tear +as the cause of failure), support some scholars’ +perception [113] that were deteriorating shear +seals (i.e., through scoring, erosion) one of the +primary causes of fluid leakages in the regula- +tor component because a deteriorated seal on +either the supply or vent spool cannot isolate +the connecting paths. This fact strengthens the +link perceived in the scenarios between fluid +leakage and damaged seals. Regulator’s inter- +nal leaks are an important failure mode to the +BOP system safety because they are undesir- +able states that can impact the system’s ex- +pected life, operational availability, and over- +all system and environmental safety [113]. The +January 4, 2023 + +scenarios RC3, RC4, and RC5 indicated leak- +ages that can occur because of a deteriorated +seal, but the scenarios are compatible with +leakages caused by damages as cut, extrud- +ing, and nibbling seals. The scenario RC2 rep- +resents a potentially dangerous failure of the +BOP system because pressure oscillations cre- +ated or not damped by a pressure regulator +cause dynamic loading of neighboring compo- +nents that may not have been considered during +their design [113]. In some cases, the scenarios +RC1, RC3, and RC5 can cause pressure oscilla- +tions and the regulator’s inability to maintain +the required pressure, as indicated in RC2, but +this depends on the leakage characteristics. +Compensated chamber solenoid valves are an +essential component of the BOP control sys- +tem, and the failure scenarios SVC1, SVC2, and +SVC3 indicated that leakages were the most fre- +quently observed failures, with the differences +between the scenarios relying mainly on main- +tenance operations. +When an operator com- +mands the BOP system to perform a specific +function, a signal is sent from the central con- +trol unit to the SEM for decoding and then +to a specific compensated chamber solenoid +valve (placed in the POD and associated with +the desired function) that opens, causing an +SPM valve to change position and allowing +high-pressure fluid stored in the accumulator +to flow [78]. Because each function has its own +compensated chamber solenoid valve and some +functions are more critical to the BOP system +than others (e.g., blind shear ram close, annu- +lar preventer close), if the CCSV fails to trig- +ger the subplate-mounted valve, the function +affected defines the risk associated with the fail- +ure. Failures in compensated chamber solenoid +valves may result in an inability to command +the other components studied and, as a result, +perform their required functions. Leaks in con- +trol system components, such as solenoid valves +and regulators, increase operating costs, while +excessive leakage can lead to a loss of hydraulic +pressure in the control circuit [10]. +The majority of failures in regulator’s fail- +ure scenarios were detected via surface func- +tional tests or inspection, whereas the main de- +tection methods in CCSV scenarios were func- +tional tests and soak tests, both of which were +performed with the BOP in the rig. The pro- +cedure to inspect the component to investigate +the cause of failure after the leakage is detected +through hydraulic tests is also common in the +highly related failures of compensated chamber +solenoid valve and regulator’s scenarios. Thus, +the scenarios of the four components point to +this as a common procedure in the oil and gas +industry, at least for failures of the components +under consideration. +The findings indicated that tests were crit- +ical for detecting failures of the four compo- +nents studied and for the safety of the drilling +phase operations and activities, as tests pro- +cedures appear in the scenarios and high re- +lated failure records of the DC matrix. Even +though the textual failure records in RAPID- +S53 are generated by a wide range of drilling +contractors, each of which has its monitoring +procedures, the scenarios indicated specific pro- +cedures capable of detecting different failure +modes. These procedures are likely to be more +effective in monitoring the capability of stud- +ied components to perform their required func- +tions. +However, many failures were detected +during tests performed with the blowout pre- +venter in the rig, which presents a challenge +for the companies because these tests are ex- +pensive to monitor the BOP system condition. +When the blowout preventer is pulled to the +rig, drilling operations should be halted, and +the companies must bear the costs of retriev- +ing and maintaining the BOP system and non- +productive time for the entire rig. Another im- +portant question raised by the scenarios is the +effectiveness of the tests or other monitoring +procedures conducted with the blowout pre- +venter in subsea to monitor the capability of +the components studied in performing their re- +quired functions, and why the majority of fail- +January 4, 2023 + +ures were detected by tests executed while the +system was on the rig. If the pressure within +the drilling system becomes uncontrollable, and +the BOP is subsea with components such as the +annular preventer and shear ram preventer un- +available to close the wellbore on demand, the +entire rig is in danger. +6. Conclusion +This paper uses a text mining approach +based on the well-known LSA method to ex- +tract information and understand the main fail- +ure scenarios of four blowout preventer’s com- +ponents: +annular preventers, shear ram pre- +venters, regulators, and compensated chamber +solenoid valves. A total of 1312 failures and cor- +rective maintenance descriptions reported by +engineers and technicians in charge of BOP sys- +tem maintenance is examined in this research +to assess the most common failures, detection +methods, repair practices, causes of failures, +challenges that drilling companies face, and +their experience with BOP system’s failures for +each component. +At least three main failure +scenarios were found for each of the four BOP’s +components studied based on unstructured fail- +ure descriptions. +Although this study only +examined failure descriptions for four compo- +nents, it represents a step toward understand- +ing the BOP system’s main failure scenarios. +This research has both theoretical and prac- +tical potentialities. In an academic sense, un- +like previous research, this study focused solely +on textual descriptions of the failures, which +contained information such as failure mode, +causes, and detection method, and contributes +to an understanding of the main failures of the +BOP system. +Given the scarcity of research +on the most common blowout preventer fail- +ures and maintenance procedures, the failure +scenarios presented in this paper may be useful +for future studies. There is also a lack of infor- +mation available on BOP test procedures that +were more effective in detecting certain types of +failures. Scholars may use the scenarios to ex- +amine the maintenance procedures associated +with corrective maintenance, to support relia- +bility and safety analysis, and risk management +with information about the failures, or as an in- +put to develop theoretical models of BOP sys- +tem condition monitoring for the components +studied. +Furthermore, the findings indicating tests +performed with the blowout preventer in the +rig as a main detection method of the fail- +ures require extra attention since this reinforces +the importance of the recent efforts of schol- +ars, agencies, and oil and gas companies to in- +crease the reliability of the BOP system. Few +studies used text mining to analyze mainte- +nance descriptions (texts that are very techni- +cal and complex), and the approach described +in this paper has the potential to mitigate +the technical vocabulary problem inherent in +maintenance data. It produced promising re- +sults when analyzing unstructured failure de- +scriptions to identify the main failure scenar- +ios. +Through customization and adaptation, +the approach presented in this study can be +used to extract relevant information of main- +tenance procedures contained in databases of +others equipment. +Drilling companies and BOP manufactur- +ers can use the failure scenarios to improve +their services and products. Even though each +drilling company has its procedures for investi- +gating anomalies detected in the BOP system +during drilling activities, managers in charge +of blowout preventer systems can use the sce- +narios to support decisions when investigat- +ing anomalies on the components studied be- +cause they indicated detection methods that +are likely more effective in detecting specific +failures. Understanding the most common fail- +ure modes, detection procedures capable of +identifying them, and their causes can also +support engineers and technicians to develop +better maintenance procedures and reliability +analysis of the blowout preventer system. The +January 4, 2023 + +findings establish a level of relevance (the sin- +gular values) between each component’s failure +scenarios, allowing drilling contractors to focus +their efforts on monitoring the failures that oc- +cur more frequently. The level of relevance is +critical to reducing the costs of the activities +carried out during the drilling phase. The fail- +ure scenarios can provide a reliable assessment +of component failures to BOP manufacturers, +which can aid in designing better products or +services. Regulatory agencies, for example, can +use the findings to encourage oil and gas com- +panies to take action, which is critical for im- +proving industry safety by lowering accident +risks while increasing equipment availability. +The findings may prompt oil and gas compa- +nies to reconsider the testing procedures used +on BOP system’s components. The testing of +the blowout preventer system is critical to in- +creasing safety, as demonstrated by the scenar- +ios. +However, the more frequently the BOP +is tested, the lower the system’s availability. +Testing the system is detrimental not only to +its availability but also because each function +test, even if it does not overload the system, +accounts for one less function in the system’s +useful life. +If a component is designed to be +used 200 times, it will lose 1/200 of its use- +ful life for each function test. When examining +components over a long period, function tests +can significantly reduce their expected service +life. For example, the service life of an annu- +lar preventer is usually 500 pressurizations or +15 years for its top cover and housing [70, 114], +so if the component is tested at least once a +week, assuming that it was not pressurized for +anything except tests, the annular will have a +service life of about ten years. The annular’s +estimated service life is five years if one pres- +surization per week due to drilling activities +is included, three years if two pressurizations +are included, and so on. +This effect is wors- +ened when investigating the impact of pressure +testing, as it overloads the system. Therefore, +drilling companies must consider the effective- +ness of the procedures they use to investigate +anomalies detected in the BOP system because +they can reduce the service life of the compo- +nents to a fraction of what it should be. As +previously stated, the failure scenarios can as- +sist them in determining which procedures are +likely to be more effective. +7. Limitations +Although this study is a step forward in using +failure records textual data, it is worth noting +some limitations. The non-planned corrective +maintenance descriptions were added to the +RAPID-S53 through an open question on the +failure event registration form, and the descrip- +tions contain some subjectivity, even though +the text is highly technical. There is a differ- +ence in the quality of the records when con- +sidering the volume of information in the de- +scriptions, which varies according to experi- +ence and engagement in reporting all the de- +tails of the failures. +The entire investigation +has been reported in some descriptions, includ- +ing details such as failure mechanisms, proper +as input to discussions about condition-based +maintenance. However, there are others cases +with very brief descriptions of corrective main- +tenance, which is detrimental to the database +and research. These two types of records are +few and have little impact on LSA results, but +drilling companies must strive to improve the +engagement of the workers responsible for re- +porting failures so that the descriptions contain +more information than they have now. +Based on a well-known text mining tech- +nique, the analysis generated knowledge about +the most common failures and corrective main- +tenance procedures related to the BOP sys- +tem’s components studied. On the other hand, +because the failure records were only collected +from the RAPID-S53, the results obtained are +restricted to a single database. Besides that, +even though the database have reports from re- +gions across all globe, the majority of failure +records are from drilling companies working in +January 4, 2023 + +the United States part of the Gulf of Mexico, +and because of that, the results of the analy- +sis may not reflect the main failure modes in +all countries. While failure descriptions for an- +nular preventers and shear ram preventers in +the US part of the Gulf of Mexico correspond +to approximately 60-65 percent of the data ex- +amined, failure descriptions for regulators and +compensated chamber solenoid valves in the +same location correspond to approximately 80 +percent. +The scenarios were presented so that the +LSA generates concepts (from higher to lower +singular values), which can be related to the +frequency of occurrence in the textual failure +records. Some failure scenarios that appear as +second, third, or fourth may pose a greater risk +to the rig’s safety than failures indicated in the +first scenario of the components. +Thus, the +severity of the failure modes is not taken into +account by the latent semantic analysis. LSA +is carried out through systematic and mathe- +matical analysis, but there is human involve- +ment in interpreting the concepts, which in- +troduces subjectivity. The relevance of analyst +knowledge to the outcome of text mining using +the LSA approach cannot be underestimated. +This limitation was addressed through a revi- +sion conducted with a team of researchers, en- +gineers, and technical workers with oil and gas +industry experience. +8. Suggestions +The study’s findings shed light on the break- +down of failure events by components’ failures +modes, detection methods, corrective actions, +suspected causes, and the frequency with which +they occur. Future research could use the re- +sults of a scientifically based procedure as a +base for reliability studies in the BOP sys- +tem’s components under consideration, as well +as the development of theoretical models of +condition monitoring, maintenance procedure +assessment, risk management, and safety anal- +ysis. The industry regulations and standard re- +quirements have been developed and evolved to +improve the safety of the drilling phase [16, 18]. +However, the findings reinforce that oil and +gas companies, regulatory agencies, technology +institutes, and scholars must research intelli- +gent and practical solutions to improve BOP +system availability while maintaining or im- +proving safety. It is critical for the future of +drilling oil and gas companies to increase the +effectiveness of monitoring procedures by pur- +suing innovative ways to monitor the BOP sys- +tem, such as new technologies, techniques, and +approaches related to condition-based mainte- +nance of the components as other intelligent so- +lutions. These efforts are essential to increase +the availability of the blowout preventer sys- +tem, and the safety of the rig crew and the en- +vironment. +In [15], for example, a state-based approach +is proposed to analyze unavailability by incor- +porating BSR preventer testing activities into +a multiphase Markov process, and investigated +the effects of testing errors and delayed re- +pairs on the likelihood of component unavail- +ability. +In [30], an integrated a fault tree +analysis updated with a near real-time failure +database is used into the operational decision- +making process to reduce non-productive time +on drilling rigs due to complex failure prop- +agation within the BOP system. +In [70], a +high-fidelity method is proposed to monitor- +ing the stress of the annular preventer’s top +cover and housing that combines the theoret- +ical calculation method (TCM), finite-element +analysis (FEA), and stress testing experiment +(STE). In [115], an adaptive model-based ap- +proach is proposed to real-time condition mon- +itoring of annular preventer’s functions by com- +bining a first-principles model with in-field data +by adapting model coefficients, interpreted as +annular preventer’s health indicators. +These +are examples of recent studies that presented +new solutions for improving drilling operations’ +safety while increasing the availability of BOP +system’s components, both of which are criti- +January 4, 2023 + +cal to the oil and gas industry’s future and sus- +tainability. Efforts in this direction are likely to +improve maintenance management and indus- +try safety by reducing accident risks. +The discussion in this study can be ex- +panded by using cuttings from the database +to other important components of the blowout +preventer system, such as pipe rams and SPM +valves, to understand their main failure scenar- +ios. Furthermore, maintenance intelligent so- +lutions, such as the gathering of video reports, +have potential to increase the volume of infor- +mation described by technicians and engineers. +In this context, the challenge of a human expert +examining each service entry can rise substan- +tially. So, the ability of text mining methods, +such as the LSA, to reduce the dimensions of +large volumes of data to a more manageable +number without losing a significant amount of +information can be explored by scholars and oil +and gas companies to extract useful informa- +tion from textual descriptions of maintenance +procedures. +Acknowledgements +The authors wish to acknowledge the finan- +cial support of Brazil’s national oil company, +Petrobrás (No.ANP: 20741-5). +References +[1] N. Khakzad, F. Khan, N. 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Hat- +tab, Model-Based Health Monitoring of Annular +Blowout Preventers, SPE Drilling & Completion +34 (04) (2019) 458–467. doi:10.2118/197041-pa. +URL https://dx.doi.org/10.2118/197041-pa +January 4, 2023 + diff --git a/QtAyT4oBgHgl3EQf7vqM/content/tmp_files/load_file.txt b/QtAyT4oBgHgl3EQf7vqM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8afd4514c11179739cc77c4d61a2fcfa9593432 --- /dev/null +++ b/QtAyT4oBgHgl3EQf7vqM/content/tmp_files/load_file.txt @@ -0,0 +1,2507 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf,len=2506 +page_content='Understanding the main failure scenarios of subsea blowout preventers systems: An approach through Latent Semantic Analysis Gustavo Jorge Martins de Aguiar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Ramon Baptista Narcizo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Rodolfo Cardoso,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Iara Tammela,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Edwin Benito Mitacc Meza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Danilo Colombo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Luiz Antônio de Oliveira Chaves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Jamile Eleutério Delesposte Federal Fluminense University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Rua Recife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Rio das Ostras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 28895-532,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Rio de Janeiro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Brazil Abstract The blowout preventer (BOP) system is one of the most important well safety barriers during the drilling phase because it can prevent the development of blowout events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This paper investigates BOP system’s main failures using an LSA-based methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A total of 1312 failure records from companies worldwide were collected from the International Association of Drilling Contrac- tors’ RAPID-S53 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The database contains recordings of halted drilling operations due to BOP system’s failures and component’s function deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The main failure scenarios of the components annular preventer, shear rams preventer, compensated chamber solenoid valve, and hy- draulic regulators were identified using the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The scenarios contained valuable information about corrective maintenance procedures, such as frequently observed failure modes, detection methods used, suspected causes, and corrective actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The findings highlighted that the major failures of the components under consideration were leakages caused by damaged elastomeric seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The majority of the failures were detected during function and pressure tests with the BOP system in the rig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This study provides an alternative safety analysis that contributes to under- standing blowout preventer system’s critical component failures by applying a methodology based on a well-established text mining technique and analyzing failure records from an international database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Keywords: Blowout preventer, Offshore drilling, Failure analysis, Latent semantic analysis, Maintenance procedures, Operations safety 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Introduction When the pressure in the underground reser- voir exceeds the pressure applied by the drilling fluid column, an uncontrolled flow of gas, oil, or other well fluids occurs through the wellbore and into the atmosphere or the sea, depend- ing on whether the operation is subsea or sur- face [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This event is known as a blowout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Though rare, blowouts are among the most feared and violent accidents in the oil and gas industry, posing a significant threat to as- sets, human lives, and the environment [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blowouts in deepwater oil and gas well explo- ration and development cause not only massive environmental disasters and property losses but also fatalities [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It is one of the major events that contribute to the risks of offshore drilling operations [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blowouts risk alone con- tributes 40 percent to 50 percent of the total risk of loss of life, environmental impact, and economic value loss for fixed platforms in the North Sea [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' January 4, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='00844v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='IR] 2 Jan 2023 The blowout preventer (BOP) system is one of the most important well safety barriers dur- ing the drilling phase because it can prevent the development of blowout events by closing and sealing the well [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Due to the risks of blowout events, blowout preventers are criti- cal to the monitoring and maintenance of well integrity and the safety of the crew and the environment [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The Gulf of Mexico, the North Sea, offshore West Africa, and offshore Brazil are experiencing increasing oil and gas exploration and production from deepwater lo- cations with water depths exceeding 300 me- ters [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Deepwater drilling activities utilize subsea blowout preventers, and the distance be- tween the BOP system and the drilling rig is approximately three kilometers [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Drilling operations face operational and technological challenges that are becoming increasingly com- plex due to the extreme variables of the deep- water drilling seabed environment, such as low temperature and high pressure [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A blowout preventer is a group of valves re- motely controlled from the rig and serves as the main barrier in well control in the event of a blowout [7, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' While the primary form of control in the drilling phase is hydrostatic pres- sure, which is counterbalanced by the weight of the drilling mud, the BOP system is considered one of the last resources capable of prevent- ing blowouts [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blowout preventers are de- signed to seal the well during emergency control events and test and training scenarios to guar- antee the system’s capability to perform the re- quired function [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Recognized industry reg- ulations and standard requirements have been implemented as guidelines for testing strategies that can be used to detect potential failures and reduce blowout preventer system unavailabil- ity [15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Despite their importance in per- forming the required function, the components are primarily monitored through testing, and numerous functional and pressure tests are re- quired to ensure the safety barriers during op- erations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The world realized the critical importance of the BOP system after the Macondo well blowout accident at the Deepwater Horizon semi-submersible drilling rig in 2010 [7, 8, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The blowout resulted in 11 deaths, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='9 million barrels of crude oil spilled, and significant pol- lution and ecological damage [8, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Aside from the estimated $42 billion in losses, the event raised concerns about the BOP system and compelled the oil and gas industry to es- tablish new maintenance and operating stan- dards to improve BOP reliability [20, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Following the accident, BOP risk analysis be- came more critical in oil exploration and pro- duction projects, and the public’s perception of the risk of offshore drilling increased due to the event’s consequences [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' However, risk management practices have not kept up with the growing complexity of drilling operations (due to the industry’s expansion into deepwa- ter areas), leading to more severe disasters [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures in equipment and components of complex systems are problems that have long demanded engineering attention for different purposes [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Deviations from the oper- ating condition of equipment or systems can cause not only financial damage but also loss of human life and environmental damage [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In this context, offshore drilling also requires a high level of safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Offshore drilling rig blowouts and explosions pose significant finan- cial, environmental, and human life risks [24, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures in the blowout preventer can affect the availability of the entire production system because of the high risks [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Making decisions in the face of indications of failure of one of the BOP system’s components is one of the industry’s most significant challenges [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In many instances, when a BOP’s failure is detected, the equipment must be taken out of operation for repair, and the return to opera- tion may take up to a week or more depending on availability on board [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Furthermore, the withdrawal of the blowout preventer stack and the marine drilling riser system due to failure January 4, 2023 is one of the most expensive downtime (non- productive time) events in offshore drilling, and millions of dollars that are lost each year due to downtime can be avoided through improve- ments in BOP system’s reliability [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This event becomes even more costly than suspen- sion due to scheduled maintenance [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [34], subsea BOP system’s failures in the Gulf of Mexico between 2007 and 2009 were investi- gated, and the study collected 156 failures that resulted in 13448 hours (560 days) of downtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Given that any problem or failure that requires the withdrawal of the blowout preventer costs approximately $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='2 million per day [23], a total of $672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='4 million was lost as a result of the 156 failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Conducting BOP system’s failure studies is critical in assisting oil and gas companies in identifying which systems and components are prone to failure and, as a result, provid- ing knowledge about failure’s scenarios for use in loss prevention decision-making [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The study of blowout preventer system failures re- quires a systematic analysis of the BOP sys- tem and an understanding of the main critical failures and their consequences if a blowout oc- curs [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' More and more companies in the oil and gas industry are striving to collect data and cre- ate databases about their operations to extract knowledge and support future management de- cisions [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failure records are essential be- cause they are used as input for data analyt- ics processes to improve industry safety, per- formance, and knowledge [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' While text de- scriptions of the interventions written by the maintenance technicians can be found in a database of maintenance records, maintenance optimization is frequently limited to informa- tion about the time of maintenance interven- tions due to the difficulty of automatically an- alyzing texts [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Natural language efforts in maintenance data pose a significant challenge due to its unstructured nature and technical language with particular terms [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Few studies have examined maintenance text records to understand equipment maintenance interventions better and improve future deci- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [41], a semi-supervised conditional random field (CRF)-based information extrac- tion approach is proposed to extracting in- formation entities from bridge inspection re- ports identifying existing deficiencies, and per- form maintenance actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [42], a text min- ing method is used to extract the most com- mon product failure stories and their causes of failure and regression analyses to look into the links between consumer repair experiences and future purchasing behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [39], a method is proposed to using textual informa- tion to identify equipment degradation states by clustering maintenance records and develop- ing a stochastic multi-state degradation model using a Convolutional Neural Network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [43], a text mining method is developed to understand defects of a secondary device in an intelligent substation that combines global vectors for word representation (GloVe) and attention-based bidirectional long short-term memory (BiLSTM-Attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [44], a text mining method is used to extracting informa- tion about failure patterns in building systems and components from CMMS databases on in- teresting parts of the database containing work orders about component’s failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [45], a hybrid natural language processing (hybrid- NLP) algorithm is used to extract entities that represent electrical equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [46], a sus- tainable fault diagnosis model based on imbal- anced text mining and natural language pro- cessing technology is used to extract fault fea- ture words from field fault data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [47], a method is developed to identifying root cause factors by extracting root cause text from un- structured data using a keyword extraction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [48], a deep learning method is proposed to processing a power grid malfunc- tion report that combines text data mining ori- ented recurrent neural networks (RNN) with long short-term memory (LSTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [49], a January 4, 2023 text mining method is used to extract addi- tional information reports and used an inte- grated spatial-temporal approach, namely the Geographically and Temporally Weighted Or- dered Logisgression (GTWOLR) to model the natural gas pipeline incident reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [50], a new methodology based on a series of steps is proposed to preprocess and decompose the service history to identify relevant words and sentences that distinguish an unhealthy wind turbine from a healthy one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This study aims to identify and analyze the most common failure scenarios of the follow- ing components of the BOP system: annu- lar, shear rams, regulator, and compensated chamber solenoid valve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Furthermore, as well, to determine how the failures were detected and which maintenance actions were taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In order to accomplish this, latent semantic analysis (LSA) was used to generate concepts that represent topics based on BOP corrective maintenance text records from an international database known as RAPID-S53, which is de- rived from the BOP Reliability Joint Industry Project managed by the International Associa- tion of Drilling Contractors (IADC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The LSA was used in this paper because it can retrieve multiple conceptual topics based on documents with similar contexts, resulting in topic group- ings of documents and terms [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Given the records’ context, these concepts can be in- terpreted as failure scenarios incorporating do- main expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Understanding the scenarios is critical for acting to improve in- dustry safety by reducing accident risks and in- creasing equipment availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' LSA has proven to be a valuable tool for analyzing reviews, products feedbacks, maintenance interventions reports, and other textual records capable of assisting decision- making [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Customers were segmented and analyzed by [54] using a combination of group RFM analysis and probabilistic latent semantic analysis models, and the results indicated that the developed approach provides insight and captures a wide range of customer preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [55], Latent Semantic Analysis, Text2Vec, and Doc2Vec techniques are used to analyze data from the Parkinson’s Progression Markers Initiative (PPMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [56], the main accident topics in a database of railroad equipment accident descriptions maintained by the Fed- eral Railroad Administration in the United States are identified using LSA and LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [51], groups of contextually similar terms from future-oriented data sources, including experts’ and the general public’s concerns about drone technology, are extracted using the LSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [57], the LSA is used to extract essential topics in a large group of paper abstracts from the field of Multiple Criteria Decision Making (MCDM), and they were able to identify principle categories and significant themes contained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The proposed research’s main contribution is that the information gathered could as- sist in the development of maintenance plan- ning actions, reliability studies, and support in identifying critical conditions for the well drilling system’s shutdown decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Further- more, text mining results provide a rich addi- tional source of data for future predictive an- alytics workflows, and leveraging unstructured data resources allows for more accurate predic- tive models [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Section 2 discusses the theoretical background and rel- evant literature on accidents in the oil and gas industry and the components and functions of blowout preventer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Section 3 describes the research methodology, which includes data collection and the use of LSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Sections 4 and 5 present the research findings and the discus- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Section 6 presents the findings’ conclu- sions and implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Finally, Section 7 dis- cusses the research’s limitations as well as sug- gestions for further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' January 4, 2023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Theoretical background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blowouts in offshore drilling The demand for oil and gas sources has increased in recent decades due to economic growth, and oil and gas remain the primary en- ergy resources [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Drilling is a crucial part of the oil and gas industry, and its success will be determined by its ability to improve its processes’ operational reliability and avail- ability significantly [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Deepwater drilling life cycle phases are well planning, drilling, and completion, with the drilling phase encompass- ing many activities such as drilling, running casing, cementing, circulation, fluid displace- ment, and clean-up [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Given that deepwater drilling entails complex operations that must be completed in short periods and that errors can cost tens of millions of dollars, balanc- ing risks, schedules, and budgets is a difficult task [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Because of the significant financial invest- ment required and the fact that drilling wells are hazardous operations, safety is a top pri- ority, and the activities are strictly regulated through regulations and laws [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Accidents during the drilling phase generally halt pro- duction and harm the image of companies in- volved, so measures to reduce the frequency and severity of accidents are critical [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Fur- thermore, as the industry approaches reservoirs at deeper water depths and in more complex ge- ological formations, drilling operations become more complex, resulting in increased risks [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The well control operation aims to keep the well integrity by addressing the procedures to be followed when formation fluids begin to flow into the wellbore [2, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blowout is the uncon- trolled flow of formation fluids to the surface, and it is one of the major events that contribute to the risks associated with offshore drilling [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A blowout occurs when a kick (the flow of formation fluid into the wellbore) is not dis- covered early enough, or when safety barriers in the well, such as blowout preventers, fail to seal [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' When the formation pressure over- comes the pressure exerted by the fluid column, such as drilling fluid, a blowout occurs [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blowouts are an expensive and feared opera- tional hazard in offshore drilling, causing de- lays as well as fires, explosions, casualties, asset damage, and environmental damage [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Because of the severe consequences of deep- water blowouts, scholars have conducted exten- sive research on them [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' April 20, 2010, well blowout, also known as the Deepwater Horizon accident, prompted questions about the safety of deepwater drilling [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The gas exploded and caught fire on the rig’s deck, killing eleven workers, and the blowout caused oil to flow for two months from the damaged well [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This environmental disaster affects local economies, sensitive coastlines, and wildlife throughout the Gulf region and still [64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' On August 16, 1984, a blowout occurred on the Enchova Cen- tral oil rig in Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The majority of the work- ers on the platform were safely rescued, but 42 people died due to a failure in the lifeboat low- ering system, and six of them died while jump- ing from a height of 30 to 40 meters into the water [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A second blowout occurred on April 24, 1988, when the BOP could not seal the well and attempts to prevent the event failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The platform was destroyed after a drill pipe was forced out of the well and struck one of the platform’s legs, causing sparks to ignite gas from the blowout [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The operators of the Montara H1 oil and gas well lost control of the well on August 21, 2009, resulting in a blowout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' There were no serious injuries or deaths due to the incident, but uncontrolled hydrocarbons spilled into the atmosphere for 75 days [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' On October 21, 1982, on Eugene Island Block 361, well-control was lost in the Gulf of Mexico, and a significant blowout and fire occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' One or more shallow gas sands with slightly abnormal pressures charged the gas flow, which flowed outside the drill pipe and upward through the annular preventer, resulting in a full-scale gas blowout [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' January 4, 2023 According to [62] analysis from data col- lected since 1956, blowouts are historically the most common type of accident in all world regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In North America, 38 accidents in- volving blowouts were recorded, accounting for 38% of all accident types in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The same pattern was observed in Europe and for North Sea operations, with 9 (28%) blowouts recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blowouts were also the most com- mon accident type in South America, Asia, and Africa, accounting, respectively, for 29 (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='8%), 12 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='2%), and 8 (57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This data em- phasizes the dominance of blowouts over other types of accidents during well drilling opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blowout preventers system A BOP is an electro-hydraulic system used to seal, control and monitor oil and gas wells [7, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It was built to handle high pressures and prevent blowouts by sealing the well with an an- nular preventer and shear rams preventers [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The two main subsystems of the subsea BOP system are the BOP stack and the control sys- tem [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A blowout preventer stack is consti- tuted of one or two annular preventers, three to six ram preventers, two connectors (one con- necting the BOP to the wellhead connector and the other connecting the lower marine riser package to the BOP), and four to ten choke and kill valves, according to [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Drilling com- panies use several annular preventers and ram preventers as system redundancies to improve the reliability of the BOP stack [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Figure 1 shows a typical configuration of a subsea BOP system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' One of the most critical barriers capable of preventing blowouts during deepwater drilling operations is the annular blowout preven- ter [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' An annular preventer is a blowout preventer that seals between the tube and the wellbore or an open hole using a shaped seal- ing elastomeric component [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The annular preventer effectively maintains a seal around the drill pipe even as it rotates during drilling, but it is not as effective as ram preventers Figure 1: Typical configuration of a subsea BOP stack [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' at sealing on an open hole, and it must be capable of closing the wellbore to comply with regulations [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The steadily increasing wall thickness, be- cause drilling ultradeep wells places significant demands on the drill string, required to absorb high tensile loads, has exceeded the capacity of some shear rams to shear drill pipe successfully in some cases [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Pipe rams, blind shear rams (BSR), and casing shear rams (CSR) are the three main types of ram preventers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Pipe rams close around a drill pipe, restricting flow be- tween the drill pipe and the wellbore [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' While blind shear rams are designed to cut the drill string as the rams close off the well, casing shear rams should be able to cut the casing as well, and they should be able to effectively seal January 4, 2023 CentralControlUnit Drilling Rig UpperAnnularPreventer LMRPConnector Subsea BluePod SubseaYellowPod LowerAnnularPreventer BlindShearRam PipeRam PipeRam PipeRam WellheadConnectorthe hole against the flow of oil and gas in emer- gencies such as potential blowouts [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Among the various devices on the BOP, blind shearing rams serve as the last line of defense, and two BSRs are used in some deepwater BOP to in- crease the chances of cutting the drill pipe and sealing the well [72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Although there are two types of subsea BOP control systems, hydraulic and multiplexed (MUX), the MUX system is the more recent and widely used because as exploration depth increased, problems with the reaction time of umbilicals used in the hydraulic control sys- tem were observed, and hydraulic lines control- ling the pilot valves were replaced with sepa- rate electric cables operating the compensated chamber solenoid valves (CCSV) [18, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Most subsea blowout preventers used for deepwater drilling are similar to those used for shallow- water drilling, but the BOP is controlled by a multiplex control system, which reduces the time it takes for the BOP functions to acti- vate [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The blowout preventer MUX control sys- tem comprises two subsystems: an electrical system and a hydraulic system that includes components such as pumps, valves, accumu- lators, fluid storage and mixing equipment, and a manifold [7, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The drilling rig’s sur- face control components include three com- puters in the central control unit (CCU), the driller’s station, and the toolpusher’s station, as well as three programmable logic controllers (PLC) [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The CCU is the brain behind the subsea BOP control system, and it uses the PLCs to send commands from surface control stations and panels to the subsea control point of distribution (POD) [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Six fiber optic re- peaters complete the communications between the CCU and the blue and yellow subsea elec- tronics modules (SEM) via two fully indepen- dent subsea umbilical cables made up of optical fibers and electrical wires to transmit signals and power [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The blue and yellow SEMs are designed to decode surface signals, and their complete independence allows for a fully redun- dant system for controlling all subsea functions and communicating with the CCU [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The decoded signal activates an electrical solenoid valve in the subsea POD, which sends a hy- draulic pilot signal to the appropriate func- tion hydraulic valve, also known as subplate- mounted valves (SPM) [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Control points of distribution (PODs) are essential to the performance of a subsea con- trol system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Each POD contains the compen- sated chamber solenoid valves, pressure trans- ducers, subplate-mounted valves, pressure reg- ulators, flowmeters, and hydraulic accumula- tors [74, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' PODs are identical and can be mounted in either the blue or yellow loca- tion [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Decoded signals operate the PODs from the SEMs, but only one of them receives hydraulic fluid for performing BOP functions, making the operation of the other POD inef- fective [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The CCSV then sends a hydraulic signal to the corresponding SPM, which is ac- tuated and sends pressurized hydraulic fluid to the BOP system’s function [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Hydraulic regulators, which reduce the pres- sure in the supplied fluid through the rigid hy- draulic line before entering the POD, are an- other important component of the BOP con- trol system [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The regulator’s spools control sizes of orifices of supply, output, and vent ports according to the pressure down- stream and upstream the valve, but there are two types of pressure regulators valves present in the BOP control system [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The first type is a manual regulating valve known as manual koomey regulators (MKR) and has the desired outlet pressure set on the surface using the ad- justment nut before it is put into operation in the BOP system [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The other is remotely ad- justable regulators known as hydraulic koomey regulators (HKR), which can be regulated re- motely from the surface via the BOP system’s control panel [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Each POD contains an accumulator used in the MUX system to store hydraulic fluid to be January 4, 2023 used in case of hydraulic flow demands and the loss of power supply to the pumps [18, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The accumulators in the BOP stack should provide enough volume and pressure of an available hy- draulic fluid to actuate the specified well con- trol equipment and enough residual pressure to maintain sealing capability [16, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' To bet- ter understand how accumulators operate, con- sider an order to close the BOP ram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' As pre- sented before, the signal will be transmitted from the central control unit (CCU) to the SEM for decoding and then to the POD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The spe- cific compensated chamber solenoid valve will then open to performing the function, trigger- ing the SPM valve to change position and al- lowing high-pressure fluid stored in the accu- mulator to pass through, closing the ram [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP components are mainly monitored through tests, and two of the most critical tests performed on the BOP are the pressure test and the function test [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A function test is an operation of a BOP system’s component that aims to verify its intended operation (that it can do what it is intended to do) and must be performed at least once a week [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Only the ability to perform the function of a BOP is tested when it is function tested, which is insufficient to ensure the safety of drilling operations because the capability of sealing the wellbore is not guaranteed [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Pressure testing the BOP entails evaluating both the capability to act the BOP function and seal off a pressure, it must also be performed regularly, with no more than 21 days between tests, and it consists of two tests: the low-pressure test and the high-pressure test [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Methodology The method was divided into three major phases: initial procedures, LSA procedures, and post-LSA procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Each major phase was made up of stages and smaller steps, and the first phase, initial procedures, consisted of collecting failure records and preparing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The second phase, LSA procedures, consists of preprocessing, which prepares the failure de- scriptions for analysis, and LSA processing of the preprocessed records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The third phase is the post-LSA procedures, which begin with creating failure scenarios using the concepts- terms and document-concepts matrices, fol- lowed by the final validation of the MFS matrix by domain experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The method was applied to BOP corrective maintenance text records from the RAPID-S53 international database, managed by the International Association of Drilling Contractors (IADC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Figure 2 shows an overview of the methodology of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Figure 2: Overview of proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Data collection The RAPID-S53 database used in this re- search results from the BOP Reliability Joint Industry Project, in which oil and gas ex- ploration companies, drilling contractors, and original equipment manufacturers (OEM) from all over the world participated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The database January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='RAPID-S53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Preprocessing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Database ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='procedures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='LSA procedures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Preprocessed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Failures records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Initial procedures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='collection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='LSA processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Data preparation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Concepts-Terms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Documents-Concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Failure records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Concepts Interpretation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Post LSA procedures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Domain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Main Failure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Experts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Scenarios Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Domain experts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='validation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Final MFS Matrixincludes global records of deviations in equip- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='ment functions and drilling operations inter- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='ruptions associated with the BOP system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' and it is maintained by the International Associa- tion of Drilling Contractors (IADC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The main objective of this database is to provide a large amount of data that individual companies can use to improve the reliability and efficiency of well control equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' There are guidelines for the register of events in the RAPID-S53 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It is necessary to generate a report for each event in which a component of the well control equipment was considered to be malfunctioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' If no immedi- ate physical corrective action is required to be applied directly to a component in order for it to operate as designed, the event does not need to be reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' However, events during main- tenance and testing should continue to be re- ported, but only defects (failures) and not any event related to preventive maintenance, such as a component preventive replacement [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The data was collected from RAPID-S53 to a worksheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' There are records of failure events from January 2012 to November 2018 from 26 different companies, 6380 registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Although the registers recorded information related to the failure event, such as the amount of us- age at the time of failure, hours of repair time, hours of non-productive time, this research fo- cused on the event description to identify and understand the main failure scenarios of the BOP system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The failure texts are 448 terms long on average, corresponding to two or three paragraphs explaining the failure observed, the components and parts affected, detection meth- ods, and corrective actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Data preparation Given the complexity of the blowout preven- ter system, which is made up of several sub- systems with numerous components and parts, it was necessary to work with subsets of the registers to improve the ability of the concepts generated by the LSA approach to describe the main failure scenarios for some components perceived more critical for the BOP system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [5], data on subsea BOPs on the Gulf of Mexico’s outer continental shelf for ten months, from July 1997 to May 1998, is gathered and the 117 failures recorded resulted in 3638 hours of downtime (non-productive time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The ram preventers account for 41% of the downtime, with a total time loss of 1505 hours, the pri- mary control system accounts for 28% (1021 hours), and the annular preventer accounts for 9% (337 hours) of the downtime caused by BOP’s failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [34], a reliability study that observed subsea BOP’s failures in the Gulf of Mexico is presented, and the data comprises 156 failures registered and 13448 hours of non- productive time from 2007 to 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failure events caused by the BOP control system, such as regulator’s failure, solenoid valve’s failure, and control fluid leak, were the primary con- tributors, accounting for 35% of the unplanned non-productive time, with a total time loss of 4712 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Annular preventers and ram pre- venters are the second and fourth most signif- icant sources of downtime events in [34], ac- counting for 17% (2345 hours) and 13% (1766 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Key components of the blowout preventer system were chosen for analysis using the data presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The data was segmented based on the components identified in the failure records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The latent semantic analysis was performed on textual descriptions of failures of the fol- lowing components: annular preventers, shear ram preventers, compensated chamber solenoid valves, and hydraulic regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It is critical to note that the database does not differentiate between blind shear ram and casing shear ram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' However, it was expected that the LSA applied to the failure event description field would al- low for the distinction of the two components’ failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Similarly, the component in- dication as regulator is used in the RAPID- S53 database to record failures of both the hy- draulic koomey regulator (HKR) and the man- January 4, 2023 ual koomey regulator (MKR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The RAPID-S53 was cut into smaller datasets according to the components, and the data associated with the components se- lected was gathered in specific worksheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The tokenization of terms (unigrams, bigrams, and trigrams) was then performed with a minimum frequency of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 percent of the total records for each component worksheet to form an initial bag-of-words (BoW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These BoWs were ana- lyzed and used to assist in creating a dictionary of terms containing bigrams and trigrams that are common in a database containing technical texts such as the RAPID-S53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The purpose is that terms such as annular element, soak test, seal plate, vent port, and weep hole, which used to appear separately, will appear in the concepts with the character underline joining the two or three words that make up the term, facilitating concept interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This list was also used to create a synonym dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Figure 3 illustrates the relationship between this study’s latent semantic analysis and data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Figure 3: Overview of the research initial procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Python, a programming language increas- ingly being used in academic research, was used in this study [83], was used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' An- other tool used that has become popular in the data science area is the Jupyter Notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The NumPy, Matplotlib, Seaborn, Scikit-learn, Pandas, Pandas-Profiling, NLTK, and SciPy li- braries were used in this research [84–90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The specific worksheets correspond to a total of 1312 failure records and 6565 hours of down- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Considering the previously stated costs of downtime, the failures documented in the work- sheets resulted in a monetary loss of more than $300 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' While annular component fail- ures account for 247 records, contributing to 2778 hours of non-productive time, shear rams’ failures account for 310 records, contributing to 1706 hours of downtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures of the regulator and CCSV, both components of the BOP control system, correspond to 421 failures (1121 hours of non-productive time) and 334 failures (960 hours of non-productive time), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Latent Semantic Analysis Many models have been developed in re- cent decades to understand the use of words found in textual documents, which is a sig- nificant challenge due to the various contexts in which words can be used [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Because a text is an unstructured form of information with high complexity, and because it is com- posed of many words or terms (such as bigrams and trigrams) combined to form different ideas throughout the text, reducing the number of terms by removing words that are not relevant to the ideas presented in the text is essential to making the analysis possible [92–94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Work- ing with thousands of dimensions can have a negative impact on textual mining results, and reducing the number of dimensions can improve accuracy and efficiency without sacrificing sig- nificant aspects of the data [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The latent semantic analysis (LSA), first introduced as an information retrieval technique [96, 97] and later defined as knowledge acquisition, induc- tion, and representation theory [98], is a model that can retrieve topics and ideas from texts records with similar contexts [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' LSA is a well-known mathematical approach for extracting and presenting textual data in a January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Generate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Term-Document Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Generate TF-IDF Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Text preprocessing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='All Documents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='All Documents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Terms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Terms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Relative Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Failure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='(Corpus) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Main Failure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Apply SVD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Scenarios Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Analyse Concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='(post-LSA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Singular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Interpretation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Values ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='TF-IDF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Failure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Vaiues ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Scenarios ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Matrices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Failure Scenarios ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Matrices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Document-Concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Concepts-Terms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Terms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Documents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Documents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Terms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Loadings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='Loadingssemantic structure [51,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The mathematical model is based on the idea that words with sim- ilar meanings are more likely to occur in similar contexts [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It acquires semantic knowledge by utilizing word associations found in training documents [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' As a result, the system’s inter- pretation of a word is determined by how the training corpus uses language and how words are associated in the training corpus [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The model generates semantic representations for words by analyzing the statistical pattern of joint occurrence of words across the training corpus [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The semantic space constructed from corpus documents contains semantic vec- tors, also known as concepts or topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Each of them has a specific value associated with each word in the set of documents, and this value can be interpreted as that word’s con- tribution to the formation of the specific con- cept [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Thus, latent semantic analysis can examine the relationships and similarities be- tween documents and the terms (the compo- sition of one or more words) contained within those documents and identify the ideas (topics) presented in texts [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' LSA consists of three major steps, accord- ing to [51] and [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The first step is to create a term frequency (TF) matrix, in which each line represents a word or term, and each col- umn represents a document or context, and individual cell entries contain the frequency with which a term occurs in a document [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some studies consider using the preprocess- ing procedures as the first step to getting bet- ter results [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The frequency of terms is then transformed to form a matrix of terms and documents with transformed values known as term frequency-inverse document frequency (TF-IDF), reflecting how important a word is to a document in a collection [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Fi- nally, to reduce the matrix’s dimensionality was used the singular-value decomposition (SVD), which is an eigenvector decomposition and fac- tor analysis technique [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This division into three major steps is quite similar to the divi- sion presented by [99], which divides into four major steps, the first three of which are iden- tical to those presented and the last of which is associated with information retrieval through vector similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Preprocessing steps, in general, involve fil- tering out documents of interest to the analyst and eliminating words and terms that are ir- relevant [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Several studies emphasize the importance of preprocessing procedures in text mining research to improve techniques used for information retrieval and topics modeling [92– 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Removal of stopwords and stemming are the most commonly used techniques for prepro- cessing textual data [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Other essential pre- processing techniques include removing unnec- essary punctuation, converting letters to lower- case, and lemmatization [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Tokenization, the first activity of the text preprocessing stage in this study, transforms these texts into vectors composed of all of the words and terms contained within them [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It was possible to identify the essential terms that will be considered in the preprocessing stage and to build a dictionary of terms and syn- onyms by using initial experimental tokeniza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Tokenization was very important for the current research because many technical terms and expressions, such as component names and maintenance procedures, were found frequently in RAPID-S53 database failure records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In this study, unnecessary punctuation was removed, letters were converted to lowercase, and stop- words were removed from the registers in this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Following that, the texts were lemma- tized with WordNet PoS tagging and the Word- Net lemmatizer [101, 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to some studies, lemmatization, despite having a higher computational cost, is more precise than stem- mization and produces better results [103, 104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Another reason for using lemmatization rather than stemming in this study is that stemming is more likely to produce incorrect or non-existent terms, which is exacerbated by the highly tech- nical nature of the texts describing the BOP January 4, 2023 system’s failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Following the preprocessing procedures is the stage of constructing the matrix of terms and documents and calculating the frequency of each term presented in the dictionary built for each failure record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The frequency value is then transformed to construct the TF-IDF matrix, which reduces the influence of words proportionally to their occurrence, given that these words do not significantly contribute to the understanding of the various topics in the records [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to [99], there are var- ious methods for calculating the entries of the TF-IDF matrix, and in this study, it was cal- culated as wij = tfij · idfi (1) where, idfi = log �1 + N 1 + ni � + 1 (2) N is the total number of records in the corpus, and ni is the number of records containing the term indicated by index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The idfi indicates the rarity of occurrence of the term indicated by index i in document j, and the higher the value obtained, the rarer it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The euclidean norm is then applied to the resulting TF-IDF vectors for each document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' As a result, the final TF- IDF vectors are denoted as vj and calculated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vj = wj � w1j2 + w2j2 + · · · + wnj2 (3) The ability of the singular values decomposi- tion technique to reduce the dimensions of large volumes of data to a more manageable number without losing a significant amount of informa- tion from the original variables is why LSA uses it [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The SVD satisfies the requirement of working with the TF-IDF matrix, which con- tains a large volume of data [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Mathematically, SVD decomposes the terms and documents matrix or the TF-IDF matrix, At×d, into the product of three other matri- ces: Gt×m, an orthogonal matrix with m repre- senting the dimensionality of the data, Sm×m, a diagonal matrix with single values ordered in descending order, and Dd×m, a transpose of the column orthogonal matrix [96–98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' That is to say, At×d = Gt×m · Sm×m · (Dd×m)T (4) Where t denotes the number of terms and d denotes the number of documents in the cor- pus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The matrices are truncated in an arbitrary number of concepts, denoted as k , to remove the noise in the original matrix and thus extract the semantic relations of the documents [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The SVD result is the best k-dimensional ap- proximation to the original matrix in terms of the least square error [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In the same latent semantic space created by singular values decomposition, each term and document is represented as a k-dimensional vector [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Each n latent semantic concept, in the interval I = [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' , k], is associated with a set of values, known as loadings, for terms and documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' So, the SVD generates two matri- ces of concepts loadings, one for the words and one for the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These terms and docu- ment loadings can be used to interpret or label each concept [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Figure 4 illustrates the LSA stages and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The optimal number of concepts is an open problem in science, and analyzing the graph of the singular values of the concepts produced is one of the most common methods for deter- mining this number [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The logic goes that the higher the singular value associated with the concept, the better it can explain the data variance [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In the context of failure records, concepts with higher singular values almost cer- tainly define the components’ main failure sce- narios (which appeared more frequently in the records).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Visualizing the elbow in the graph or when there are accelerating decreasing returns is one possible criterion for defining the con- cepts under consideration [105, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Several studies have investigated the terms that con- tribute the most to concepts (via term load- ings) in conjunction with the singular values technique to uncover topics hidden in textual January 4, 2023 Figure 4: Overview of LSA and post-LSA procedures and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' records or documents [109, 110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Given the im- possibility of analyzing all of the concepts gen- erated by the SVD, using singular values as a cut-off rule to define the number of concepts that should be analyzed is an interesting solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Constructing scenarios involves concept in- terpretation using the term loadings of the Concept-Term (CT) matrix and the documents loadings of the Document-Concept (DC) ma- trix [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The high-loading terms and docu- ments were analyzed to identify failure scenar- ios, which are composed of information regard- ing corrective maintenance operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Defin- ing the appropriate term and document loading threshold value is critical and can impact the interpretation process, but there are no estab- lished methods for determining the value [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to [111], researchers must empiri- cally determine the appropriate loading thresh- old values for each scenario based on coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Each concept may have a different threshold value for its terms or documents, but high load- ing values are required to ensure the scenario is reasonably concrete [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Given that the corpus used in this study was composed of failure records texts, the generated concepts’ high loading terms were highly tech- nical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The majority of these terms were compo- nent names, detection methods, observed fail- ures, and corrective actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' So, the domain ex- perts responsible for interpreting the concepts and constructing the failure scenarios did a sim- ple classification of the terms, which can sup- port the construction of failure scenarios stage as an initial step of the interpretation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Figure 5 shows an example of a high loading terms classification for a theoretical concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The classification is not necessary for interpret- ing the concepts, but it does help with interpre- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Results To extract the main failure scenarios, the maximum number of concepts, k, was empir- ically set as ten since this number of concepts is higher than the number of main failure sce- narios expected by the authors for each BOP component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It was then used the singular val- January 4, 2023 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 6 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 Value Annular Value 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Blind Shear Ram 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 Singular 4 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 3 - 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 2 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C1 C2 C3 C4 C6 C7 C8 C9 C5 C10 LSA Concepts LSA Concepts 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 - 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 6 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 - Value 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 Regulator + Compensated Chamber Solenoid Valve 57 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 ular 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='5 - 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 LSA Concepts LSA ConceptsFigure 5: Example of terms classification of a theoreti- cal concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' ues elbow graph combined with the high load- ing terms and documents contextual analysis to determine the number of concepts to be an- alyzed by domain experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Figure 6 shows sin- gular values graphs for each BOP component to the ten concepts with higher singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Following the analyses, eight concepts asso- ciated with BOP stack components were cho- sen, four annular preventer’s concepts and four shear ram preventer’s concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' As with the BOP stack’s components, eight concepts re- lated to BOP control system’ components have been selected, five regulator’s concepts, and three compensated chamber solenoid valve’s concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' When analyzing the concepts, it was evaluated that a maximum of twenty-five terms was sufficient for understanding each concept’s failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' All concepts are given names (denoted by the combination of the component name abbreviation and the number of the order in which it was generated, for example, AC1 for the annular preventer’s concept one), and their highest loading terms are reported in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The concepts were independently interpreted by four experts of BOP system’s functions, fail- ures, and maintenance operations based on the CT matrix terms loadings of the terms that contribute the most to each concept and the DC matrix highly related failure records, which have high loadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The results were then compared, and the interpretation was similar for the majority of the scenarios constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some minor disagreements were solved through discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The following failure scenarios rep- resent the concepts’ final interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario AC1: The annular’s leakage fail- ure is the subject of this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Aside from the leakage failure, the records highly associ- ated with this scenario show that scratches on the annular piston were observed during inspec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some records described them as light scratches, while others were more specific, stat- ing that they did not exceed an acceptable depth defined by the original equipment man- ufacturer (OEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The most affected compo- nents are the head seals, chamber seals, piston seals, and annular sealing element (packer el- ement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The leaks were caused by seal cuts, pinching, and wear, according to the records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Because the seals are made of elastomeric ma- terial (rubber), seal damage and wear can re- sult in fluid leaks due to leakage paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' There appears to be no difference in the likelihood of failure occurrence for both lower and up- per annular preventers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures were detected through surface pressure tests (the term surface refers to the fact that the tests were performed with the BOP in the rig).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to the records highly connected to this scenario, after the leakage was noticed, the annular was disas- sembled at the surface to investigate and iden- tify the cause of the incident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The corrective action is the installation of a new component during maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The observed failure may affect the capability of the BOP system to seal the well in an emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario AC2: The scenario is about pro- trusion failure of the annular’s sealing spher- ical element due to the passage of drill pipes and their joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The annular element pro- January 4, 2023 BOP Annular System, Subsystem, and Components Element Piston Seal Pressure Test Inspect Corrective Replace action Detection Method Damage Leak Observed FailureFigure 6: Singular values graphs Table 1: BOP Stack components related concepts and their highest loading terms Concept Singular Values Terms that contribute the most to each concept Annular’s failures concepts AC1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='304 seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' annular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' leak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' mainte- nance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' piston;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' upper_annular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' inspect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' lower_annular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' root_cause;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' instal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' observe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' close;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' AC2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='333 element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' annular_element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' root_cause;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' main- tenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' rubber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' upper_annular_element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' tool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' drill_pipe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' attempt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' joint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pull;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' protrude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' closure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' packer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' AC3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='840 piston;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' score;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' maintenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' adapter_ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' inspect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' sent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' rubber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' disassembly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' annular_element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' bore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replacement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' seal_area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' swarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' AC4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='509 seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' lower_annular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' roll;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' wellbore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' upper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' cap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' o-ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' mud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' housing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' adapter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' instal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fluid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pull;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' rubber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' weep_hole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' packer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' anti_extrusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' remove;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' leak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' polypak_seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' adapter_ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Shear ram’s failures concepts SRC1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='485 seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' bonnet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' leak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' blind_shear_ram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' door;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' maintenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' inspect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' close;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' o-ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' ram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' damage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' remove;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' hinge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' SRC2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='717 bolt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' blade;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' inspect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' upper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' crack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' block;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' ram_block;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' low;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' maintenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' non_destructive_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' low;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' rubber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' shear_ram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' blind_shear_ram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' prior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' attempt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' wellbore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' complete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' packer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' high;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' sur- face;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' initial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' cavity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' good;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' SRC4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='927 door;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' hinge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' aft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' assembly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' piston;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' forward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' damage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' body;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' close_chamber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' remove;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' notice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' crew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' poslock;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' cham- ber_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' plug;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' shear_ram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' cavity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fwd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' trudes into the wellbore area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The test drill pipe joints can wedge it because it is made of an elastomeric material (rubber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures were discovered due to the impossibility of pass- ing drill pipes and their joints or testing tools through the annular element and pressure tests performed during operation (subsea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The de- tection of the failure through the impossibil- ity of passing drill pipe joints with the annular in operation was more common than through pressure tests in the records highly associated with this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The corrective action for this scenario is the replacement of the compo- nent during maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario AC3: This scenario involves an- nular’s piston damage such as pitting and scor- ing (the terms, scuffing, and scratches were used as synonyms too).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Pitting failure is a type of corrosion that results in oxidation marks on the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The scoring is usually found on January 4, 2023 the upper section of the piston, where it comes into contact with the adapter ring seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Ac- cording to some highly associated records, the scoring occurrence is related to piston damage that resulted in a raised edge between the pis- ton and adapter ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Other records indicated that foreign material was found on the cham- ber head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The majority of the failures were ob- served during routine maintenance operations through surface inspection with annular disas- sembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The corrective action is the replace- ment of the component during maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The drill’s rotation generates acceptable metal- lic debris during drilling operations, and there is also ferrous debris leftover from milling or cutting the casing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' When drilling mud exits the well and returns to the surface, rock debris, metallic and ferrous particles pass through the BOP stack, potentially entering the cavities of the annular and shear ram preventers and dam- aging parts such as the piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A few records related to this scenario indicated that the cause of failure was that there is nothing to protect the piston from scratches and also that the ma- terial removed from scratches was found at the top when the piston was returning to the open position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario AC4: This scenario is about an- nular’s leakage failure caused by rolling seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A leak path is formed when the seal rolls out of the seal profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to failure records, this failure occurs more frequently in the an- nular cap seal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' While some failure records ar- gue that this failure is a recurring issue with the cap seal and a design flaw, others claim that failures occurred due to maintenance er- rors, such as installing the incorrect seal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The seal rolling identification by disassembling and inspecting the annular after one pressure test- ing with leaks appears frequent in the highly associated failure records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Leakage frequently occurs through a weep hole, which is a hole placed downstream of a seal to open a leakage path and accuse the seal of losing integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The component replacement is the corrective action for this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario SRC1: This scenario involves leakage failures caused by damaged seals, such as o-rings, on the blind shear ram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Seal damage and wear can result in leakage paths and, as a result, fluid leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures were detected us- ing functional tests and pressure tests, the ma- jority of which were performed at the surface and the stump test of the blind shear ram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Af- ter visually detecting the leak, a common pro- cedure is to bleed off the pressure and remove the BSR door from the body, and then dam- aged seals, such as pinched, and leakage paths are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some highly related records show that the problem occurred more frequently with backup o-rings and polypak seals, but other records show that the failure can affect other seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to one of these maintenance records, the bonnet studs were coated with grit and dirt around the cylinder head, with even more debris on the inner piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The correc- tive action for this scenario is the component replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Another common procedure ap- pears to be the execution of chamber tests to verify the BSR’s capability to perform its re- quired function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The leak was more common in the blind shear ram door or door hinge than in others parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario SRC2: The fractures or cracks in the blind shear ram blade’s bolts are the sub- jects of this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures were detected by non-destructive tests and inspection of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to the highly associated records, this failure is an ongoing issue, and it probably is a design or material issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The cor- rective action is the replacement of the cracked blade bolts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The presence of the term end of well indicates that this failure scenario is typ- ically detected at the ending of drilling opera- tions procedures, such as surface tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario SRC3: This scenario is about the failure of blind shear ram’s seals, which affects the component’s capability to maintain the pressure levels required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' High-pressure tests ex- ecuted with the BOP system in the rig detected January 4, 2023 the failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A common procedure after detect- ing a failure is to bleed off the pressure and disassemble the BSR to investigate the cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to the records, the failure was de- tected only during high-pressure tests, not dur- ing low-pressure tests that yielded successful re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In this scenario, the corrective action is to replace the seals in the BSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to some records, this failure was caused by a de- sign flaw or a manufacturing error, while others report it was caused by seal wear and tear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario SRC4: The leakage failure in the blind shear ram’s doors, bonnets, and hinges is the subject of this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some highly related records indicated that these failures oc- curred due to problems with parts such as bolts and seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The records reported that the oc- currence was caused by wear in the compo- nents and parts, but it is worth noting that a few pointed to maintenance errors as the root cause of the failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The majority of the fail- ures were identified during routine maintenance operations by performing a chamber test on the BSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In this scenario, the corrective action is to replace the damaged components or parts dur- ing maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Another common procedure appears to be the execution of chamber tests to determine whether or not the component’s leakage has been resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario RC1: This scenario is about leak- age failure in the HKR and MKR regulators due to damaged seals, such as o-rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The ma- jority of the failures were detected during sur- face soak tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' After visually detecting a leak coming from the vent tube, a common proce- dure is to bleed off the pressure and disassemble the regulator, where damaged seals and leak- age paths were discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The corrective ac- tion for this scenario is to replace or rebuild the damaged component during maintenance using a repair kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Finally, the regulator is tested, and if no leaks are found, the maintenance is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to the strongly related records, the failures were caused by wear and tear in the regulators and their parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario RC2: This scenario describes a failure caused by the HKR regulators’ inability to reach and (or) maintain the required pres- sure, despite increase and decrease commands executed on the control panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Highly related failure records indicated that pressure fluctua- tions above the required set pressure occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures were detected during surface functional tests as well as with the BOP in subsea op- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to the records, after the pressure oscillations were detected, the regu- lator was disassembled to investigate possible causes of the failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The corrective action for this scenario is the replacement or the repair (overhaul) of the regulator during maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Wear and tear in components, such as o-rings, was suspected as the cause of the failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario RC3: This scenario involves reg- ulator’s leakage failure in both the HKR and the MKR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures were detected during func- tional tests at the surface and soak tests, with fluid leaking through the vent tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This sce- nario is similar to scenario RC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These leaks were detected by passing fluid through the weep hole, which is a hole placed downstream of a seal to open a leak path and accuse the spring housing seal of failing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The vent hole and vent tube are connected to the spring housing com- ponent, and also any damage to the seals, such as o-rings, can result in fluid entering the spring housing and being detected as a leak when flow- ing through the spring house seal’s weep hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The corrective action for this scenario is the re- placement of the damaged components, such as o-rings and spring housing seals, during main- tenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario RC4: This scenario is similar to the scenario RC3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The differences between the scenarios are due to the characteristics of the leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The vast majority of the cases involved drips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Soak tests and surface function tests were used to detect the failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The compo- nent replacement during maintenance was the corrective action for this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario RC5: This scenario describes January 4, 2023 Table 2: BOP Control System components related concepts and their highest loading terms Concept Singular Values Terms that contribute the most to each concept Regulator’s failures concepts RC1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='379 regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' leak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' note;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' maintenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' bop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' blue_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' instal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' yellow_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' soak_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' remove;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fluid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' o_ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' valve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' observe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' rebuilt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' oem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' RC2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='358 pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' maintenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' increase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' function_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' regu- late;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' decrease;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' overhaul;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' swap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' subsea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' maintain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' yellow_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pilot_pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' root_cause;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' check;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' spare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' supply_regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' rov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' RC3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='092 soak_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' leak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' root_cause;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replace_new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' blue_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' oem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' mainte- nance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' note;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' assembly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' manifold_regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' yellow_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' supply_regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' spring_housing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' manual_regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vent_tube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' function_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' crew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' spare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' detect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' start;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' weep_hole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' con- trol_POD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' RC4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='945 valve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' constantly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vent_tube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' manifold_regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' soak_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replace_new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' root_cause;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' drip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pilot_regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' stop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' notice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' blue_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' increase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' RC5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='755 note;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' yellow_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' damage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' manifold_regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' o_ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' shear_seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vent_port;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' seal_plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' crew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' blue_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' soak_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' re- place_new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' come;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' annular_regulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' internal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' inspect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' subsea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' low;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' revision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' regulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Compensated chamber solenoid valve’s failures concepts SVC1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='738 CCSV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' leak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' valve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' solenoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' maintenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' note;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' soak_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' remove;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' yellow_POD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' replace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fluid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' instal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' descrip- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' observe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' inspect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' rebuilt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' o_ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' SVC2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='999 valve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' maintenance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' built;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' built_incorrectly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' event;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' re- ceive_oem_unused;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' straight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' sustain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' receive_oem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' dedicate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' end_of_well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test_bench;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' oem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' apply;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' prior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' warehouse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' de- scription;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' root_cause;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' vent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fluid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' SVC3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='578 solenoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' shear_seal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' seal_plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' common;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' troubleshoot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' leak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' BOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' re- place_oem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' multiple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' disassemble;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' bench_test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' solenoid_bank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' reacti- vation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' report;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' cameron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' fail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' currently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' rig;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' run;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' investigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' drain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' wear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' receive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' January 4, 2023 HKR regulator leakage caused by damaged components such as shear seals, o-rings, and seal plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The failures were detected during surface soak tests, with the leaking occurring through the vent port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Following the detec- tion, the regulator was disassembled in order to determine the cause of the failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to some highly associated records, they found damages in the shear seals, such as cut, ex- truding, and nibbling o-rings, shear seals, and scratches in the seal plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The corrective ac- tion for this scenario is to replace damaged components during maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario SVC1: The failure of compen- sated chamber solenoid valves due to leakage is the subject of this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Soak tests and functional tests, both performed at the surface, were used to detect the failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to highly related records, leakage frequently oc- curs in the vent tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' After visually detecting a leak from the vent tube, a common procedure is to remove and disassemble the CCSV for inves- tigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The immediate corrective action for this scenario is to replace the component, which can be a rebuilt or new valve, and test to see if the failure has been resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The records with a high score in this scenario indicated a variety of failure causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' They report that the leaks were caused by wear and tear on the seal plates and damage to the seals, including the seal assembly, o-rings, and shear seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some of the seal plates had been worn and scuffed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario SVC2: This scenario is similar to scenario SVC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to the highly related records, when pressure was applied, the valve would not seal and allow fluid to pass straight through to the vent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The tests were conducted at the surface, probably near the well’s end or before the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scenario SVC3: This scenario is about CCSV’s leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Leaks were discovered during functional tests performed at the surface due to a stream of fluid passing through the common vent or common drain on the solenoid bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' According to the highly associated records, a common procedure is troubleshooting to iden- tify the specific compensated chamber solenoid valve leaking and then disassembling to inves- tigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some reported finding scratches and wear signals on the seal plate and shear seal and leaks caused by wear and tear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In this sce- nario, the corrective action is to replace the en- tire CCSV or just the seals during maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Discussion The failure scenarios reported in this re- search evidenced that critical failure modes in BOP system’s components can occur for vari- ous reasons and can be detected using a vari- ety of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some scenarios were similar to others, but they differed in minor ways, such as the failures observed and detection methods or detection conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' For example, some an- nular leakage failures occurred due to different component problems, such as damaged head seals, chamber seals, or piston seals, and these failures were sometimes detected through pres- sure tests and other times through functional tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Despite the use of the singular values elbow graph, domain experts were essential to solve the problem of analyzing failure scenarios that were very similar to others and concepts that lacked essential parameters, such as fail- ure modes, by helping to determine the cut-off number of concepts to be considered for analy- sis in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It is worth noting that some of the excluded concepts had similar terms and high loading failure records to the selected con- cepts for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' They may represent a varia- tion of the failure scenario, taking into account a less frequent dimension, such as one different detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' LSA results highlighted the scenarios of fail- ure that were more frequent in the RAPID-S53 non-planned corrective maintenance records of the BOP system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Three or more failure scenar- ios reported more frequently in the database were found for the components under consid- eration, as expected by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Also was January 4, 2023 possible to identify the affected systems, sub- systems, components, detection methods used, observed failures, and immediate corrective ac- tions taken to solve the problem for all scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Aside from the fact that the components interact with each other, findings show that failures and maintenance actions differ depend- ing on the component and even within the same component due to the type of failure observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This characteristic of the results is most likely due to the complexity of the blowout preven- ter system and its components, which contain a large number of subcomponents and parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' To fully comprehend the findings of this study, it is essential to assess the similarities and differences between the failure scenarios and the findings of other studies related to the selected BOP system’s components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' An- nular failure scenarios reported leakage failures caused by damaged or rolled seals, annular’s piston damage, and protrusion of the annu- lar’s sealing spherical element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [5], was ob- served that six of the 12 annular preventer’s failures collected in the study were internal leakage (leakage through a closed annular) fail- ures, while the other six were failures to fully open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In a more recent study, out of a total of 24 annular preventer’s failures collected, failed to fully open and internal leakage were the most common failure modes, accounting for 15 of the failures [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These observed major fail- ure modes are present in the failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The internal leakage failure mode is included in the leakage failure, and the protrusion of the annular’s sealing spherical element, which is part of scenario AC2, is equivalent to the fail- ure to fully open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' One internal leakage failure collected in [112] had a similar context to sce- nario AC3 because after opening the upper an- nular preventer, some metal swarf was observed stuck between the annular piston and the hous- ing, and the corrective action was to remove the piston, adaptor ring, and packing element and replace all with new parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [34], many inter- nal leakage failures that occurred through the weep holes due to damaged seals, as seen in sce- narios AC1 and AC4, were reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' They also reported one undetailed annular failure discov- ered during testing before running the BOP, which was corrected by replacing the piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Similar to the failures in scenario AC3, this fail- ure was probably discovered during inspections of the disassembled annular preventer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The detection method varies depending on the failure observed, but in general, in annu- lar preventer’s failure scenarios, surface pres- sure tests and subsea pressure tests are the most frequent, followed by inspection and sur- face functional tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Several failure records as- sociated with the concepts reported using hy- draulic tests in conjunction with the disassem- bled annular inspection to identify the causes of leakage failures, a fact which [34] also ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Most fully open failures observed in re- liability studies were due to the inability to pass drill pipes or testing tools through the annu- lar spherical element, necessitating an increase in tool load (over-pull), and leakage failures are primarily identified through subsea pres- sure tests [34, 112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Furthermore, the most common maintenance corrective actions for an- nular preventer’s failure scenarios were compo- nent replacement, but in some cases, the entire annular preventer was replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These main- tenance procedures were observed for failures collected in other reliability studies from 1999 to 2019 [5, 34, 112], suggesting that having a standby annular preventer stored is less ex- pensive than the non-productive time costs of awaiting repair affected components in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The shear rams’ failures included both blind shear ram’s and casing shear ram’s failures, but the high loading terms and records for each con- cept only included BSR’s failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' CSR’s fail- ures were almost certainly reported much less frequently than BSR’s failures, and this could be examined further in a more focused study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' When the pressure within the drilling system becomes uncontrollable, the blind shear ram in January 4, 2023 a BOP system is the critical last line of de- fense, and if the BSR is available on-demand, a blowout will not occur [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Thus, the BSR is a critical component to the safety of drilling operations and the blowout preventer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failure modes of leakage due to damaged seals, damaged components such as doors and hinges, fractures and crackings in the bolts, and in- ability to maintain required pressure levels due to damaged seals were all present in the blind shear rams’ scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The two previous relia- bility studies [34, 112] collected and analyzed a few blind shear ram’s failures, and when the two studies were combined, a total of six blind shear ram’s failures were gathered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Five of the six failures were caused by leakages in the BSRs, four of which were internal leakages (through a closed ram), and one was an exter- nal leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' One internal leakage was reported without the affected component or the causes specified, so this failure could be included in scenarios SRC1, SRC3, or SRC4, depending on the real unreported causes and detection meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Another internal leakage occurred due to lateral t-seal failure, which allows for the exis- tence of leakage paths and is compatible with scenario SRC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A failure in the bonnet seals caused one other internal leakage, and the ex- ternal leakage was caused by a failure in the bonnet door seals, both of which are similar to the highly related failures observed in sce- nario SRC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The fifth reported leakage failure was leakage through the BSR EVO lock motor while it was in the unlock position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The sixth failure occurred when a BSR failed to close due to the hydraulic hose not being connected to the manifold, and the failure occurred during BOP’s maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' None of the previously reported failures are compatible with SRC2, which involves fractures or cracks in the shear ram blade’ bolts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Surface pressure tests are the most frequently reported method of detecting BSR’s failures in the blind shear ram’s preventer failure scenar- ios, followed by inspection and surface func- tional tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The use of hydraulic tests to iden- tify leaks, followed by disassembling the blind shear ram’s for inspection to determine the causes of the failures, is a common procedure in the scenarios’ highly related failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Three of the five leakage failures identified in the previous reliability studies [34, 112] were de- tected during the BOP installation testing, one through pressure tests and one during the exe- cution of a function test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In three of the leakage failures, the blowout preventer was pulled to re- pair and replace the affected components, one did not specify the corrective actions, and one reported that the failure was corrected thirteen days later when the BOP was on the rig for other reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The scenarios indicated more failures in tests performed with the blowout preventer in the rig and did not address the need to pull the BOP from subsea to the rig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' However, the failures reported in the reliability studies, in which the BOP was pulled, and the drilling company lost productive time, high- light the importance of the blind shear ram pre- venter to drilling operations safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This study’s regulator’s failure scenarios in- dicated leakage due to damaged seals, compo- nent’s failures such as seal plates and spring housing, and failures due to the regulator’s in- ability to reach and (or) maintain the required pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The findings, such as scenarios RC1 and RC2 (which suggested seal wear and tear as the cause of failure), support some scholars’ perception [113] that were deteriorating shear seals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=', through scoring, erosion) one of the primary causes of fluid leakages in the regula- tor component because a deteriorated seal on either the supply or vent spool cannot isolate the connecting paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This fact strengthens the link perceived in the scenarios between fluid leakage and damaged seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Regulator’s inter- nal leaks are an important failure mode to the BOP system safety because they are undesir- able states that can impact the system’s ex- pected life, operational availability, and over- all system and environmental safety [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The January 4, 2023 scenarios RC3, RC4, and RC5 indicated leak- ages that can occur because of a deteriorated seal, but the scenarios are compatible with leakages caused by damages as cut, extrud- ing, and nibbling seals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The scenario RC2 rep- resents a potentially dangerous failure of the BOP system because pressure oscillations cre- ated or not damped by a pressure regulator cause dynamic loading of neighboring compo- nents that may not have been considered during their design [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In some cases, the scenarios RC1, RC3, and RC5 can cause pressure oscilla- tions and the regulator’s inability to maintain the required pressure, as indicated in RC2, but this depends on the leakage characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Compensated chamber solenoid valves are an essential component of the BOP control sys- tem, and the failure scenarios SVC1, SVC2, and SVC3 indicated that leakages were the most fre- quently observed failures, with the differences between the scenarios relying mainly on main- tenance operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' When an operator com- mands the BOP system to perform a specific function, a signal is sent from the central con- trol unit to the SEM for decoding and then to a specific compensated chamber solenoid valve (placed in the POD and associated with the desired function) that opens, causing an SPM valve to change position and allowing high-pressure fluid stored in the accumulator to flow [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Because each function has its own compensated chamber solenoid valve and some functions are more critical to the BOP system than others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=', blind shear ram close, annu- lar preventer close), if the CCSV fails to trig- ger the subplate-mounted valve, the function affected defines the risk associated with the fail- ure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Failures in compensated chamber solenoid valves may result in an inability to command the other components studied and, as a result, perform their required functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Leaks in con- trol system components, such as solenoid valves and regulators, increase operating costs, while excessive leakage can lead to a loss of hydraulic pressure in the control circuit [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The majority of failures in regulator’s fail- ure scenarios were detected via surface func- tional tests or inspection, whereas the main de- tection methods in CCSV scenarios were func- tional tests and soak tests, both of which were performed with the BOP in the rig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The pro- cedure to inspect the component to investigate the cause of failure after the leakage is detected through hydraulic tests is also common in the highly related failures of compensated chamber solenoid valve and regulator’s scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Thus, the scenarios of the four components point to this as a common procedure in the oil and gas industry, at least for failures of the components under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The findings indicated that tests were crit- ical for detecting failures of the four compo- nents studied and for the safety of the drilling phase operations and activities, as tests pro- cedures appear in the scenarios and high re- lated failure records of the DC matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Even though the textual failure records in RAPID- S53 are generated by a wide range of drilling contractors, each of which has its monitoring procedures, the scenarios indicated specific pro- cedures capable of detecting different failure modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These procedures are likely to be more effective in monitoring the capability of stud- ied components to perform their required func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' However, many failures were detected during tests performed with the blowout pre- venter in the rig, which presents a challenge for the companies because these tests are ex- pensive to monitor the BOP system condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' When the blowout preventer is pulled to the rig, drilling operations should be halted, and the companies must bear the costs of retriev- ing and maintaining the BOP system and non- productive time for the entire rig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Another im- portant question raised by the scenarios is the effectiveness of the tests or other monitoring procedures conducted with the blowout pre- venter in subsea to monitor the capability of the components studied in performing their re- quired functions, and why the majority of fail- January 4, 2023 ures were detected by tests executed while the system was on the rig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' If the pressure within the drilling system becomes uncontrollable, and the BOP is subsea with components such as the annular preventer and shear ram preventer un- available to close the wellbore on demand, the entire rig is in danger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Conclusion This paper uses a text mining approach based on the well-known LSA method to ex- tract information and understand the main fail- ure scenarios of four blowout preventer’s com- ponents: annular preventers, shear ram pre- venters, regulators, and compensated chamber solenoid valves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' A total of 1312 failures and cor- rective maintenance descriptions reported by engineers and technicians in charge of BOP sys- tem maintenance is examined in this research to assess the most common failures, detection methods, repair practices, causes of failures, challenges that drilling companies face, and their experience with BOP system’s failures for each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' At least three main failure scenarios were found for each of the four BOP’s components studied based on unstructured fail- ure descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Although this study only examined failure descriptions for four compo- nents, it represents a step toward understand- ing the BOP system’s main failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This research has both theoretical and prac- tical potentialities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In an academic sense, un- like previous research, this study focused solely on textual descriptions of the failures, which contained information such as failure mode, causes, and detection method, and contributes to an understanding of the main failures of the BOP system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Given the scarcity of research on the most common blowout preventer fail- ures and maintenance procedures, the failure scenarios presented in this paper may be useful for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' There is also a lack of infor- mation available on BOP test procedures that were more effective in detecting certain types of failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Scholars may use the scenarios to ex- amine the maintenance procedures associated with corrective maintenance, to support relia- bility and safety analysis, and risk management with information about the failures, or as an in- put to develop theoretical models of BOP sys- tem condition monitoring for the components studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Furthermore, the findings indicating tests performed with the blowout preventer in the rig as a main detection method of the fail- ures require extra attention since this reinforces the importance of the recent efforts of schol- ars, agencies, and oil and gas companies to in- crease the reliability of the BOP system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Few studies used text mining to analyze mainte- nance descriptions (texts that are very techni- cal and complex), and the approach described in this paper has the potential to mitigate the technical vocabulary problem inherent in maintenance data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It produced promising re- sults when analyzing unstructured failure de- scriptions to identify the main failure scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Through customization and adaptation, the approach presented in this study can be used to extract relevant information of main- tenance procedures contained in databases of others equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Drilling companies and BOP manufactur- ers can use the failure scenarios to improve their services and products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Even though each drilling company has its procedures for investi- gating anomalies detected in the BOP system during drilling activities, managers in charge of blowout preventer systems can use the sce- narios to support decisions when investigat- ing anomalies on the components studied be- cause they indicated detection methods that are likely more effective in detecting specific failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Understanding the most common fail- ure modes, detection procedures capable of identifying them, and their causes can also support engineers and technicians to develop better maintenance procedures and reliability analysis of the blowout preventer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The January 4, 2023 findings establish a level of relevance (the sin- gular values) between each component’s failure scenarios, allowing drilling contractors to focus their efforts on monitoring the failures that oc- cur more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The level of relevance is critical to reducing the costs of the activities carried out during the drilling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The fail- ure scenarios can provide a reliable assessment of component failures to BOP manufacturers, which can aid in designing better products or services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Regulatory agencies, for example, can use the findings to encourage oil and gas com- panies to take action, which is critical for im- proving industry safety by lowering accident risks while increasing equipment availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The findings may prompt oil and gas compa- nies to reconsider the testing procedures used on BOP system’s components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The testing of the blowout preventer system is critical to in- creasing safety, as demonstrated by the scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' However, the more frequently the BOP is tested, the lower the system’s availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Testing the system is detrimental not only to its availability but also because each function test, even if it does not overload the system, accounts for one less function in the system’s useful life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' If a component is designed to be used 200 times, it will lose 1/200 of its use- ful life for each function test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' When examining components over a long period, function tests can significantly reduce their expected service life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' For example, the service life of an annu- lar preventer is usually 500 pressurizations or 15 years for its top cover and housing [70, 114], so if the component is tested at least once a week, assuming that it was not pressurized for anything except tests, the annular will have a service life of about ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The annular’s estimated service life is five years if one pres- surization per week due to drilling activities is included, three years if two pressurizations are included, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This effect is wors- ened when investigating the impact of pressure testing, as it overloads the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Therefore, drilling companies must consider the effective- ness of the procedures they use to investigate anomalies detected in the BOP system because they can reduce the service life of the compo- nents to a fraction of what it should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' As previously stated, the failure scenarios can as- sist them in determining which procedures are likely to be more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Limitations Although this study is a step forward in using failure records textual data, it is worth noting some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The non-planned corrective maintenance descriptions were added to the RAPID-S53 through an open question on the failure event registration form, and the descrip- tions contain some subjectivity, even though the text is highly technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' There is a differ- ence in the quality of the records when con- sidering the volume of information in the de- scriptions, which varies according to experi- ence and engagement in reporting all the de- tails of the failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The entire investigation has been reported in some descriptions, includ- ing details such as failure mechanisms, proper as input to discussions about condition-based maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' However, there are others cases with very brief descriptions of corrective main- tenance, which is detrimental to the database and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These two types of records are few and have little impact on LSA results, but drilling companies must strive to improve the engagement of the workers responsible for re- porting failures so that the descriptions contain more information than they have now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Based on a well-known text mining tech- nique, the analysis generated knowledge about the most common failures and corrective main- tenance procedures related to the BOP sys- tem’s components studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' On the other hand, because the failure records were only collected from the RAPID-S53, the results obtained are restricted to a single database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Besides that, even though the database have reports from re- gions across all globe, the majority of failure records are from drilling companies working in January 4, 2023 the United States part of the Gulf of Mexico, and because of that, the results of the analy- sis may not reflect the main failure modes in all countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' While failure descriptions for an- nular preventers and shear ram preventers in the US part of the Gulf of Mexico correspond to approximately 60-65 percent of the data ex- amined, failure descriptions for regulators and compensated chamber solenoid valves in the same location correspond to approximately 80 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The scenarios were presented so that the LSA generates concepts (from higher to lower singular values), which can be related to the frequency of occurrence in the textual failure records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Some failure scenarios that appear as second, third, or fourth may pose a greater risk to the rig’s safety than failures indicated in the first scenario of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Thus, the severity of the failure modes is not taken into account by the latent semantic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' LSA is carried out through systematic and mathe- matical analysis, but there is human involve- ment in interpreting the concepts, which in- troduces subjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The relevance of analyst knowledge to the outcome of text mining using the LSA approach cannot be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' This limitation was addressed through a revi- sion conducted with a team of researchers, en- gineers, and technical workers with oil and gas industry experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Suggestions The study’s findings shed light on the break- down of failure events by components’ failures modes, detection methods, corrective actions, suspected causes, and the frequency with which they occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Future research could use the re- sults of a scientifically based procedure as a base for reliability studies in the BOP sys- tem’s components under consideration, as well as the development of theoretical models of condition monitoring, maintenance procedure assessment, risk management, and safety anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The industry regulations and standard re- quirements have been developed and evolved to improve the safety of the drilling phase [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' However, the findings reinforce that oil and gas companies, regulatory agencies, technology institutes, and scholars must research intelli- gent and practical solutions to improve BOP system availability while maintaining or im- proving safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' It is critical for the future of drilling oil and gas companies to increase the effectiveness of monitoring procedures by pur- suing innovative ways to monitor the BOP sys- tem, such as new technologies, techniques, and approaches related to condition-based mainte- nance of the components as other intelligent so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These efforts are essential to increase the availability of the blowout preventer sys- tem, and the safety of the rig crew and the en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [15], for example, a state-based approach is proposed to analyze unavailability by incor- porating BSR preventer testing activities into a multiphase Markov process, and investigated the effects of testing errors and delayed re- pairs on the likelihood of component unavail- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [30], an integrated a fault tree analysis updated with a near real-time failure database is used into the operational decision- making process to reduce non-productive time on drilling rigs due to complex failure prop- agation within the BOP system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [70], a high-fidelity method is proposed to monitor- ing the stress of the annular preventer’s top cover and housing that combines the theoret- ical calculation method (TCM), finite-element analysis (FEA), and stress testing experiment (STE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In [115], an adaptive model-based ap- proach is proposed to real-time condition mon- itoring of annular preventer’s functions by com- bining a first-principles model with in-field data by adapting model coefficients, interpreted as annular preventer’s health indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' These are examples of recent studies that presented new solutions for improving drilling operations’ safety while increasing the availability of BOP system’s components, both of which are criti- January 4, 2023 cal to the oil and gas industry’s future and sus- tainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Efforts in this direction are likely to improve maintenance management and indus- try safety by reducing accident risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' The discussion in this study can be ex- panded by using cuttings from the database to other important components of the blowout preventer system, such as pipe rams and SPM valves, to understand their main failure scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Furthermore, maintenance intelligent so- lutions, such as the gathering of video reports, have potential to increase the volume of infor- mation described by technicians and engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' In this context, the challenge of a human expert examining each service entry can rise substan- tially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' So, the ability of text mining methods, such as the LSA, to reduce the dimensions of large volumes of data to a more manageable number without losing a significant amount of information can be explored by scholars and oil and gas companies to extract useful informa- tion from textual descriptions of maintenance procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' Acknowledgements The authors wish to acknowledge the finan- cial support of Brazil’s national oil company, Petrobrás (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content='ANP: 20741-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAyT4oBgHgl3EQf7vqM/content/2301.00844v1.pdf'} +page_content=' References [1] N.' 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b/VtAzT4oBgHgl3EQfYPxj/content/tmp_files/2301.01331v1.pdf.txt @@ -0,0 +1,877 @@ +arXiv:2301.01331v1 [math.CO] 3 Jan 2023 +Local Configurations in Union-Closed Families +Jonad Pulaj and Kenan Wood∗ +January 2023 +Abstract +The Frankl or Union-Closed Sets conjecture states that for any finite union- +closed family of sets F containing some nonempty set, there is some element i +in the ground set U(F) := � +S∈F S of F such that i is in at least half of the sets +in F. In this work, we find new values and bounds for the least integer m such +that any family containing m distinct k-sets of an n-set X satisfies Frankl’s +conjecture with an element of X. Additionally, we answer an older question of +Vaughan [12] regarding symmetry in union-closed families and we give a proof +of a recent question posed by Ellis, Ivan and Leader [5]. Finally, we introduce +novel local configuration criteria to prove the conjecture for many, previously +unknown classes of families. +1 +Introduction +Frankl’s or the Union-Closed Sets conjecture is an open, well-known problem in ex- +tremal set theory. A finite family of finite sets F is union-closed if for every A, B ∈ F, +it follows that A∪B ∈ F. Frankl’s conjecture states that for any union-closed family +F containing some nonempty set, there is some element i in the ground set, or uni- +verse, of F defined as U(F) := � +S∈F S such that i is in at least half of the sets in +F. +Of the well-known techniques to tackle Frankl’s conjecture, this work is concerned +with the approach of local configurations [2], a method that aims to prove the con- +jecture for any union-closed family F satisfying some local conditions with respect +to some fixed ground set X ⊆ U(F). We believe recent developments [10, 11], in- +cluding the work presented here, provide a new impetus into this line of research and +its implications for Frankl’s conjecture. In addition, some questions related to local +configurations may be of independent interest since they are not implied by Frankl’s +conjecture. +∗Department of Mathematics and Computer Science, Davidson College, Davidson, NC 28036, +{jopulaj, kewood}@davidson.edu +1 + +The genesis of local configurations began with the well-known observations that +any union-closed family containing a 1-set or a 2-set satisfies Frankl’s conjecture with +an element from the 1-set or 2-set (where a k-set is a set with k elements). Poonen [9] +provided a complete characterization of families A such that every union-closed family +containing A satisfies Frankl’s conjecture with an element from U(A). Such families +A are called Frankl-Complete (FC). Families A that are not FC, are called Non- +FC. As a consequence he showed that there is a union-closed family F containing +a 3-set A such that every element of A is in strictly less than half the sets of F; +that is, {A} is Non-FC. Using Poonen’s Theorem and machine-assisted techniques, +Morris [8] and Vaughan [12] were able to characterize many FC-families on at most six +elements. More recently, Pulaj [10] exhibited the first efficient algorithm to completely +characterize FC-families on at most 10-elements. +For a positive integer k, we define [k] = {1, . . . , k} and let P([k]) denote the +power set of [k]. For 3 ≤ k < n, define FC(k, n) to be the least integer m such +that any A ⊆ P([n]) containing m distinct k-sets is FC. Morris [8] proved that +FC(3, n) ≥ ⌊n/2⌋ + 1 for all n ≥ 4 and conjectured that equality holds. +Pulaj +[11] then proved Morris’s conjecture showing FC(3, n) = ⌊n/2⌋ + 1 for all n ≥ 4. +Morris also proved that FC(4, 5) = 5 and 7 ≤ FC(4, 6) ≤ 8, while Mari´c, Vuˇckovi´c +and ˇZivkovi´c [7] provided a complete classification of all FC-families on six elements, +showing FC(4, 6) = 7. +Our main contributions in this work are as follows. First, we algorithmically show: +• FC(4, 7) = 10, FC(4, 8) = 12, FC(5, 7) = 14, FC(6, 8) = 26; +• FC(4, 9) ≥ 14, FC(4, 10) ≥ 16, FC(5, 8) ≥ 21, FC(5, 9) ≥ 31, FC(5, 10) ≥ +44, FC(6, 9) ≥ 42, FC(6, 10) ≥ 71, FC(7, 10) ≥ 85. +We also prove the following new upper bounds for general n: +• FC(4, n) ≤ 1 + +� 11 +1680 · n(n − 1)(n − 2)(n − 3) +� +for n > 8; +• FC(5, n) ≤ 1 + +� 13 +2520 · n(n − 1)(n − 2)(n − 3)(n − 4) +� +for n > 7; +• FC(6, n) ≤ 1 + +� +5 +4032 · n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) +� +for n > 8. +As a consequence, we obtain FC(4, 9) ≤ 21, FC(5, 8) ≤ 36, and FC(6, 9) ≤ 76. In +contrast to previous works [10, 11, 3] where exact integer programming is used for +verification of computational results, in our current work we use a SMT (Satisfiability +Modulo Theory) solver for verification as suggested in [7]. Tools like SMTCoq [4] pave +the way for further verification in interactive theorem provers. +Second, we answer a question of Vaughan [12] in the positive that simplifies Poo- +nen’s characterization of FC-families according to the symmetry of a given family, in +particular, its automorphism group. For families A and B, define A ⊎ B := {A ∪ B : +A ∈ A, B ∈ B}. For an element i, let Ai := {A ∈ A : i ∈ A}. We explicitly state +Poonen’s Theorem below. +2 + +Theorem 1.1 (Poonen). Let A be a union-closed family of sets with ∅ ∈ A and +U(A) = [n]. Then the following are equivalent: +1. A is an FC-family. That is, for all union-closed F ⊇ A, there is some i ∈ U(A) +such that |Fi| ≥ |F|/2. +2. There exists some c ∈ Rn +≥0 satisfying � +i∈[n] ci = 1 such that for any union-closed +B ⊆ P([n]) with A ⊎ B = B, we have +� +i∈[n] +ci|Bi| ≥ |B|/2. +The set of all c allowed in (2) is a polyhedron denoted P A. In 2002, Vaughan [12] +asked whether or not P A ̸= ∅ implies there is some c ∈ P A such that ci = cj whenever +there is an automorphism of A mapping i to j. We prove that this implication does, +indeed, hold. +Additionally, we highlight the utility of FC-families by answering in the positive +the following recently posed question by Ellis, Ivan and Leader [5]. Let n ≥ 4 and +choose some R ⊂ Zn with |R| = 3. Does the union-closed family generated by all +translates of R × {0} or {0} × R by elements of Zn × Zn necessarily satisfy Frankl’s +conjecture? +Finally, we prove an intriguing result that constructs a new type of local config- +uration. Given a Non-FC family A, our theorem gives a method of restricting the +possible union-closed families F ⊇ A such that |Fi| < |F|/2 for all i ∈ U(A) by con- +sidering the structure of the families {S ∩C : S ∈ F} and A for a fixed set C ⊇ U(A), +which we stress is a local condition with respect to C. To our knowledge, this is the +first result that allows us to prove that a large collection of union-closed families that +contain a Non-FC family satisfy Frankl’s conjecture. +The rest of this paper is organized as follows. In section 2, we give many new values +and bounds for FC(k, n), along with an interesting structural conjecture. Section 3 +settles two open questions; one is an older question of Vaughan and one is a more +recent question of Ellis, Ivan and Leader. Finally, section 4 gives a novel approach +to local configurations, which we use to prove Frankl’s conjecture for many new +previously unknown classes of families. +2 +FC-values and FC-bounds +In this section, we give many new values and bounds implying Frankl-Completeness +and conjecture a striking structural pattern regarding maximal Non-FC families. +Definition 2.1. Two families of sets A and B are isomorphic, written A ∼= B, +provided there is some bijection φ : U(A) → U(B) such that B = {φ(S) : S ∈ A}. +The map φ is called an isomorphism. +3 + +We now introduce our main tool for determining exact values of FC(k, n), Al- +gorithm 1. The method getNonIsomorphicFamilies(n, k, m) returns a set of rep- +resentatives from each isomorphism class of families A of m distinct k-sets with +U(A) = [n]. The method isFC(F) uses Pulaj’s algorithm [10] to return true if F is +FC, and false otherwise. +Algorithm 1: getNFC(n, k, m) +Input: Positive integers n, k, m where n ≥ k ≥ 3 +Output: A set of all pair-wise nonisomorphic Non-FC families of m distinct +k-sets with universe [n] +1 if km < n or m > +�n +k +� +then +2 +return ∅ +3 +4 NFC ← ∅ +5 FC ← ∅ +6 +7 if k(m − 1) < n then +8 +for F ∈ getNonIsomorphicFamilies(n, k, m) do +9 +if not isFC(F) then +10 +NFC ← NFC ∪ {F} +11 +return NFC +12 +13 J ← {i ∈ Z | max{k, n − k} ≤ i ≤ n} +14 for F ∈ � +i∈J getNFC(i, k, m − 1) do +15 +for S ⊆ [n] such that |S| = k and U(F ∪ {S}) = [n] do +16 +if ∀A ∈ NFC ∪ FC: F ∪ {S} ̸∼= A then +17 +if F ∪ {S} contains a proper FC-family then +18 +continue +19 +if isFC(F ∪ {S}) then +20 +FC ← FC ∪ {F ∪ {S}} +21 +else +22 +NFC ← NFC ∪ {F ∪ {S}} +23 return NFC +Algorithm 1 is a recursive algorithm designed to determine all isomorphism classes +of Non-FC families of m distinct k-sets with universe [n], while disregarding families +containing a proper FC-family. The isomorphism checks in line 16 are performed by +4 + +computing a canonical form1 of the family F such that any family isomorphic to F +has an identical canonical form, checking if that canonical form has been computed +before, and if not, storing its canonical form. The proper FC-containment check in +line 17 is computed in a similar fashion by computing the canonical form of subfamilies +of F with one fewer member-set. In our implementation, we manually start at the +bottom of the call stack to avoid recomputation. +Additionally, for the purpose of ensuring the correctness of each isFC() com- +putation, we use the SMT solver Z3 [1] within the SMT python library, pySMT [6]. +For verifying Non-FC families, we check the infeasibility of the terminating set of +constraints produced by the isFC() algorithm. For FC families, (using Pulaj’s no- +tation) we check the infeasibility of the linear integer system defining X(A, c), where +c is the vector in Zn found by the algorithm that is proposed to satisfy X(A, c) = ∅. +Lemma 2.1. Algorithm 1 correctly finds a desired collection of Non-FC families. +Proof. For termination, notice that in each recursive call, we must have n ≥ k > +0. +Also, the m argument is decreased by 1 at every call, so if Algorithm 1 did +not terminate, km ≥ n at every iteration; however, m must be zero at some point +assuming no termination. This is a contradiction because n > 0. Therefore Algorithm +1 terminates. +For correctness, observe that the theorem is true if either km < n or k(m−1) < n +or m > +�n +k +� +. We first prove the following claim. +Claim: Let J = {i ∈ Z | max{k, n − k} ≤ i ≤ n}. Assume getNFC(i, k, m − 1) is +correct for all i ∈ J. Then getNFC(n, k, m) is correct. +Proof of claim. We may assume k(m − 1) ≥ n and m ≤ +�n +k +� +. Consider the execu- +tion of getNFC(n, k, m). Observe that anytime a family is added to NFC, we always +first verify that it is Non-FC. Hence every family in NFC is Non-FC. Suppose F +is a Non-FC family with universe [n] containing m distinct k-sets. Let S ∈ F; let +F ′ = F −{S} with i := |U(F)|. Since S ∈ F, we know |S| = k, so that n−k ≤ i and +k ≤ i ≤ n (because m ≥ 2). Hence i ∈ J, which implies that getNFC(n, k, m) iterates +through all families in getNFC(i, k, m − 1). By assumption, one of these families, say +G′, is isomorphic to F ′. Since there is an isomorphism φ : U(F ′) → U(G′), the family +G := G′ ∪ {φ(S ∩ U(F ′)) ∪ (S − U(F ′))} is isomorphic to F. Also G is added to NFC +since F ∼= G is Non-FC, as desired. Thus getNFC(n, k, m) is correct. +We proceed by induction on n. Note that n ≥ k, so the base case is n = k. If +m = 1, the the result follows by inspecting lines 7-11 in Algorithm 1. If m ≥ 2, then +m > +�n +k +� += 1, so getNFC(n, k, m) correctly returns. +For the induction step on n, suppose n ≥ k + 1 and getNFC(n′, k, m′) is correct +for all k ≤ n′ < n and m′. Then getNFC(i, k, m − 1) correctly returns for all i ∈ +1We use SageMath’s canonical label() method within the IncidenceStructure class. +5 + +J−{n}. To show getNFC(n, k, m) correctly returns, we use induction on m. If m = 1, +then certainly getNFC(n, k, m) is correct. Suppose m ≥ 2 and getNFC(n, k, m − 1) +correctly returns. This shows that getNFC(i, k, m − 1) is correct for all i ∈ J. Hence +the theorem follows from the above claim. +Using Algorithm 1, which can be easily extended to determine the exact value of +FC(k, n) for small values of k and n, we have determined the following.2 +Theorem 2.2. FC(4, 7) = 10. +Theorem 2.3. FC(4, 8) = 12. +Theorem 2.4. FC(5, 7) = 14. +Theorem 2.5. FC(6, 8) = 26. +The system used to verify all our results (including those in section 4) has an Intel +Xeon Processor E5-2620 v4 with 16 cores, each running at 2.1GHz; the system has +128GB of memory and two NUMA nodes. Theorems 2.2, 2.4, 2.5 have been verified +within at most a couple hours, but Theorem 2.3 took us more than 26 days to verify. +Let +�S +k +� +be the set of all k-subsets of a set S. Define a strict total order <, called +the lexicographic order, on the set +�[n] +k +� +by A < B if min(A∆B) ∈ A. In this order, +for fixed n and k and for any S ∈ +�[n] +k +� +, define [S] := {A ∈ +�[n] +k +� +| A ≤ S}. Let +{Sn,k +i +}i≥1 = +�[n] +k +� +, where Sn,k +i +< Sn,k +j +for all 1 ≤ i < j. If n and k are clear, we simply +write Si instead of Sn,k +i +. +The following conjecture seems to be very promising based off of our experimental +results. +Conjecture 1. For fixed n > k ≥ 3, if [Sm] is an FC-family for some positive integer +m and has universe size n, then FC(k, n) ≤ m. +This conjecture has been verified for all n > k ≥ 3 such that FC(k, n) is known +(it is trivial for k = 3 and any n ≥ 4); there is always a maximum Non-FC family of +the form [Sm] for some m. If the conjecture is true, then we could easily find all exact +values of FC(k, n) for n ≤ 10; all we would need to do in that case is to find an integer +m such that [Sm] is FC and [Sm−1] is Non-FC, giving us a result of FC(k, n) = m. +It can be shown using the above method that the following FC lower bounds are +also exact values, assuming Conjecture 1. Most of these have been verified to be tight +bounds within pySMT, except lower bounds of FC(k, 10). +Theorem 2.6. FC(4, 9) ≥ 14, FC(4, 10) ≥ 16, FC(5, 8) ≥ 21, FC(5, 9) ≥ 31, +FC(5, 10) ≥ 44, FC(6, 9) ≥ 42, FC(6, 10) ≥ 71, FC(7, 10) ≥ 85. The remaining +values of FC(k, n) for 5 ≤ k < n ≤ 10 are undefined. +2All code used in this paper can be accessed here: +https://github.com/KenanWood/Local- +Configurations-in-Union-Closed-Families +6 + +k\n +4 +5 +6 +7 +8 +9 +10 +3 +3 +3 +4 +4 +5 +5 +6 +4 +5 +7 +10 +12 +14 +16 +5 +14 +21 +31 +44 +6 +26 +42 +71 +7 +85 +Table 1: FC-values +Proof. For each pair (k, n) ∈ {(4, 9), (4, 10), (5, 8), (5, 9), (5, 10), (6, 9), (6, 10), (7, 10)}, +the family [Sn,k +m−1] as defined above is Non-FC, where m is the proposed lower bound. +For the remaining pairs (k, n) when 5 ≤ k < n ≤ 10, we can easily show that the +family +�[n] +k +� +is Non-FC. +Assuming Conjecture 1 is true, Table 1 shows a complete classification of FC- +values for (k, n) ∈ {3, . . . , 7} × {4, . . . , 10}, where no entry at (k, n) indicates that +FC(k, n) is undefined. +To find upper bounds of FC(k, n), we generalize and tighten a result of Morris +[8]. Morris showed that FC(4, n) ≤ +7 +360n4, though without an explicit proof. The +following theorem improves and generalizes this bound, which yields improved explicit +upper bounds on FC(k, n) for 4 ≤ k ≤ 6. +Theorem 2.7. If m0 = FC(k, n0) ≤ +�n0 +k +� +, then +FC(k, n) ≤ 1 + +� +(m0 − 1) +n0 · · · (n0 − k + 1) · n · · · (n − k + 1) +� +≤ +�n +k +� +for all n > n0. +Proof. Suppose m0 = FC(k, n0) ≤ +�n0 +k +� +. Let n > n0 and m := 1+ +� +(m0 − 1) · (n0−k)! +n0! +· +n! +(n−k)! +� +, +noting that m = 1 + +� +(m0−1) +n0···(n0−k+1) · n · · · (n − k + 1) +� +. +Since n > n0, we know +(n0−k)! +n0! +· +n! +(n−k)! > 1. This implies +m ≤ +� +1 + +��n0 +k +� +− 1 +� +· (n0 − k)! +n0! +· +n! +(n − k)! +� += +� +1 + +�n0 +k +� +· (n0 − k)! +n0! +· +n! +(n − k)! − (n0 − k)! +n0! +· +n! +(n − k)! +� +≤ +� +n0! +k!(n0 − k)! · (n0 − k)! +n0! +· +n! +(n − k)! +� += +�n +k +� +. +7 + +Next, let A be a family of m distinct k-sets with a universe of size at most n. Define +A0 := A; for i ≥ 0, recursively define Ai+1 := Ai if |U(Ai)| < n − i, and otherwise, +Ai+1 := Ai − Ai +x, where x ∈ U(Ai) minimizes |Ai +x|. It follows that |U(Ai)| ≤ n − i +for all i ≥ 0 by induction. Since FC(k, n0) = m0, it suffices to prove |An−n0| ≥ m0. +To this end, for any i ≥ 0, the pigeonhole principle shows that there is some +x ∈ U(Ai) such that |Ai +x| ≤ +k|Ai| +|U(Ai)|. If |U(Ai)| = n − i, then +|Ai+1| ≥ |Ai| − +k|Ai| +|U(Ai)| += |Ai| +� +1 − +k +n − i +� += |Ai| +�n − i − k +n − i +� +. +Otherwise, |Ai+1| = |Ai| by construction, so that the inequality still holds since +n−i−k +n−i +< 1. In writing +|An−n0| = |A0| · +n−n0−1 +� +i=0 +|Ai+1| +|Ai| , +we obtain +|An−n0| ≥ m · +n−n0−1 +� +i=0 +�n − i − k +n − i +� +≥ +� +1 + (m0 − 1) · (n0 − k)! +n0! +· +n! +(n − k)! +� +· +�(n − k)! +n! +· +n0! +(n0 − k)! +� +> m0 − 1, +so that |An−n0| ≥ m0. Thus A ⊇ An−n0 is an FC-family and the result follows. +Corollary 2.7.1. The following bounds hold: +• FC(4, n) ≤ 1 + +� 11 +1680 · n(n − 1)(n − 2)(n − 3) +� +for n > 8; +• FC(5, n) ≤ 1 + +� 13 +2520 · n(n − 1)(n − 2)(n − 3)(n − 4) +� +for n > 7; +• FC(6, n) ≤ 1 + +� +5 +4032 · n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) +� +for n > 8. +Proof. This is an immediate consequence of Theorem 2.7 along with Theorems 2.3, +2.4, and 2.5. +Corollary 2.7.2. FC(4, 9) ≤ 21, FC(5, 8) ≤ 36, and FC(6, 9) ≤ 76. +Proof. This is an immediate consequence of Corollary 2.8.1. +8 + +3 +Symmetry in FC-families +In this section, we answer two previously unsolved questions regarding symmetry in +union-closed families with respect to local configurations. +3.1 +Automorphisms in FC-families +Given a union-closed family A containing ∅ with U(A) = [n], let B(A) be the set +of all union-closed B ⊆ P([n]) such that A ⊎ B = B. Recall that P A = {c ∈ Rn +≥0 : +� +i∈[n] ci = 1 ∧ ∀B ∈ B(A), � +i∈[n] ci|Bi| ≥ |B|/2}. Then By Poonen’s Theorem 1.1, +A is FC if and only if P A ̸= ∅. As outlined in our introduction, the following is a +generalization of Vaughan’s [12] question. +Question 1. Given a union-closed family A containing ∅, if P A is nonempty, then +is there always some c ∈ P A such that ci = cj whenever there is an automorphism of +A that maps i to j? +We prove that this question holds in the following theorem. First, let Aut(A) +denote the set of all automorphisms of A and note that Aut(A) is a group under +function composition. +Theorem 3.1. Let A be a union-closed family containing ∅. If P A is nonempty, then +there is some c ∈ P A such that ci = cj whenever there is an automorphism of A that +maps i to j. +Proof. Without loss of generality, assume U(A) = [n]. Suppose x ∈ P A. For any +φ ∈ Aut(A), we first show that (xφ(i))i∈[n] ∈ P A; it suffices to show that for any B ∈ +B(A), we have � +i∈[n] xφ(i)|Bi| ≥ |B|/2. Consider the image φ(B) = {φ(S) : S ∈ B}. +If A′ ∈ A and B′ ∈ φ(B), then there are A ∈ A and B ∈ B such that A′ = φ(A) +and B′ = φ(B); the first holds since φ−1 ∈ Aut(A), so that A = φ−1(A′) ∈ A, and +the second is by construction of φ(B); this shows A′ ∪ B′ = φ(A ∪ B) ∈ φ(B), which +implies φ(B) ∈ B(A). Since x ∈ P A, then � +i∈[n] xi|φ(B)i| ≥ |φ(B)|/2. Since φ is a +bijection, � +i∈[n] xφ(i)|φ(B)φ(i)| ≥ |φ(B)|/2, which shows � +i∈[n] xφ(i)|Bi| ≥ |B|/2. Thus +(xφ(i))i∈[n] ∈ P A for any φ ∈ Aut(A). +Consider the convex combination of elements of P A, +c = +1 +| Aut(A)| · +� +φ∈Aut(A) +(xφ(i))i∈[n] += +1 +| Aut(A)| · + + +� +φ∈Aut(A) +xφ(i) + + +i∈[n] +. +For each i ∈ [n], let +si = +� +φ∈Aut(A) +xφ(i). +9 + +Then we obtain +c = +1 +| Aut(A)| · (si)i∈[n]. +For any φ0 ∈ Aut(A), every automorphism in Aut(A) can be written as a unique +left composition with φ0; that is, Aut(A) = {φ ◦ φ0 : φ ∈ Aut(A)}. Then, for every +i, j ∈ [n] such that there is some automorphism φ0 ∈ Aut(A) mapping i to j, we +know +si = +� +φ∈Aut(A) +x[φ◦φ0](i) = +� +φ∈Aut(A) +xφ(j) = sj, +so that ci = cj as well. Since c is a convex combination of points in a polyhedron, +c ∈ P A. +3.2 +A Result on Transitive Families of 3-sets +In an Abelian group (G, +), if R ⊆ G and g ∈ G, we define the translation of R by g +in G as the set +g + R := {g + r : r ∈ R}. +The set of all translations (by some element g ∈ G) of R in G is denoted T(R). Given +a family A, the union-closure of A, or the family generated by A, is defined as the +union-closed family ⟨A⟩ := {� +S∈A′ S : A′ ⊆ A}. +Given some 3-set R ⊂ Zn, the authors of [5] ask and were unable to verify that +the family generated by A = T(R × {0}) ∪ T({0} × R) ⊆ Z2 +n necessarily satisfies +Frankl’s conjecture. The authors remark that this family is transitive; that is, for any +x, y ∈ Z2 +n, there is an automorphism φ ∈ Aut(A) such that φ(x) = y. +Let A be a family of sets. Let d(x) = |Ax| be the degree of x in A. The family +A is regular if d(x) = d(y) for all x, y ∈ U(A), in which case the degree of A is the +common degree. +Lemma 3.2. Let A be a regular family of 3-sets with degree k ≥ 2 and universe of +size n ≥ 4. Then A is FC. +Proof. Since � +i∈U(A) d(i) = � +A∈A |A|, we obtain kn = 3m, so that m = kn/3 ≥ +2n/3, where m = |A|. Since FC(3, n) ≤ 2n/3 for all n ≥ 5, this shows A is FC if +n ≥ 5. If n = 4, then m ≥ 4 (in fact equality holds). In this case, since FC(3, 4) = 3, +we know A is FC. +Theorem 3.3. Let R ⊂ Zn be a 3-set, where n ≥ 4. Then the family A = T(R × +{0}) ∪ T({0} × R) ⊆ Z2 +n is FC, and thus, ⟨A⟩ satisfies Frankl’s conjecture. +Proof. It is clear that A is a regular family of 3-sets with degree at least two. Fur- +thermore, |U(A)| ≥ 4. The result follows from Lemma 3.2. +10 + +4 +Deducing Frankl’s Conjecture From New Local +Conditions +In this section, we give a sufficiency condition for when a family F satisfies Frankl’s +conjecture from its local structure with respect to some C ⊆ U(F). We use this result +to determine many new local configurations. +The following lemma is central to this section. +We should note that this is a +generalization of an idea presented in [5]. In particular, the construction of graph G +and map f was inspired by the work of Ellis, Ivan, and Leader [5]. +Lemma 4.1. Let F be a union-closed family containing some nonempty set and ∅. +Let C ⊆ U(F) be nonempty. Then there is some i ∈ C such that |Fi| ≥ |F|/2 if there +is some weight function w : C → R>0 such that for all X ′ ⊆ X , we have +|X ′| ≤ |{F ∪ S : F ∈ F ∩ P(C), S ∈ X ′, w(S) + w(F ∪ S) ≥ w(C)}|, +where X = {S ∩ C : S ∈ F, w(S ∩ C) < w(C)/2}. +Proof. Suppose there is some nonempty C ⊆ U(A) and weight function w : C → R>0 +such that for all X ′ ⊆ X , we have +|X ′| ≤ |{F ∪ S : F ∈ F ∩ P(C), S ∈ X ′, w(S) + w(F ∪ S) ≥ w(C)}|. +Construct a bipartite graph G with partite sets X and Y := {S ∩ C : S ∈ F, w(S ∩ +C) > w(C)/2}, where S ∈ X is joined with T ∈ Y via an edge if w(S)+w(T) ≥ w(C) +and there is some F ∈ F with F ⊆ C such that T = F ∪ S. It is clear that if S ∈ X +and F ∈ F with F ⊆ C, then F ∪ S ∈ {F ′ ∩ C : F ′ ∈ F} since F is union- +closed. Thus the above condition is equivalent to saying for all X ′ ⊆ X , we have +|X ′| ≤ |NG(X ′)|. By Hall’s Theorem, G has an X -saturating matching m : X → Y. +Let X F = {S ∈ F : w(S∩C) < w(C)/2} and let YF = {S ∈ F : w(S∩C) > w(C)/2}. +Define a map f : X F → YF by +f(F) = m(F ∩ C) ∪ (F − C) +for all F ∈ X F. We first verify that f is well-defined. Suppose F ∈ X F. Then by +definition of edges in G, f(F) ∩ C ∈ Y. There is some F ′ ∈ F with F ′ ⊆ C such that +m(F ∩ C) = (F ∩ C) ∪ F ′; this implies +f(F) = (F ∩ C) ∪ F ′ ∪ (F − C) += F ∪ F ′ ∈ F +by union-closure of F. Hence f(F) ∈ YF. Now we show f is injective. If f(A) = f(B) +for some A, B ∈ X F, then A − C = B − C since Y ⊆ P(C); furthermore, since m is a +11 + +matching, and thus an injection, A∩C = B ∩C, showing f is injective. Additionally, +by definition of the edges of G, we know that for each S ∈ X F, +w(S ∩ C) + w(f(S) ∩ C) ≥ w(C). +Thus, we know +� +S∈F +(w(S ∩ C) − w(C)/2) ≥ 1 +2 +� +S∈X F +(w(S ∩ C) + w(f(S) ∩ C) − w(C)) +≥ 0. +It is easy to show with a simple combinatorial argument that +1 +w(C) +� +i∈C +w(i) · |Fi| ≥ |F| +2 +⇔ +� +i∈C +w(i) +w(C) · |Fi| ≥ |F| +2 . +Since � +i∈C w(i)/w(C) = 1, we have a weighted average of all |Fi| over i ∈ C being +at least |F|/2. This shows that there is some i ∈ C such that |Fi| ≥ |F|/2. +For the following theorem, we let X (F, C, w) := {S ∩ C : S ∈ F, w(S ∩ C) < +w(C)}. When applying this theorem, since for any nonempty C and F that is union- +closed, F ∩ P(C) is union-closed, we only need to consider families A ⊆ P(C) that +are union-closed. +Theorem 4.2. Let C be a nonempty set; let w : C → R>0 be a weight function on +C. Let X ⊆ P(C) such that for each S ∈ X , we have w(S) < w(C)/2; let A ⊆ P(C) +with ∅ ∈ A. Suppose for all X ′ ⊆ X that +|X ′| ≤ |{A ∪ S : A ∈ A, S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)}|. +Then for any union-closed family F such that X (F, C, w) ⊆ X and A ⊆ F, there is +some i ∈ C such that |Fi| ≥ |F|/2. +Proof. Let F be a union-closed family such that X (F, C, w) ⊆ X and A ⊆ F. +Observe that for any X ′ ⊆ X (F, C, w) ⊆ X , +{A ∪ S : A ∈ A, S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)} +is a subset of +{A ∪ S : A ∈ F ∩ P(C), S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)}, +since A ⊆ F ∩ P(C). By Lemma 4.1, there is some i ∈ C such that |Fi| ≥ |F|/2. +12 + +We say that any family A satisfying the consequence of Theorem 4.2 is quasi-FC +with respect to (C, w, X ). +We remark that a special case of Theorem 4.2 is when X = {S ⊆ C : w(S) < +w(C)/2}. In this case, any union-closed family F containing A satisfies X (F, C, w) ⊆ +X . Then if the conditions of the theorem are met, we obtain that A is an FC-family. +A natural question at this point is the converse in this special case. If the converse is +true, then Theorem 4.2 is simply a strictly more general version of Poonen’s Theorem. +Given C, w, X , A, let Y := {A ∪ S : A ∈ A, S ∈ X , w(S) + w(A ∪ S) ≥ w(C)}. +Let IP(C, w, X , A) denote the following binary linear program: +Max. +� +S∈X +xS − +� +T∈Y +yT +� +S∈X +xS ≥ +� +T∈Y +yT + 1 +xS ≤ yA∪S +∀S ∈ X , ∀A ∈ A, w(S) + w(A ∪ S) ≥ w(C) +xS ∈ {0, 1} +∀S ∈ X +yT ∈ {0, 1} +∀T ∈ Y +Theorem 4.3. Let C be a nonempty set; let w : C → R>0; let X ⊆ P(C) such that +for each S ∈ X , we have w(S) < w(C)/2; let A ⊆ P(C). Then the conditions of +Theorem 4.2 are satisfied if and only if IP(C, w, X , A) is infeasible. +Proof. Suppose the conditions of Theorem 4.2 do not hold. +Then there is some +X ′ ⊆ X such that +|X ′| ≥ |{A ∪ S : A ∈ A, S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)}| + 1. +Let Y′ = {A∪S : A ∈ A, S ∈ X ′, w(S)+w(A∪S) ≥ w(C)}. Let xS = +� +1 +if S ∈ X ′ +0 +otherwise +and yT = +� +1 +if T ∈ Y′ +0 +otherwise for S ∈ X and T ∈ Y. +By assumption, � +S∈X xS ≥ +1 + � +T∈Y yT. Consider any S ∈ X , A ∈ A with w(S) + w(A ∪ S) ≥ w(C). If S ̸∈ X ′, +then xS = 0 so that xS ≤ yA∪S; otherwise, A ∪ S ∈ Y′, so that yA∪S = 1 and again +xS ≤ yA∪S, as desired. Thus (x, y) is feasible in IP(C, w, X , A). +Suppose (x, y) is feasible in IP(C, w, X , A). Let X ′ = {S ∈ X : xS = 1} and +Y′ = {T ∈ Y : yT = 1}. It suffices to show +{A ∪ S : A ∈ A, S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)} ⊆ Y′ +since the feasibility of (x, y) in IP(C, w, X , A) implies |X ′| > |Y′|. Consider any +A ∈ A, S ∈ X ′ such that w(S) + w(A ∪ S) ≥ w(C); let T = A ∪ S. Observe xS = 1, +so that yT = 1 since xS ≤ yA∪S. Thus T ∈ Y′. +13 + +With these tools in mind, we find the following results, which may be indepen- +dently verified with Theorem 4.3. Given a family A, recall that the union-closure of +A is defined as the union-closed family ⟨A⟩ := {� +S∈A′ S : A′ ⊆ A}. +Theorem 4.4. Let C = [3] and A = ⟨{{1, 2, 3}}⟩. Then any union-closed F ⊇ A +such that |{S ∩ C : S ∈ F, |S ∩ C| = 1}| = 0 satisfies Frankl’s conjecture. +Proof. Let w : C → R>0 be defined by w(x) = 1 for all x ∈ C. If X = {∅}, then +we find that IP(C, w, X , A) is infeasible. The result follows from Theorems 4.2 and +4.3. +Theorem 4.5. Let C = [4] and A = ⟨{{1, 2, 3}, {2, 3, 4}}⟩. Then any union-closed +F ⊇ A such that |{S ∩ C : S ∈ F, |S ∩ C| = 1}| ≤ 2 satisfies Frankl’s conjecture. +Proof. Let w : C → R>0 be defined by w(x) = 1 for all x ∈ C. We must only +consider X = {∅, {1}, {2}}, X = {∅, {1}, {4}}, and X = {∅, {2}, {3}}. In any case, +IP(C, w, X , A) is infeasible, showing A is quasi-FC with respect to (C, w, X ) by +Theorems 4.2 and 4.3. +Theorem 4.6. Let C = [5], A = ⟨{{1, 2, 3}, {1, 4, 5}}⟩, w(x) = 1, X = {∅, {1, 2}, {2, 3}, +{1, 3}, {4}, {5}, {4, 5}}. Then A is quasi-FC with respect to (C, w, X ). +Proof. By Theorems 4.2 and 4.3, the result follows by showing IP(C, w, X , A) is +infeasible. +Theorem 4.7. Let C = [6] and A = ⟨{{1, 2, 3}, {4, 5, 6}}⟩. Then any union-closed +F ⊇ A such that {S ∩ C : S ∈ F, |S ∩ C| = 1} = ∅ and |{S ∩ C : S ∈ F, |S ∩ C| = +2}| ≤ 5 satisfies Frankl’s conjecture. Additionally if w(x) = 1 and X = +{∅, {3, 4}, {4, 6}, {2, 3}, {1, 2}, {2, 6}, {4, 5}, {3, 6}, {2, 5}, {2, 4}, {5, 6}, {1, 5}, {1, 3}}, +then A is quasi-FC with respect to (C, w, X ). +Proof. The first part can be proved using a brute-force check of of all possible X with +∅, five 2-subsets of C and no singletons, and choosing w(x) = 1 for all x ∈ C. The +second part is a simple application of Theorems 4.2 and 4.3. +A natural question regarding quasi-FC families is the converse of Theorem 4.2. We +do not have a complete answer to this question, but we exhibit a counterexample to a +natural attempt to the converse below. In particular, if c ∈ P A, so that A is FC, then +A may not be quasi-FC with respect to (C, w, X ), where C = U(A) and w(x) = cx +and X = {S ⊆ C : w(S) < w(C)/2}. Let C = [5], c = (1/4, 1/4, 1/6, 1/6, 1/6), +w(x) = cx, A = ⟨{{1, 2, 3}, {1, 2, 4}, {1, 2, 5}}⟩, and X = {S ⊆ C : w(S) < w(C)/2}. +Then c ∈ P A, yet A is not quasi-FC with respect to (C, w, X ). +14 + +5 +Conclusion +In this work, we were able to find many new values of FC(k, n) for k ≥ 4, of which +only two were previously known. We did this within a safe and exact computational +framework using SMT solving to verify results, so our results can be trusted and +independently verified. Additionally we answered two previously unsolved questions. +One was an older question of Vaughan that shows the dimension of Poonen’s poly- +hedron P A can be reduced from |U(A)| to the number of orbits of Aut(A) through +a projection, which shrinks the search space and reduces computational work. We +also answered a question of Ellis, Ivan and Leader [5] related to union-closed families +generated by 3-sets. Our solution highlights the continual importance of FC-families +in that they provide simple solutions to difficult problems related to the Union-Closed +Sets conjecture. +We believe several directions merit further attention, including Conjecture 1 and +Theorem 4.3. Can we extend our technique to prove Frankl’s conjecture for union- +closed families F such that X (F, C, w) ⊆ X without the restriction that F contains +A? In addition, it would be interesting to known whether Theorem 4.2 is a strict +generalization of Poonen’s Theorem. Affirmative answers to these questions would +be a significant step towards a deeper understanding of local configurations. +References +[1] Nikolaj Bjørner and Leonardo de Moura. Z3: An Efficient SMT Solver. Tools and +Algorithms for the Construction and Analysis of Systems, 4963:337–340, 2008. +[2] Henning Bruhn and Oliver Schaudt. +The Journey of the Union-Closed Sets +Conjecture. Graph. Comb., 31(6):2043–2074, nov 2015. +[3] Leon Eifler, Ambros Gleixner, and Jonad Pulaj. A Safe Computational Frame- +work for Integer Programming Applied to Chv´atal’s Conjecture. ACM Trans. +Math. Softw., 48(2), may 2022. +[4] Burak Ekici, Alain Mebsout, Cesare Tinelli, Chantal Keller, Guy Katz, Andrew +Reynolds, and Clark W. Barrett. SMTCoq: Plug-In for Integrating SMT Solvers +into Coq. In Rupak Majumdar and Viktor Kuncak, editors, Computer Aided Ver- +ification - 29th International Conference, CAV 2017, Heidelberg, Germany, July +24-28, 2017, Proceedings, Part II, volume 10427 of Lecture Notes in Computer +Science, pages 126–133. Springer, 2017. +[5] David Ellis, Maria-Romina Ivan, and Imre Leader. Small sets in union-closed +families. arXiv preprint arXiv:2201.11484, 2022. +[6] Marco Gario, Andrea Micheli, and Bruno Kessler. PySMT: a Solver-Agnostic +Library for Fast Prototyping of SMT-Based Algorithms. 2015. +15 + +[7] Mari´c, Vuˇckovi´c, and ˇZivkovi´c. Fully Automatic, Verified Classification of all +Frankl-Complete (FC(6)) Set Families. arXiv preprint arXiv:1902.08765, 2019. +[8] Robert Morris. FC-families and improved bounds for Frankl’s conjecture. Euro- +pean Journal of Combinatorics, 27(2):269–282, 2006. +[9] Bjorn Poonen. Union-Closed Families. Journal of Combinatorial Theory, Series +A, 59(2):253–268, 1992. +[10] Jonad Pulaj. Cutting Planes for Families Implying Frankl’s Conjecture. Mathe- +matics of Computation, 89(322):829–857, 2019. +[11] Jonad Pulaj. +Characterizing 3-Sets in Union-Closed Families. +Experimental +Mathematics, pages 1–12, 2021. +[12] Theresa P. Vaughan. A Note on the Union-Closed Sets Conjecture. Journal of +Combinatorial Mathematics and Combinatorial Computing, 45:97–110, 2002. +16 + diff --git a/VtAzT4oBgHgl3EQfYPxj/content/tmp_files/load_file.txt b/VtAzT4oBgHgl3EQfYPxj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf7c1fe9b7dd4f2ecff5c213c2c3df8aaa30fd77 --- /dev/null +++ b/VtAzT4oBgHgl3EQfYPxj/content/tmp_files/load_file.txt @@ -0,0 +1,508 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf,len=507 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='01331v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='CO] 3 Jan 2023 Local Configurations in Union-Closed Families Jonad Pulaj and Kenan Wood∗ January 2023 Abstract The Frankl or Union-Closed Sets conjecture states that for any finite union- closed family of sets F containing some nonempty set, there is some element i in the ground set U(F) := � S∈F S of F such that i is in at least half of the sets in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In this work, we find new values and bounds for the least integer m such that any family containing m distinct k-sets of an n-set X satisfies Frankl’s conjecture with an element of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Additionally, we answer an older question of Vaughan [12] regarding symmetry in union-closed families and we give a proof of a recent question posed by Ellis, Ivan and Leader [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Finally, we introduce novel local configuration criteria to prove the conjecture for many, previously unknown classes of families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 1 Introduction Frankl’s or the Union-Closed Sets conjecture is an open, well-known problem in ex- tremal set theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' A finite family of finite sets F is union-closed if for every A, B ∈ F, it follows that A∪B ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Frankl’s conjecture states that for any union-closed family F containing some nonempty set, there is some element i in the ground set, or uni- verse, of F defined as U(F) := � S∈F S such that i is in at least half of the sets in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Of the well-known techniques to tackle Frankl’s conjecture, this work is concerned with the approach of local configurations [2], a method that aims to prove the con- jecture for any union-closed family F satisfying some local conditions with respect to some fixed ground set X ⊆ U(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We believe recent developments [10, 11], in- cluding the work presented here, provide a new impetus into this line of research and its implications for Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In addition, some questions related to local configurations may be of independent interest since they are not implied by Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' ∗Department of Mathematics and Computer Science, Davidson College, Davidson, NC 28036, {jopulaj, kewood}@davidson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='edu 1 The genesis of local configurations began with the well-known observations that any union-closed family containing a 1-set or a 2-set satisfies Frankl’s conjecture with an element from the 1-set or 2-set (where a k-set is a set with k elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Poonen [9] provided a complete characterization of families A such that every union-closed family containing A satisfies Frankl’s conjecture with an element from U(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Such families A are called Frankl-Complete (FC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Families A that are not FC, are called Non- FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' As a consequence he showed that there is a union-closed family F containing a 3-set A such that every element of A is in strictly less than half the sets of F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' that is, {A} is Non-FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Using Poonen’s Theorem and machine-assisted techniques, Morris [8] and Vaughan [12] were able to characterize many FC-families on at most six elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' More recently, Pulaj [10] exhibited the first efficient algorithm to completely characterize FC-families on at most 10-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For a positive integer k, we define [k] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' , k} and let P([k]) denote the power set of [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For 3 ≤ k < n, define FC(k, n) to be the least integer m such that any A ⊆ P([n]) containing m distinct k-sets is FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Morris [8] proved that FC(3, n) ≥ ⌊n/2⌋ + 1 for all n ≥ 4 and conjectured that equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Pulaj [11] then proved Morris’s conjecture showing FC(3, n) = ⌊n/2⌋ + 1 for all n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Morris also proved that FC(4, 5) = 5 and 7 ≤ FC(4, 6) ≤ 8, while Mari´c, Vuˇckovi´c and ˇZivkovi´c [7] provided a complete classification of all FC-families on six elements, showing FC(4, 6) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Our main contributions in this work are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' First, we algorithmically show: FC(4, 7) = 10, FC(4, 8) = 12, FC(5, 7) = 14, FC(6, 8) = 26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(4, 9) ≥ 14, FC(4, 10) ≥ 16, FC(5, 8) ≥ 21, FC(5, 9) ≥ 31, FC(5, 10) ≥ 44, FC(6, 9) ≥ 42, FC(6, 10) ≥ 71, FC(7, 10) ≥ 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We also prove the following new upper bounds for general n: FC(4, n) ≤ 1 + � 11 1680 · n(n − 1)(n − 2)(n − 3) � for n > 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(5, n) ≤ 1 + � 13 2520 · n(n − 1)(n − 2)(n − 3)(n − 4) � for n > 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(6, n) ≤ 1 + � 5 4032 · n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) � for n > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' As a consequence, we obtain FC(4, 9) ≤ 21, FC(5, 8) ≤ 36, and FC(6, 9) ≤ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In contrast to previous works [10, 11, 3] where exact integer programming is used for verification of computational results, in our current work we use a SMT (Satisfiability Modulo Theory) solver for verification as suggested in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Tools like SMTCoq [4] pave the way for further verification in interactive theorem provers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Second, we answer a question of Vaughan [12] in the positive that simplifies Poo- nen’s characterization of FC-families according to the symmetry of a given family, in particular, its automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For families A and B, define A ⊎ B := {A ∪ B : A ∈ A, B ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For an element i, let Ai := {A ∈ A : i ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We explicitly state Poonen’s Theorem below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1 (Poonen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let A be a union-closed family of sets with ∅ ∈ A and U(A) = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then the following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' A is an FC-family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' That is, for all union-closed F ⊇ A, there is some i ∈ U(A) such that |Fi| ≥ |F|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' There exists some c ∈ Rn ≥0 satisfying � i∈[n] ci = 1 such that for any union-closed B ⊆ P([n]) with A ⊎ B = B, we have � i∈[n] ci|Bi| ≥ |B|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The set of all c allowed in (2) is a polyhedron denoted P A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In 2002, Vaughan [12] asked whether or not P A ̸= ∅ implies there is some c ∈ P A such that ci = cj whenever there is an automorphism of A mapping i to j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We prove that this implication does, indeed, hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Additionally, we highlight the utility of FC-families by answering in the positive the following recently posed question by Ellis, Ivan and Leader [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let n ≥ 4 and choose some R ⊂ Zn with |R| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Does the union-closed family generated by all translates of R × {0} or {0} × R by elements of Zn × Zn necessarily satisfy Frankl’s conjecture?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Finally, we prove an intriguing result that constructs a new type of local config- uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Given a Non-FC family A, our theorem gives a method of restricting the possible union-closed families F ⊇ A such that |Fi| < |F|/2 for all i ∈ U(A) by con- sidering the structure of the families {S ∩C : S ∈ F} and A for a fixed set C ⊇ U(A), which we stress is a local condition with respect to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' To our knowledge, this is the first result that allows us to prove that a large collection of union-closed families that contain a Non-FC family satisfy Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In section 2, we give many new values and bounds for FC(k, n), along with an interesting structural conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Section 3 settles two open questions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' one is an older question of Vaughan and one is a more recent question of Ellis, Ivan and Leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Finally, section 4 gives a novel approach to local configurations, which we use to prove Frankl’s conjecture for many new previously unknown classes of families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 2 FC-values and FC-bounds In this section, we give many new values and bounds implying Frankl-Completeness and conjecture a striking structural pattern regarding maximal Non-FC families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Two families of sets A and B are isomorphic, written A ∼= B, provided there is some bijection φ : U(A) → U(B) such that B = {φ(S) : S ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The map φ is called an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 3 We now introduce our main tool for determining exact values of FC(k, n), Al- gorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The method getNonIsomorphicFamilies(n, k, m) returns a set of rep- resentatives from each isomorphism class of families A of m distinct k-sets with U(A) = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The method isFC(F) uses Pulaj’s algorithm [10] to return true if F is FC, and false otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Algorithm 1: getNFC(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' m) Input: Positive integers n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' m where n ≥ k ≥ 3 Output: A set of all pair-wise nonisomorphic Non-FC families of m distinct k-sets with universe [n] 1 if km < n or m > �n k � then 2 return ∅ 3 4 NFC ← ∅ 5 FC ← ∅ 6 7 if k(m − 1) < n then 8 for F ∈ getNonIsomorphicFamilies(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' m) do 9 if not isFC(F) then 10 NFC ← NFC ∪ {F} 11 return NFC 12 13 J ← {i ∈ Z | max{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n − k} ≤ i ≤ n} 14 for F ∈ � i∈J getNFC(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' m − 1) do 15 for S ⊆ [n] such that |S| = k and U(F ∪ {S}) = [n] do 16 if ∀A ∈ NFC ∪ FC: F ∪ {S} ̸∼= A then 17 if F ∪ {S} contains a proper FC-family then 18 continue 19 if isFC(F ∪ {S}) then 20 FC ← FC ∪ {F ∪ {S}} 21 else 22 NFC ← NFC ∪ {F ∪ {S}} 23 return NFC Algorithm 1 is a recursive algorithm designed to determine all isomorphism classes of Non-FC families of m distinct k-sets with universe [n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' while disregarding families containing a proper FC-family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The isomorphism checks in line 16 are performed by 4 computing a canonical form1 of the family F such that any family isomorphic to F has an identical canonical form, checking if that canonical form has been computed before, and if not, storing its canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The proper FC-containment check in line 17 is computed in a similar fashion by computing the canonical form of subfamilies of F with one fewer member-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In our implementation, we manually start at the bottom of the call stack to avoid recomputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Additionally, for the purpose of ensuring the correctness of each isFC() com- putation, we use the SMT solver Z3 [1] within the SMT python library, pySMT [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For verifying Non-FC families, we check the infeasibility of the terminating set of constraints produced by the isFC() algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For FC families, (using Pulaj’s no- tation) we check the infeasibility of the linear integer system defining X(A, c), where c is the vector in Zn found by the algorithm that is proposed to satisfy X(A, c) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Algorithm 1 correctly finds a desired collection of Non-FC families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For termination, notice that in each recursive call, we must have n ≥ k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Also, the m argument is decreased by 1 at every call, so if Algorithm 1 did not terminate, km ≥ n at every iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' however, m must be zero at some point assuming no termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' This is a contradiction because n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Therefore Algorithm 1 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For correctness, observe that the theorem is true if either km < n or k(m−1) < n or m > �n k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We first prove the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Claim: Let J = {i ∈ Z | max{k, n − k} ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Assume getNFC(i, k, m − 1) is correct for all i ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then getNFC(n, k, m) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We may assume k(m − 1) ≥ n and m ≤ �n k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Consider the execu- tion of getNFC(n, k, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Observe that anytime a family is added to NFC, we always first verify that it is Non-FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Hence every family in NFC is Non-FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose F is a Non-FC family with universe [n] containing m distinct k-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let S ∈ F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' let F ′ = F −{S} with i := |U(F)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since S ∈ F, we know |S| = k, so that n−k ≤ i and k ≤ i ≤ n (because m ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Hence i ∈ J, which implies that getNFC(n, k, m) iterates through all families in getNFC(i, k, m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' By assumption, one of these families, say G′, is isomorphic to F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since there is an isomorphism φ : U(F ′) → U(G′), the family G := G′ ∪ {φ(S ∩ U(F ′)) ∪ (S − U(F ′))} is isomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Also G is added to NFC since F ∼= G is Non-FC, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Thus getNFC(n, k, m) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We proceed by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Note that n ≥ k, so the base case is n = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If m = 1, the the result follows by inspecting lines 7-11 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If m ≥ 2, then m > �n k � = 1, so getNFC(n, k, m) correctly returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For the induction step on n, suppose n ≥ k + 1 and getNFC(n′, k, m′) is correct for all k ≤ n′ < n and m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then getNFC(i, k, m − 1) correctly returns for all i ∈ 1We use SageMath’s canonical label() method within the IncidenceStructure class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 5 J−{n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' To show getNFC(n, k, m) correctly returns, we use induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If m = 1, then certainly getNFC(n, k, m) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose m ≥ 2 and getNFC(n, k, m − 1) correctly returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' This shows that getNFC(i, k, m − 1) is correct for all i ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Hence the theorem follows from the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Using Algorithm 1, which can be easily extended to determine the exact value of FC(k, n) for small values of k and n, we have determined the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(4, 7) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(4, 8) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(5, 7) = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(6, 8) = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The system used to verify all our results (including those in section 4) has an Intel Xeon Processor E5-2620 v4 with 16 cores, each running at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1GHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' the system has 128GB of memory and two NUMA nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='5 have been verified within at most a couple hours, but Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3 took us more than 26 days to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let �S k � be the set of all k-subsets of a set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Define a strict total order <, called the lexicographic order, on the set �[n] k � by A < B if min(A∆B) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In this order, for fixed n and k and for any S ∈ �[n] k � , define [S] := {A ∈ �[n] k � | A ≤ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let {Sn,k i }i≥1 = �[n] k � , where Sn,k i < Sn,k j for all 1 ≤ i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If n and k are clear, we simply write Si instead of Sn,k i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The following conjecture seems to be very promising based off of our experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For fixed n > k ≥ 3, if [Sm] is an FC-family for some positive integer m and has universe size n, then FC(k, n) ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' This conjecture has been verified for all n > k ≥ 3 such that FC(k, n) is known (it is trivial for k = 3 and any n ≥ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' there is always a maximum Non-FC family of the form [Sm] for some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If the conjecture is true, then we could easily find all exact values of FC(k, n) for n ≤ 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' all we would need to do in that case is to find an integer m such that [Sm] is FC and [Sm−1] is Non-FC, giving us a result of FC(k, n) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' It can be shown using the above method that the following FC lower bounds are also exact values, assuming Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Most of these have been verified to be tight bounds within pySMT, except lower bounds of FC(k, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(4, 9) ≥ 14, FC(4, 10) ≥ 16, FC(5, 8) ≥ 21, FC(5, 9) ≥ 31, FC(5, 10) ≥ 44, FC(6, 9) ≥ 42, FC(6, 10) ≥ 71, FC(7, 10) ≥ 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The remaining values of FC(k, n) for 5 ≤ k < n ≤ 10 are undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 2All code used in this paper can be accessed here: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='com/KenanWood/Local- Configurations-in-Union-Closed-Families 6 k\\n 4 5 6 7 8 9 10 3 3 3 4 4 5 5 6 4 5 7 10 12 14 16 5 14 21 31 44 6 26 42 71 7 85 Table 1: FC-values Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For each pair (k, n) ∈ {(4, 9), (4, 10), (5, 8), (5, 9), (5, 10), (6, 9), (6, 10), (7, 10)}, the family [Sn,k m−1] as defined above is Non-FC, where m is the proposed lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For the remaining pairs (k, n) when 5 ≤ k < n ≤ 10, we can easily show that the family �[n] k � is Non-FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Assuming Conjecture 1 is true, Table 1 shows a complete classification of FC- values for (k, n) ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' , 7} × {4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' , 10}, where no entry at (k, n) indicates that FC(k, n) is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' To find upper bounds of FC(k, n), we generalize and tighten a result of Morris [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Morris showed that FC(4, n) ≤ 7 360n4, though without an explicit proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The following theorem improves and generalizes this bound, which yields improved explicit upper bounds on FC(k, n) for 4 ≤ k ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If m0 = FC(k, n0) ≤ �n0 k � , then FC(k, n) ≤ 1 + � (m0 − 1) n0 · · · (n0 − k + 1) · n · · · (n − k + 1) � ≤ �n k � for all n > n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose m0 = FC(k, n0) ≤ �n0 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let n > n0 and m := 1+ � (m0 − 1) · (n0−k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n−k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' � , noting that m = 1 + � (m0−1) n0···(n0−k+1) · n · · · (n − k + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since n > n0, we know (n0−k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n−k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' This implies m ≤ � 1 + ��n0 k � − 1 � (n0 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' � = � 1 + �n0 k � (n0 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' − (n0 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' � ≤ � n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n0 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' · (n0 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' � = �n k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 7 Next, let A be a family of m distinct k-sets with a universe of size at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Define A0 := A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' for i ≥ 0, recursively define Ai+1 := Ai if |U(Ai)| < n − i, and otherwise, Ai+1 := Ai − Ai x, where x ∈ U(Ai) minimizes |Ai x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' It follows that |U(Ai)| ≤ n − i for all i ≥ 0 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since FC(k, n0) = m0, it suffices to prove |An−n0| ≥ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' To this end, for any i ≥ 0, the pigeonhole principle shows that there is some x ∈ U(Ai) such that |Ai x| ≤ k|Ai| |U(Ai)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If |U(Ai)| = n − i, then |Ai+1| ≥ |Ai| − k|Ai| |U(Ai)| = |Ai| � 1 − k n − i � = |Ai| �n − i − k n − i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Otherwise, |Ai+1| = |Ai| by construction, so that the inequality still holds since n−i−k n−i < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In writing |An−n0| = |A0| · n−n0−1 � i=0 |Ai+1| |Ai| , we obtain |An−n0| ≥ m · n−n0−1 � i=0 �n − i − k n − i � ≥ � 1 + (m0 − 1) · (n0 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' � �(n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' n0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' (n0 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' � > m0 − 1, so that |An−n0| ≥ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Thus A ⊇ An−n0 is an FC-family and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The following bounds hold: FC(4, n) ≤ 1 + � 11 1680 · n(n − 1)(n − 2)(n − 3) � for n > 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(5, n) ≤ 1 + � 13 2520 · n(n − 1)(n − 2)(n − 3)(n − 4) � for n > 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(6, n) ≤ 1 + � 5 4032 · n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) � for n > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' This is an immediate consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='7 along with Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='4, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' FC(4, 9) ≤ 21, FC(5, 8) ≤ 36, and FC(6, 9) ≤ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' This is an immediate consequence of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 8 3 Symmetry in FC-families In this section, we answer two previously unsolved questions regarding symmetry in union-closed families with respect to local configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1 Automorphisms in FC-families Given a union-closed family A containing ∅ with U(A) = [n], let B(A) be the set of all union-closed B ⊆ P([n]) such that A ⊎ B = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Recall that P A = {c ∈ Rn ≥0 : � i∈[n] ci = 1 ∧ ∀B ∈ B(A), � i∈[n] ci|Bi| ≥ |B|/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then By Poonen’s Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1, A is FC if and only if P A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' As outlined in our introduction, the following is a generalization of Vaughan’s [12] question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Given a union-closed family A containing ∅, if P A is nonempty, then is there always some c ∈ P A such that ci = cj whenever there is an automorphism of A that maps i to j?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We prove that this question holds in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' First, let Aut(A) denote the set of all automorphisms of A and note that Aut(A) is a group under function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let A be a union-closed family containing ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If P A is nonempty, then there is some c ∈ P A such that ci = cj whenever there is an automorphism of A that maps i to j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Without loss of generality, assume U(A) = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose x ∈ P A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For any φ ∈ Aut(A), we first show that (xφ(i))i∈[n] ∈ P A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' it suffices to show that for any B ∈ B(A), we have � i∈[n] xφ(i)|Bi| ≥ |B|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Consider the image φ(B) = {φ(S) : S ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If A′ ∈ A and B′ ∈ φ(B), then there are A ∈ A and B ∈ B such that A′ = φ(A) and B′ = φ(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' the first holds since φ−1 ∈ Aut(A), so that A = φ−1(A′) ∈ A, and the second is by construction of φ(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' this shows A′ ∪ B′ = φ(A ∪ B) ∈ φ(B), which implies φ(B) ∈ B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since x ∈ P A, then � i∈[n] xi|φ(B)i| ≥ |φ(B)|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since φ is a bijection, � i∈[n] xφ(i)|φ(B)φ(i)| ≥ |φ(B)|/2, which shows � i∈[n] xφ(i)|Bi| ≥ |B|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Thus (xφ(i))i∈[n] ∈ P A for any φ ∈ Aut(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Consider the convex combination of elements of P A, c = 1 | Aut(A)| · � φ∈Aut(A) (xφ(i))i∈[n] = 1 | Aut(A)| · \uf8eb \uf8ed � φ∈Aut(A) xφ(i) \uf8f6 \uf8f8 i∈[n] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For each i ∈ [n], let si = � φ∈Aut(A) xφ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 9 Then we obtain c = 1 | Aut(A)| · (si)i∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For any φ0 ∈ Aut(A), every automorphism in Aut(A) can be written as a unique left composition with φ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' that is, Aut(A) = {φ ◦ φ0 : φ ∈ Aut(A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then, for every i, j ∈ [n] such that there is some automorphism φ0 ∈ Aut(A) mapping i to j, we know si = � φ∈Aut(A) x[φ◦φ0](i) = � φ∈Aut(A) xφ(j) = sj, so that ci = cj as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since c is a convex combination of points in a polyhedron, c ∈ P A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 A Result on Transitive Families of 3-sets In an Abelian group (G, +), if R ⊆ G and g ∈ G, we define the translation of R by g in G as the set g + R := {g + r : r ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The set of all translations (by some element g ∈ G) of R in G is denoted T(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Given a family A, the union-closure of A, or the family generated by A, is defined as the union-closed family ⟨A⟩ := {� S∈A′ S : A′ ⊆ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Given some 3-set R ⊂ Zn, the authors of [5] ask and were unable to verify that the family generated by A = T(R × {0}) ∪ T({0} × R) ⊆ Z2 n necessarily satisfies Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The authors remark that this family is transitive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' that is, for any x, y ∈ Z2 n, there is an automorphism φ ∈ Aut(A) such that φ(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let A be a family of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let d(x) = |Ax| be the degree of x in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The family A is regular if d(x) = d(y) for all x, y ∈ U(A), in which case the degree of A is the common degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let A be a regular family of 3-sets with degree k ≥ 2 and universe of size n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then A is FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since � i∈U(A) d(i) = � A∈A |A|, we obtain kn = 3m, so that m = kn/3 ≥ 2n/3, where m = |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since FC(3, n) ≤ 2n/3 for all n ≥ 5, this shows A is FC if n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If n = 4, then m ≥ 4 (in fact equality holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In this case, since FC(3, 4) = 3, we know A is FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let R ⊂ Zn be a 3-set, where n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then the family A = T(R × {0}) ∪ T({0} × R) ⊆ Z2 n is FC, and thus, ⟨A⟩ satisfies Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' It is clear that A is a regular family of 3-sets with degree at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Fur- thermore, |U(A)| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The result follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 10 4 Deducing Frankl’s Conjecture From New Local Conditions In this section, we give a sufficiency condition for when a family F satisfies Frankl’s conjecture from its local structure with respect to some C ⊆ U(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We use this result to determine many new local configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The following lemma is central to this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We should note that this is a generalization of an idea presented in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In particular, the construction of graph G and map f was inspired by the work of Ellis, Ivan, and Leader [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let F be a union-closed family containing some nonempty set and ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let C ⊆ U(F) be nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then there is some i ∈ C such that |Fi| ≥ |F|/2 if there is some weight function w : C → R>0 such that for all X ′ ⊆ X , we have |X ′| ≤ |{F ∪ S : F ∈ F ∩ P(C), S ∈ X ′, w(S) + w(F ∪ S) ≥ w(C)}|, where X = {S ∩ C : S ∈ F, w(S ∩ C) < w(C)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose there is some nonempty C ⊆ U(A) and weight function w : C → R>0 such that for all X ′ ⊆ X , we have |X ′| ≤ |{F ∪ S : F ∈ F ∩ P(C), S ∈ X ′, w(S) + w(F ∪ S) ≥ w(C)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Construct a bipartite graph G with partite sets X and Y := {S ∩ C : S ∈ F, w(S ∩ C) > w(C)/2}, where S ∈ X is joined with T ∈ Y via an edge if w(S)+w(T) ≥ w(C) and there is some F ∈ F with F ⊆ C such that T = F ∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' It is clear that if S ∈ X and F ∈ F with F ⊆ C, then F ∪ S ∈ {F ′ ∩ C : F ′ ∈ F} since F is union- closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Thus the above condition is equivalent to saying for all X ′ ⊆ X , we have |X ′| ≤ |NG(X ′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' By Hall’s Theorem, G has an X -saturating matching m : X → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let X F = {S ∈ F : w(S∩C) < w(C)/2} and let YF = {S ∈ F : w(S∩C) > w(C)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Define a map f : X F → YF by f(F) = m(F ∩ C) ∪ (F − C) for all F ∈ X F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We first verify that f is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose F ∈ X F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then by definition of edges in G, f(F) ∩ C ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' There is some F ′ ∈ F with F ′ ⊆ C such that m(F ∩ C) = (F ∩ C) ∪ F ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' this implies f(F) = (F ∩ C) ∪ F ′ ∪ (F − C) = F ∪ F ′ ∈ F by union-closure of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Hence f(F) ∈ YF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Now we show f is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If f(A) = f(B) for some A, B ∈ X F, then A − C = B − C since Y ⊆ P(C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' furthermore, since m is a 11 matching, and thus an injection, A∩C = B ∩C, showing f is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Additionally, by definition of the edges of G, we know that for each S ∈ X F, w(S ∩ C) + w(f(S) ∩ C) ≥ w(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Thus, we know � S∈F (w(S ∩ C) − w(C)/2) ≥ 1 2 � S∈X F (w(S ∩ C) + w(f(S) ∩ C) − w(C)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' It is easy to show with a simple combinatorial argument that 1 w(C) � i∈C w(i) · |Fi| ≥ |F| 2 ⇔ � i∈C w(i) w(C) · |Fi| ≥ |F| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Since � i∈C w(i)/w(C) = 1, we have a weighted average of all |Fi| over i ∈ C being at least |F|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' This shows that there is some i ∈ C such that |Fi| ≥ |F|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' For the following theorem, we let X (F, C, w) := {S ∩ C : S ∈ F, w(S ∩ C) < w(C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' When applying this theorem, since for any nonempty C and F that is union- closed, F ∩ P(C) is union-closed, we only need to consider families A ⊆ P(C) that are union-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let C be a nonempty set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' let w : C → R>0 be a weight function on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let X ⊆ P(C) such that for each S ∈ X , we have w(S) < w(C)/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' let A ⊆ P(C) with ∅ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose for all X ′ ⊆ X that |X ′| ≤ |{A ∪ S : A ∈ A, S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then for any union-closed family F such that X (F, C, w) ⊆ X and A ⊆ F, there is some i ∈ C such that |Fi| ≥ |F|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let F be a union-closed family such that X (F, C, w) ⊆ X and A ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Observe that for any X ′ ⊆ X (F, C, w) ⊆ X , {A ∪ S : A ∈ A, S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)} is a subset of {A ∪ S : A ∈ F ∩ P(C), S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)}, since A ⊆ F ∩ P(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='1, there is some i ∈ C such that |Fi| ≥ |F|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 12 We say that any family A satisfying the consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 is quasi-FC with respect to (C, w, X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We remark that a special case of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 is when X = {S ⊆ C : w(S) < w(C)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In this case, any union-closed family F containing A satisfies X (F, C, w) ⊆ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then if the conditions of the theorem are met, we obtain that A is an FC-family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' A natural question at this point is the converse in this special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If the converse is true, then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 is simply a strictly more general version of Poonen’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Given C, w, X , A, let Y := {A ∪ S : A ∈ A, S ∈ X , w(S) + w(A ∪ S) ≥ w(C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let IP(C, w, X , A) denote the following binary linear program: Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' � S∈X xS − � T∈Y yT � S∈X xS ≥ � T∈Y yT + 1 xS ≤ yA∪S ∀S ∈ X , ∀A ∈ A, w(S) + w(A ∪ S) ≥ w(C) xS ∈ {0, 1} ∀S ∈ X yT ∈ {0, 1} ∀T ∈ Y Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let C be a nonempty set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' let w : C → R>0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' let X ⊆ P(C) such that for each S ∈ X , we have w(S) < w(C)/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' let A ⊆ P(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 are satisfied if and only if IP(C, w, X , A) is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then there is some X ′ ⊆ X such that |X ′| ≥ |{A ∪ S : A ∈ A, S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)}| + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let Y′ = {A∪S : A ∈ A, S ∈ X ′, w(S)+w(A∪S) ≥ w(C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let xS = � 1 if S ∈ X ′ 0 otherwise and yT = � 1 if T ∈ Y′ 0 otherwise for S ∈ X and T ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' By assumption, � S∈X xS ≥ 1 + � T∈Y yT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Consider any S ∈ X , A ∈ A with w(S) + w(A ∪ S) ≥ w(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If S ̸∈ X ′, then xS = 0 so that xS ≤ yA∪S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' otherwise, A ∪ S ∈ Y′, so that yA∪S = 1 and again xS ≤ yA∪S, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Thus (x, y) is feasible in IP(C, w, X , A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Suppose (x, y) is feasible in IP(C, w, X , A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let X ′ = {S ∈ X : xS = 1} and Y′ = {T ∈ Y : yT = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' It suffices to show {A ∪ S : A ∈ A, S ∈ X ′, w(S) + w(A ∪ S) ≥ w(C)} ⊆ Y′ since the feasibility of (x, y) in IP(C, w, X , A) implies |X ′| > |Y′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Consider any A ∈ A, S ∈ X ′ such that w(S) + w(A ∪ S) ≥ w(C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' let T = A ∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Observe xS = 1, so that yT = 1 since xS ≤ yA∪S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Thus T ∈ Y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 13 With these tools in mind, we find the following results, which may be indepen- dently verified with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Given a family A, recall that the union-closure of A is defined as the union-closed family ⟨A⟩ := {� S∈A′ S : A′ ⊆ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let C = [3] and A = ⟨{{1, 2, 3}}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then any union-closed F ⊇ A such that |{S ∩ C : S ∈ F, |S ∩ C| = 1}| = 0 satisfies Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let w : C → R>0 be defined by w(x) = 1 for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' If X = {∅}, then we find that IP(C, w, X , A) is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The result follows from Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let C = [4] and A = ⟨{{1, 2, 3}, {2, 3, 4}}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then any union-closed F ⊇ A such that |{S ∩ C : S ∈ F, |S ∩ C| = 1}| ≤ 2 satisfies Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let w : C → R>0 be defined by w(x) = 1 for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We must only consider X = {∅, {1}, {2}}, X = {∅, {1}, {4}}, and X = {∅, {2}, {3}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In any case, IP(C, w, X , A) is infeasible, showing A is quasi-FC with respect to (C, w, X ) by Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let C = [5], A = ⟨{{1, 2, 3}, {1, 4, 5}}⟩, w(x) = 1, X = {∅, {1, 2}, {2, 3}, {1, 3}, {4}, {5}, {4, 5}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then A is quasi-FC with respect to (C, w, X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' By Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3, the result follows by showing IP(C, w, X , A) is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let C = [6] and A = ⟨{{1, 2, 3}, {4, 5, 6}}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then any union-closed F ⊇ A such that {S ∩ C : S ∈ F, |S ∩ C| = 1} = ∅ and |{S ∩ C : S ∈ F, |S ∩ C| = 2}| ≤ 5 satisfies Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Additionally if w(x) = 1 and X = {∅, {3, 4}, {4, 6}, {2, 3}, {1, 2}, {2, 6}, {4, 5}, {3, 6}, {2, 5}, {2, 4}, {5, 6}, {1, 5}, {1, 3}}, then A is quasi-FC with respect to (C, w, X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The first part can be proved using a brute-force check of of all possible X with ∅, five 2-subsets of C and no singletons, and choosing w(x) = 1 for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' The second part is a simple application of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' A natural question regarding quasi-FC families is the converse of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We do not have a complete answer to this question, but we exhibit a counterexample to a natural attempt to the converse below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In particular, if c ∈ P A, so that A is FC, then A may not be quasi-FC with respect to (C, w, X ), where C = U(A) and w(x) = cx and X = {S ⊆ C : w(S) < w(C)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Let C = [5], c = (1/4, 1/4, 1/6, 1/6, 1/6), w(x) = cx, A = ⟨{{1, 2, 3}, {1, 2, 4}, {1, 2, 5}}⟩, and X = {S ⊆ C : w(S) < w(C)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Then c ∈ P A, yet A is not quasi-FC with respect to (C, w, X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 14 5 Conclusion In this work, we were able to find many new values of FC(k, n) for k ≥ 4, of which only two were previously known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We did this within a safe and exact computational framework using SMT solving to verify results, so our results can be trusted and independently verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Additionally we answered two previously unsolved questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' One was an older question of Vaughan that shows the dimension of Poonen’s poly- hedron P A can be reduced from |U(A)| to the number of orbits of Aut(A) through a projection, which shrinks the search space and reduces computational work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We also answered a question of Ellis, Ivan and Leader [5] related to union-closed families generated by 3-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Our solution highlights the continual importance of FC-families in that they provide simple solutions to difficult problems related to the Union-Closed Sets conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' We believe several directions merit further attention, including Conjecture 1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Can we extend our technique to prove Frankl’s conjecture for union- closed families F such that X (F, C, w) ⊆ X without the restriction that F contains A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' In addition, it would be interesting to known whether Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content='2 is a strict generalization of Poonen’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Affirmative answers to these questions would be a significant step towards a deeper understanding of local configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' References [1] Nikolaj Bjørner and Leonardo de Moura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Z3: An Efficient SMT Solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Tools and Algorithms for the Construction and Analysis of Systems, 4963:337–340, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' [2] Henning Bruhn 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FC-families and improved bounds for Frankl’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Euro- pean Journal of Combinatorics, 27(2):269–282, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' [9] Bjorn Poonen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Union-Closed Families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Journal of Combinatorial Theory, Series A, 59(2):253–268, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' [10] Jonad Pulaj.' metadata={'source': 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+page_content=' [12] Theresa P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Vaughan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' A Note on the Union-Closed Sets Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' Journal of Combinatorial Mathematics and Combinatorial Computing, 45:97–110, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAzT4oBgHgl3EQfYPxj/content/2301.01331v1.pdf'} diff --git a/VtE2T4oBgHgl3EQfXwe7/content/tmp_files/2301.03848v1.pdf.txt b/VtE2T4oBgHgl3EQfXwe7/content/tmp_files/2301.03848v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..79d0fa4909c038f68b4d168dcd6827c906739c38 --- /dev/null +++ b/VtE2T4oBgHgl3EQfXwe7/content/tmp_files/2301.03848v1.pdf.txt @@ -0,0 +1,230 @@ +THE USE OF NEW TECHNOLOGIES TO SUPPORT PUBLIC +ADMINISTRATION. +SENTIMENT ANALYSIS AND THE CASE OF THE APP ’IO’ +Vincenzo Miracula +Dipartimento di Fisica e Astronomia “E. Majorana” +Università di Catania +Catania +vincenzo.miracula@phd.unict.it +Antonio Picone +Dipartimento di Fisica e Astronomia “E. Majorana” +Università di Catania +Catania +antonio.picone@phd.unict.it +ABSTRACT +App IO is an app developed for the Italian PA. It is definitely useful for citizens to interact with the +PA and to get services that were not digitized yet. Nevertheless, it was not perceived in a good way by +the citizens and it has been criticized. As we wanted to find the root that caused all these bad reviews +we scraped feedback from mobile app stores using custom-coded automated tools and - after that - +we trained two machine learning models to perform both sentiment analysis and emotion detection to +understand what caused the bad reviews. +Keywords Public Administration, Artificial Intelligence, Digital Transformation, Decision Making +1 +Introduction +Since 2005, there has been an increasing development of digitization within the public administration that sees the +introduction of the use of technology as a privileged tool in the management of administrative activities. The main +objective is to promote digitization in administrations in order to achieve greater efficiency in their activities in internal +relations, between different administrations, and between the latter and private individuals. The entry of artificial +intelligence into public action, however, needs to be accompanied by an adequate regulatory framework to guarantee +the rights of those administered. +The notion of digital transformation has gained significant attention in the literature[1]. Although approaches to the +definition of digital transformation vary[2], most authors suggest that digital transformation involves the use of ICT +technology to create fundamentally new capabilities in business, public administration[3] and people’s lives[4]. The +theory emphasizes the importance of digitization to optimize the public value of government services for citizens[5] +as well as to increase the efficiency of government functions by implementing lean government models[6]. Digital +technologies stimulate citizen participation[7] and support customer involvement in the co-production and co-creation +of public value[8]. The digital transformation thus has a significant impact not only on the accessibility and quality of +public services but also on the way other functions of public administration are carried out[9] (like policy development, +regulation, and enforcement, etc.). +A strong emphasis in the current phase is placed on government solutions as a platform[10] and, consequently, on data- +driven governance. In the academic literature, there are also other approaches to the staging of digital transformation +in public administration. Janowski[11], for instance, suggests that digital transformation is not a one-off event or +project with a clear start and end date, but rather an ongoing process involving changes both in internal processes and +procedures and in the way the public administration communicates with its main beneficiaries. +It is assumed that the results of this process will be significant, but so far no uniform approaches have been implemented +to measure these results. Although digital transformation is an evolving process, the measures used to evaluate this +process should also evolve according to the stage of the digital transformation. For instance, in the early stages of +e-government, indicators such as the ratio of e-services provided online are considered relevant. +arXiv:2301.03848v1 [cs.SI] 10 Jan 2023 + +The use of new technologies to support Public Administration. +In the more mature stages of digital transformation, the effects are different: smart government leads to the replacement +of official portals by automated interactions and, thus, results in the reduction of the types of public services[9]. This +contribution intends to present new tools for measuring the results of digital transformation through the use of sentiment +analysis and social network analysis. +2 +The use of the IO app in public administration +As of 18 April 2020, the IO app is downloadable from the online stores for Android and Apple. The app, conceived +and developed by the Digital Transformation Team, was still in Beta version in April 2020, with a limited number of +services available, but making it downloadable was an important first step to start testing it on a large scale. The IO app, +in fact, is the platform designed to allow all citizens to have a new and single telematic access point to the services, +information, and communications of the public administration, thus enabling them to use national and local public +services from their smartphones in a simple, modern and secure way. +Two years after its release on online stores, there have been many developments, mainly related to the fact that the app +was chosen as a tool to access some of the services activated in the months of the Covid-19 pandemic, such as the +holiday bonus, the ’Cashback’ program and the green-pass. The regulatory basis of the IO project is Art. 64 bis of the +new Digital Administration Code[12] (CAD), which provides a single point of telematic access to public administration +services and highlights a fundamental change in the relationship between citizens and PA, centered on three key aspects: +simplicity, speed, and transparency. The ultimate goal is to bring the citizen closer to the administration, making +simple mechanisms that often still prove to be complex and cumbersome. The IO app is, therefore, the tool designed to +concretely enable Digital Citizenship, providing citizens with a direct connection (through their smartphone) with PA +services and communications, a single access point for the provision and use of public services. +As can be understood, for now, it is always the citizen who has to express his or her willingness to interact with the Public +Administration through the IO app. By using the ’IO’ app, the citizen agrees to interface with the Public administration +he or she needs to get in touch with according to the provisions of the 2019 Three-Year Plan for Information Technology +in Public Administration. +In the document ’Strategy for Technological Innovation and the Digitisation of the Country’[13] (also called, Plan 2025) +presented on 17 December 2019 by Minister Pisano, the IO app is included among the top 20 Actions to Transform +the Country. In fact, page 15 of the document states that "Italy, at least digitally, should be one big municipality that +treats all its citizens equally", and again, "IO, is the public services app that transforms the relationship between citizens +and Public Administration, putting people at the center and erasing complexity: a single interface to access all public +services directly from your smartphone after identifying yourself with your digital identity." +3 +Methodology +In order to extract data for the purposes of our analysis, we decided to extract user reviews from the two main stores for +mobile devices[14]: AppStore for Apple iOS users and Play Store for Android users. To date, both account for 95% of +total downloads, so they constitute a representative sample of the thinking of the users of such applications in the Italian +context. +In order to extract the reviews, code was written in Python, so as to automate the extraction of the reviews, not only of the +text but also of the author of the review and the stars that were assigned (from 1 to 5). This process is called ’scraping’, +an English term that literally means ’scraping’. Since there is no official API, which would allow programmatic access, +the only alternative for data extraction is scraping. This technique consists of accessing the various tags of the HTML +source code in order to extract the desired fields. This approach certainly requires higher skills than the use of an API +and a further cleaning of the data, sometimes soiled by the presence of HTML tags that must be removed, in order to +obtain text in its classic form. +The dataset, consisting of the cleaned textual reviews and the metadata mentioned, was saved in serialized JSON +(JavaScript Object Notation) format, for easy reading by both a human being and a computer, as further analysis of +sentiment and emotions followed, obtained through the use of machine learning techniques applied to Natural Language +Processing. In particular, BERT (Bidirectional Encoder Representations from Transformers) was used, a type of neural +network, based on Transformers, devised by Google[15], thanks to whose attention mechanism, each word within +the same sentence is related both to the word that follows it, and to the previous one, allowing the entire context of a +sentence to be maintained. +We therefore, on the basis of an existing model for Italian, carried out what is called fine-tuning, i.e. starting from a +general model, not specific to any task, the model was trained with our data to perform and perform better in the two +2 + +The use of new technologies to support Public Administration. +tasks specific to our objective. In this way, we obtained a sentiment analysis model capable of discerning sentences +into two categories: positive and negative; furthermore, as already mentioned, a model for emotion detection was also +realized. There has long been debate as to which emotions are worth training our model on it is necessary to have a +psychological background on the matter and the literature on the subject has proved to be of fundamental importance. +Paul Ekman[16] is an American psychologist who came up with the universal theory of emotions, according to which +there are seven primary emotions: fear, anger, joy, sadness, contempt, disgust, and surprise; but these have been +reformulated and amalgamated to become six in its latest official version, namely: fear, anger, joy, sadness, disgust, and +surprise. However, according to a recent study[17], it is proposed that human beings have four basic emotions: fear, +anger, joy, and sadness. Over the course of time, other authors have also proposed the above emotions as the four basic +emotions[18],[19]. In light of these studies, our model follows in the footsteps and ideas of the most recent studies on +the subject, thus managing to categorize input sentences into the four categories of emotions mentioned above. +4 +Network analysis +Social media platforms have become key channels for communication and information dissemination, making it +increasingly important to understand how information spreads within these networks. In general, in social network +analysis, nodes are people and ties are all the social connections between them - for example, friendship, marital/family +ties, or financial ties. +Social network analysis (SNA) is a field of study involving the use of statistical and mathematical techniques to analyze +and understand relationships and patterns within a network of individuals or organizations. It is often used to identify +key actors and understand the dynamics of social, professional, and communication networks. The objective of social +network analysis is to understand a community by mapping the relationships that connect it as a network and then +trying to identify key individuals and groups within the network and/or associations between individuals. +Twitter is an excellent platform to observe these behaviors. For this reason, we decided to collect tweets to try to +understand what are the dynamics that lead to the formation of communities on Twitter, mapping the relationships +that connect people as a network and then trying to bring out the key individuals, and groups within the network and +associations between individuals. +In an SNA, a network is typically represented as a graph, with nodes representing individual actors and links representing +relationships between them (e.g. friendship, collaboration). Several metrics can be used to analyze these networks, +including measures of centrality (e.g. degree) measures of community structure (e.g. modularity), and measures of +network evolution (e.g. preferential attachment). +SNA has a wide range of applications, including studying the spread of information and ideas, identifying influencers, +understanding the structure and dynamics of social networks, and predicting the formation of new relationships. It is +used in fields such as sociology, psychology, anthropology, communication, and information technology and has also +been applied to the analysis of networks in business, politics, and public health. +5 +Results +Social networks represent an emerging challenging sector where the natural language expressions of people can be easily +reported through short but meaningful text messages. Key information that can be grasped from social environments +relates to the polarity of text messages. +The first result of our research is related to the citizen’s perception of the IO app. We then extracted 62986 reviews from +the various stores and through the use of Natural Language Processing[20] techniques such as sentiment analysis and +emotion detection, we have that 75.91% of the reviews have negative sentiment and 60.5% negative emotions (34.3% +sadness, 26.2% anger). +The second result relates to SNA. From our analysis, the SNA presents 5 different and distinct communities calculated +according to Louvain’s algorithm[21]. The community detection thus constructed shows moderately connected +communities with a modularity value of 0.6. We also note that the graph presents itself according to a Barabasi- +Albert[22] with 62986 nodes with an average degree (i.e. the average number of links) of 1.37 and a density of +0.573(1). +These results, as shown in Fig. 1 lead to the hypothesis that individuals struggle to change their opinion and sentiment +toward the use of the IO app (and by extension towards the digitization path undertaken by the PA). +3 + +The use of new technologies to support Public Administration. +(H) +Table 1: Network results +Statistics +Value +Number of nodes +62986 +Mean degree +1.37 +Density +0.573 +Modularity +0.6 +Figure 1: network analysis IO App. +6 +Conclusion +This paper brings attention to how computational social science and network science can both explain the complex +dynamics of controversial and challenging topics. The digital ecosystem not only evolves social network communication +but also provides the social researcher with useful data to explain social-complex dynamics. +Using natural language processing techniques, such as sentiment analysis and emotion detection, we proposed a new +way to measure the impact of digital transformation and how the resulting analysis can be applied to provide legislators +with meta-evaluation tools, in this case starting with how the app is perceived by citizens. +We noticed that many of the comments found within the dataset we created are not related to the app itself, a large +number of them are negative towards the government and the PA itself. The comments are therefore just a proxy for the +citizen to criticize policy choices. +References +[1] Vesna Bosilj Vukši´c, Lucija Ivanˇci´c, and Dalia Suša Vugec. A preliminary literature review of digital transformation +case studies. International Journal of Computer and Information Engineering, 12(9):737–742, 2018. +[2] João Reis, Marlene Amorim, Nuno Melão, and Patrícia Matos. Digital transformation: a literature review and +guidelines for future research. In World conference on information systems and technologies, pages 411–421. +Springer, 2018. +4 + +The use of new technologies to support Public Administration. +[3] Colin Lankshear and Michele Knobel. Digital literacies: Concepts, policies and practices, volume 30. Peter +Lang, 2008. +[4] George Westerman, Claire Calméjane, Didier Bonnet, Patrick Ferraris, Andrew McAfee, et al. Digital transfor- +mation: A roadmap for billion-dollar organizations. MIT Center for digital business and capgemini consulting, +1:1–68, 2011. +[5] Frank Bannister and Regina Connolly. Ict, public values and transformative government: A framework and +programme for research. Government Information Quarterly, 31(1):119–128, 2014. +[6] Marijn Janssen and Elsa Estevez. Lean government and platform-based governance—doing more with less. +Government Information Quarterly, 30:S1–S8, 2013. +[7] Luis Felipe Luna-Reyes. Opportunities and challenges for digital governance in a world of digital participation. +Information polity, 22(2-3):197–205, 2017. +[8] Antonio Cordella and Andrea Paletti. Icts and value creation in public sector: Manufacturing logic vs service +logic. Information Polity, 23(2):125–141, 2018. +[9] Elena Dobrolyubova. Measuring outcomes of digital transformation in public administration: Literature review +and possible steps forward. NISPAcee Journal of Public Administration and Policy, 14(1):61–86, 2021. +[10] Tim O’Reilly. Government as a platform. Innovations: Technology, Governance, Globalization, 6(1):13–40, 2011. +[11] Tomasz Janowski. Digital government evolution: From transformation to contextualization, 2015. +[12] Consiglio dei Ministri. Codice dell’amministrazione digitale, 2022. +[13] Ministro per l’Innovazione tecnologica e la digitalizzazione. Piano nazionale innovazione 2025, 2022. +[14] C Scott Hemphill. Intellectual property and competition law. Forthcoming, Oxford Handbook of Intellectual +Property Law (Rochelle C. Dreyfuss & Justine Pila eds. 2017), 2017. +[15] Ulrike von Luxburg, I Guyon, S Bengio, H Wallach, R Fergus, SVN Vishwanathan, and R Garnett. Advances in +neural information processing systems 30. In 31st annual conference on neural information processing systems +(NIPS 2017), Long Beach, California, USA, pages 4–9, 2017. +[16] Paul Ekman. An argument for basic emotions. Cognition & emotion, 6(3-4):169–200, 1992. +[17] Rachael E Jack, Oliver GB Garrod, and Philippe G Schyns. Dynamic facial expressions of emotion transmit an +evolving hierarchy of signals over time. Current biology, 24(2):187–192, 2014. +[18] Simeng Gu, Fushun Wang, Tifei Yuan, Benyu Guo, and Jason H Huang. Differentiation of primary emotions +through neuromodulators: review of literature. International Journal of Neurology Research, 1(2):43–50, 2015. +[19] Fushun Wang and Alfredo Pereira. Neuromodulation, emotional feelings and affective disorders. Mens sana +monographs, 14(1):5, 2016. +[20] KR1442 Chowdhary. Natural language processing. Fundamentals of artificial intelligence, pages 603–649, 2020. +[21] Vincent D Blondel, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre. Fast unfolding of communi- +ties in large networks. Journal of statistical mechanics: theory and experiment, 2008(10):P10008, 2008. +[22] Réka Albert and Albert-László Barabási. Statistical mechanics of complex networks. Reviews of modern physics, +74(1):47, 2002. +5 + diff --git a/VtE2T4oBgHgl3EQfXwe7/content/tmp_files/load_file.txt b/VtE2T4oBgHgl3EQfXwe7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..659299842e840dd88470b30f38e8b49f00417335 --- /dev/null +++ b/VtE2T4oBgHgl3EQfXwe7/content/tmp_files/load_file.txt @@ -0,0 +1,183 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf,len=182 +page_content='THE USE OF NEW TECHNOLOGIES TO SUPPORT PUBLIC ADMINISTRATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' SENTIMENT ANALYSIS AND THE CASE OF THE APP ’IO’ Vincenzo Miracula Dipartimento di Fisica e Astronomia “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Majorana” Università di Catania Catania vincenzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='miracula@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='unict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='it Antonio Picone Dipartimento di Fisica e Astronomia “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Majorana” Università di Catania Catania antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='picone@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='unict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='it ABSTRACT App IO is an app developed for the Italian PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' It is definitely useful for citizens to interact with the PA and to get services that were not digitized yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Nevertheless, it was not perceived in a good way by the citizens and it has been criticized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' As we wanted to find the root that caused all these bad reviews we scraped feedback from mobile app stores using custom-coded automated tools and - after that - we trained two machine learning models to perform both sentiment analysis and emotion detection to understand what caused the bad reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Keywords Public Administration, Artificial Intelligence, Digital Transformation, Decision Making 1 Introduction Since 2005, there has been an increasing development of digitization within the public administration that sees the introduction of the use of technology as a privileged tool in the management of administrative activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The main objective is to promote digitization in administrations in order to achieve greater efficiency in their activities in internal relations, between different administrations, and between the latter and private individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The entry of artificial intelligence into public action, however, needs to be accompanied by an adequate regulatory framework to guarantee the rights of those administered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The notion of digital transformation has gained significant attention in the literature[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Although approaches to the definition of digital transformation vary[2], most authors suggest that digital transformation involves the use of ICT technology to create fundamentally new capabilities in business, public administration[3] and people’s lives[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The theory emphasizes the importance of digitization to optimize the public value of government services for citizens[5] as well as to increase the efficiency of government functions by implementing lean government models[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Digital technologies stimulate citizen participation[7] and support customer involvement in the co-production and co-creation of public value[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The digital transformation thus has a significant impact not only on the accessibility and quality of public services but also on the way other functions of public administration are carried out[9] (like policy development, regulation, and enforcement, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' A strong emphasis in the current phase is placed on government solutions as a platform[10] and, consequently, on data- driven governance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In the academic literature, there are also other approaches to the staging of digital transformation in public administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Janowski[11], for instance, suggests that digital transformation is not a one-off event or project with a clear start and end date, but rather an ongoing process involving changes both in internal processes and procedures and in the way the public administration communicates with its main beneficiaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' It is assumed that the results of this process will be significant, but so far no uniform approaches have been implemented to measure these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Although digital transformation is an evolving process, the measures used to evaluate this process should also evolve according to the stage of the digital transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' For instance, in the early stages of e-government, indicators such as the ratio of e-services provided online are considered relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='03848v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='SI] 10 Jan 2023 The use of new technologies to support Public Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In the more mature stages of digital transformation, the effects are different: smart government leads to the replacement of official portals by automated interactions and, thus, results in the reduction of the types of public services[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' This contribution intends to present new tools for measuring the results of digital transformation through the use of sentiment analysis and social network analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' 2 The use of the IO app in public administration As of 18 April 2020, the IO app is downloadable from the online stores for Android and Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The app, conceived and developed by the Digital Transformation Team, was still in Beta version in April 2020, with a limited number of services available, but making it downloadable was an important first step to start testing it on a large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The IO app, in fact, is the platform designed to allow all citizens to have a new and single telematic access point to the services, information, and communications of the public administration, thus enabling them to use national and local public services from their smartphones in a simple, modern and secure way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Two years after its release on online stores, there have been many developments, mainly related to the fact that the app was chosen as a tool to access some of the services activated in the months of the Covid-19 pandemic, such as the holiday bonus, the ’Cashback’ program and the green-pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The regulatory basis of the IO project is Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' 64 bis of the new Digital Administration Code[12] (CAD), which provides a single point of telematic access to public administration services and highlights a fundamental change in the relationship between citizens and PA, centered on three key aspects: simplicity, speed, and transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The ultimate goal is to bring the citizen closer to the administration, making simple mechanisms that often still prove to be complex and cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The IO app is, therefore, the tool designed to concretely enable Digital Citizenship, providing citizens with a direct connection (through their smartphone) with PA services and communications, a single access point for the provision and use of public services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' As can be understood, for now, it is always the citizen who has to express his or her willingness to interact with the Public Administration through the IO app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' By using the ’IO’ app, the citizen agrees to interface with the Public administration he or she needs to get in touch with according to the provisions of the 2019 Three-Year Plan for Information Technology in Public Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In the document ’Strategy for Technological Innovation and the Digitisation of the Country’[13] (also called, Plan 2025) presented on 17 December 2019 by Minister Pisano, the IO app is included among the top 20 Actions to Transform the Country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In fact, page 15 of the document states that "Italy, at least digitally, should be one big municipality that treats all its citizens equally", and again, "IO, is the public services app that transforms the relationship between citizens and Public Administration, putting people at the center and erasing complexity: a single interface to access all public services directly from your smartphone after identifying yourself with your digital identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='" 3 Methodology In order to extract data for the purposes of our analysis, we decided to extract user reviews from the two main stores for mobile devices[14]: AppStore for Apple iOS users and Play Store for Android users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' To date, both account for 95% of total downloads, so they constitute a representative sample of the thinking of the users of such applications in the Italian context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In order to extract the reviews, code was written in Python, so as to automate the extraction of the reviews, not only of the text but also of the author of the review and the stars that were assigned (from 1 to 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' This process is called ’scraping’, an English term that literally means ’scraping’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Since there is no official API, which would allow programmatic access, the only alternative for data extraction is scraping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' This technique consists of accessing the various tags of the HTML source code in order to extract the desired fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' This approach certainly requires higher skills than the use of an API and a further cleaning of the data, sometimes soiled by the presence of HTML tags that must be removed, in order to obtain text in its classic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The dataset, consisting of the cleaned textual reviews and the metadata mentioned, was saved in serialized JSON (JavaScript Object Notation) format, for easy reading by both a human being and a computer, as further analysis of sentiment and emotions followed, obtained through the use of machine learning techniques applied to Natural Language Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In particular, BERT (Bidirectional Encoder Representations from Transformers) was used, a type of neural network, based on Transformers, devised by Google[15], thanks to whose attention mechanism, each word within the same sentence is related both to the word that follows it, and to the previous one, allowing the entire context of a sentence to be maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' We therefore, on the basis of an existing model for Italian, carried out what is called fine-tuning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' starting from a general model, not specific to any task, the model was trained with our data to perform and perform better in the two 2 The use of new technologies to support Public Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' tasks specific to our objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In this way, we obtained a sentiment analysis model capable of discerning sentences into two categories: positive and negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' furthermore, as already mentioned, a model for emotion detection was also realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' There has long been debate as to which emotions are worth training our model on it is necessary to have a psychological background on the matter and the literature on the subject has proved to be of fundamental importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Paul Ekman[16] is an American psychologist who came up with the universal theory of emotions, according to which there are seven primary emotions: fear, anger, joy, sadness, contempt, disgust, and surprise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' but these have been reformulated and amalgamated to become six in its latest official version, namely: fear, anger, joy, sadness, disgust, and surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' However, according to a recent study[17], it is proposed that human beings have four basic emotions: fear, anger, joy, and sadness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Over the course of time, other authors have also proposed the above emotions as the four basic emotions[18],[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In light of these studies, our model follows in the footsteps and ideas of the most recent studies on the subject, thus managing to categorize input sentences into the four categories of emotions mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' 4 Network analysis Social media platforms have become key channels for communication and information dissemination, making it increasingly important to understand how information spreads within these networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In general, in social network analysis, nodes are people and ties are all the social connections between them - for example, friendship, marital/family ties, or financial ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Social network analysis (SNA) is a field of study involving the use of statistical and mathematical techniques to analyze and understand relationships and patterns within a network of individuals or organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' It is often used to identify key actors and understand the dynamics of social, professional, and communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The objective of social network analysis is to understand a community by mapping the relationships that connect it as a network and then trying to identify key individuals and groups within the network and/or associations between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Twitter is an excellent platform to observe these behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' For this reason, we decided to collect tweets to try to understand what are the dynamics that lead to the formation of communities on Twitter, mapping the relationships that connect people as a network and then trying to bring out the key individuals, and groups within the network and associations between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' In an SNA, a network is typically represented as a graph, with nodes representing individual actors and links representing relationships between them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' friendship, collaboration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Several metrics can be used to analyze these networks, including measures of centrality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' degree) measures of community structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' modularity), and measures of network evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' preferential attachment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' SNA has a wide range of applications, including studying the spread of information and ideas, identifying influencers, understanding the structure and dynamics of social networks, and predicting the formation of new relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' It is used in fields such as sociology, psychology, anthropology, communication, and information technology and has also been applied to the analysis of networks in business, politics, and public health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' 5 Results Social networks represent an emerging challenging sector where the natural language expressions of people can be easily reported through short but meaningful text messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Key information that can be grasped from social environments relates to the polarity of text messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The first result of our research is related to the citizen’s perception of the IO app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' We then extracted 62986 reviews from the various stores and through the use of Natural Language Processing[20] techniques such as sentiment analysis and emotion detection, we have that 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='91% of the reviews have negative sentiment and 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='5% negative emotions (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='3% sadness, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='2% anger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The second result relates to SNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' From our analysis, the SNA presents 5 different and distinct communities calculated according to Louvain’s algorithm[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The community detection thus constructed shows moderately connected communities with a modularity value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' We also note that the graph presents itself according to a Barabasi- Albert[22] with 62986 nodes with an average degree (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' the average number of links) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='37 and a density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='573(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' These results, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' 1 lead to the hypothesis that individuals struggle to change their opinion and sentiment toward the use of the IO app (and by extension towards the digitization path undertaken by the PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' 3 The use of new technologies to support Public Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' (H) Table 1: Network results Statistics Value Number of nodes 62986 Mean degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='37 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='573 Modularity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content='6 Figure 1: network analysis IO App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' 6 Conclusion This paper brings attention to how computational social science and network science can both explain the complex dynamics of controversial and challenging topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The digital ecosystem not only evolves social network communication but also provides the social researcher with useful data to explain social-complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' Using natural language processing techniques, such as sentiment analysis and emotion detection, we proposed a new way to measure the impact of digital transformation and how the resulting analysis can be applied to provide legislators with meta-evaluation tools, in this case starting with how the app is perceived by citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' We noticed that many of the comments found within the dataset we created are not related to the app itself, a large number of them are negative towards the government and the PA itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' The comments are therefore just a proxy for the citizen to criticize policy choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' References [1] Vesna Bosilj Vukši´c, Lucija Ivanˇci´c, and Dalia Suša Vugec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE2T4oBgHgl3EQfXwe7/content/2301.03848v1.pdf'} +page_content=' A preliminary literature review of digital transformation case studies.' metadata={'source': 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91125, USA +4Department of Physics, Kyungpook National University, Daegu, South Korea +5MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, Singapore UMI 3654, Singapore +6National Institute of Education, Nanyang Technological University, Singapore 637616, Singapore +7Quantum Science and Engineering Centre (QSec), Nanyang Technological University, Singapore +A paradigm for quantum synchronization is the quantum analog of the Stuart–Landau oscillator, +which corresponds to a van der Pol oscillator in the limit of weak (i.e. vanishingly small) nonlin- +earity. Due to this limitation, the quantum Stuart–Landau oscillator fails to capture interesting +nonlinearity-induced phenomena such as relaxation oscillations. +To overcome this deficiency we +propose an alternative model which approximates the van der Pol oscillator to finitely large non- +linearities while remaining numerically tractable. This allows us to uncover interesting phenomena +in the deep-quantum strongly-nonlinear regime with no classical analog, such as the persistence of +amplitude death on resonance. We also report nonlinearity-induced position correlations in reac- +tively coupled quantum oscillators. Such coupled oscillations become more and more correlated with +increasing nonlinearity before reaching some maximum. Again, this behavior is absent classically. +We also show how strong nonlinearity can enlarge the synchronization bandwidth in both single +and coupled oscillators. This effect can be harnessed to induce mutual synchronization between two +oscillators initially in amplitude death. +Introduction.—Mathematical modelling has shown us +how the immense variety and beauty of nature can +be governed by nonlinear differential equations [1–4]. +Such equations, owing to their nonlinearity, are difficult +to analyze and their application to physical processes +has come to be known as nonlinear science [5, 6]. +In +physics, interest in nonlinear phenomena has spread to +quantum-mechanical systems. Effects such as chaos [7– +10], stochastic resonance [11–14], and coherence reso- +nance [12, 15], are some of the better known exam- +ples. Besides fundamental research, there are also several +promising applications of nonlinear dissipation, e.g. sta- +bilizing bosonic qubits for fault-tolerant quantum com- +puting [16–18], and enhancing the sensitivity of quantum +sensors [19, 20]. +A relative newcomer to the study of nonlinear effects +in quantum systems is synchronization [21]. Its most ele- +mentary form consists of applying a sinusoidal force, say +with amplitude f, and frequency Ωd, to a self-sustained +oscillator. +Synchronization is then the modification of +the oscillator frequency to Ωd. A prototypical model is +the driven van der Pol (vdP) oscillator [22], defined by +phase-space coordinates (x, y) satisfying +x′ = y , +y′ = f cos(Ωdt) − ω2 +0 x − µ (x2 − q2) y , +(1) +where primes denote differentiation with respect to the +argument (in this case t, representing time). In the ab- +sence of forcing (f = 0) the oscillator is characterized +by ω0, and a nonlinearity parameter µ which controls +how much the oscillator is damped towards an ampli- +tude of order |q|. An important feature of the undriven +vdP system is the existence of a supercritical Hopf bifur- +cation at µ = 0, via which a stable limit cycle appears +for µ > 0 [22]. +At µ = 0, (1) is entirely linear. This motivates one to +consider the quasilinear limit of (1), defined by µ −→ 0+. +In this limit the vdP oscillator is well approximated by +the Stuart–Landau (SL) oscillator, the steady state of +which is rotationally symmetric in phase space (a circular +limit cycle). This makes the SL oscillator much simpler +to analyze, and has thus served as a starting point in the +literature on quantum synchronization for continuous- +variable systems, e.g. Refs. [23–38]. +The trade-off of +course, is that effects taking place at finite values of µ are +excluded. A prominent example is relaxation oscillations +in the undriven vdP oscillator1 [3, 22, 43]. More effects +start to appear if driving is included, such as quasiperi- +odicity and chaos [44–46], both of which are absent in +the driven SL oscillator. +In this work, we investigate the effects of nonlinear- +ity in quantum oscillators by considering a more general +model based on the classical Duffing–van der Pol (DvdP) +oscillator. This adds ζ x3 to y′ where ζ is another nonlin- +earity parameter. To overcome the inadequacy of the SL +model we propose a quantum DvdP oscillator in which +1 In fact, vdP intended for (1) to model relaxation oscillations in +an electrical circuit [39]. +To observe relaxation oscillations in +quantum theory one needs to quantize the exact vdP model, and +it is only relatively recently that such efforts have been made [40– +42]. +arXiv:2301.02948v1 [quant-ph] 8 Jan 2023 + +2 +the vdP and Duffing nonlinearities (respectively µ and +ζ) are nonvanishing, but also not arbitrarily large. Our +model is accurate up to order (µ/ω0)2, at which the dis- +tinct signatures of strong nonlinearity appear, such as +relaxation oscillations [41]. Our approach has the bene- +fit of capturing novel nonlinear effects while evading the +large computational cost of simulating quantum systems +with very strong nonlinear dissipations. +We show that for a single oscillator with periodic forc- +ing there exists a critical Duffing nonlinearity, above +which further increases in ζ enlarges the synchroniza- +tion bandwidth (the amount of detuning the forcing can +tolerate from the oscillator and still entrain it). This re- +sult is similar to the synchronization enhancement from +the classical literature [47], but now generalized to quan- +tum oscillators.2 In contrast, the vdP nonlinearity ac- +tivates genuine quantum effects. Coupling two vdP os- +cillators dissipatively may lead to either amplitude death +(the cessation of oscillations), or mutual synchronization. +Classically, amplitude death occurs only when the two +oscillators are sufficiently detuned [49]. Interestingly, we +find this need not be the case for quantum oscillators. +We show that two quantum vdP oscillators possessing +relatively small limit cycles and nonvanishing nonlinear- +ities can exhibit amplitude death even with zero detun- +ing. Larger limit cycles on the other hand can mutually +synchronize from a state of amplitude death if their non- +linearity is increased. +We also consider reactively coupled oscillators. Two +such SL oscillators cannot develop positional correla- +tions, and hence do not synchronize. This is true regard- +less of whether the oscillators are classical or quantum. +We show here that at finitely large nonlinearity, posi- +tion correlations behave rather differently between the +classical and quantum oscillators: Two reactively cou- +pled quantum vdP oscillators can undergo nonlinearity- +induced correlations whereby their position correlation +increases as they become more nonlinear. In contrast, we +find that making the analogous classical oscillators more +nonlinear monotonically reduces their position correla- +tion. The nonlinearity-induced correlations in the quan- +tum vdP oscillators are thus a consequence of both their +quantum nature and strong nonlinearity. +Model.—For simplicity we consider here a dimension- +less DvdP model in terms of the nonlinearity parameters +λ ≡ µq2/ω0r2 and β ≡ ζq2/ω2 +0r2 in which r is a dimen- +sionless scale parameter: +˜x′ = ˜y , +˜y′ = F cos(ωd˜t ) − ˜x − λ(˜x − r2)˜x′ − β ˜x3. (2) +Note that ˜x ≡ xr/q is now a function of ˜t = ω0t, and +2 It is also worth mentioning that nonlinear oscillators are of inter- +est to quantum information too, when they are coupled to qubits. +In this context the Duffing nonlinearity has been shown to both +increase and stabilize the oscillator-qubit entanglement [48]. +we have also included a dimensionless external force pa- +rameterized by F = fr/ω2 +0q and ωd = Ωd/ω0. From the +approximate analysis of (2), the leading contribution to +the oscillator frequency is quadratic in λ, and linear in +β [50], given by ω ≈ 1 + r2(3β/2 − λ2r2/16). This moti- +vates a Bogoliubov–Krylov time-average of the equations +of motion up to these orders, giving [50, 51] +α′ =i F +2 cos(ωdt) − i α − i3β +2 |α|2α + λ +2 (r2 − |α|2)α ++ iλ2 +8 +� +r4 − 6r2|α|2 + 11 +2 |α|4 +� +α, +(3) +where α = (˜x + i ˜y)/2. For F = 0, (3) predicts a limit- +cycle amplitude of 2|α| = 2r with the expected frequency +shifts due to λ and β. Additionally, note the first-order +averaging in λ for β = 0 yields the SL equation. Our +approximate model captures the effects of strong vdP +nonlinearity of order λ2. +We seek a quantum master +equation ρ′ = Lρ such that ⟨ˆa⟩′ = Tr[ˆa Lρ] (with [ˆa, ˆa†] = +ˆ1) agrees with (3) in the mean-field limit [41]. It can then +be shown that this is satisfied by the Lindbladian [50, 52– +54] +L = −i [ ˆH,·] + λr2D[ˆa†] + λ +2 D[ˆa2] , +(4) +where +ˆH = +� +1 − λ2r4 +8 +� +ˆa†ˆa + 3λ2r2 +8 +ˆa†2ˆa2 − 11λ2 +48 +ˆa†3ˆa3 ++ 3β +4 ˆa†2ˆa2 − F +2 cos(ωdt)(ˆa + ˆa†) . +(5) +We have also defined D[ˆc] ≡ ˆc· ˆc†−(ˆc†ˆc·+· ˆc†ˆc)/2 for any +ˆc, and a dot denotes the position of ρ when acted upon +by a superoperator. We remark that both the higher- +order Kerr terms and the nonlinear two-photon dissipa- +tion in our proposed model can be implemented in circuit +QED [17, 55]. The tunability of the limit cycle radius r +allows us to access different parameter regimes of the +quantum oscillator, in particular the quantum (r ≪ 1), +and semiclassical (r ≈ 1) regimes. +We have included +the second-order contributions in λ in our model for its +nonlinearity-tuning capability, since the terms linear in λ +neither affect the limit-cycle amplitude nor phase dynam- +ics. The Duffing nonlinearity translates to a Kerr term +in L. This model can be considered as an alternative to +the quantum SL oscillator, but with flexibility in tuning +the nonlinearity. +All our numerical results for a given +parameter set are obtained with a sufficiently large trun- +cation of the Hilbert space by ensuring the corresponding +steady-state power spectrum converge. +Nonlinearity-enhanced +synchronization.—We +study +first the frequency locking of the approximate quantum +DvdP oscillator to a periodic force [(4) and (5)]. The syn- +chronization bandwidth is the range of ωd for which the +oscillator frequency is locked to the driving frequency at + +3 +steady state. This is achieved when |ωd−˜ω| = 0, where ˜ω +the observed frequency of the driven oscillator, obtained +from the peak of its spectrum averaged over one period +of the drive [41]. +Here we find the Duffing nonlinearity to enhance quan- +tum synchronization: For a range of λ and a fixed r, +increasing β past a critical value widens the synchroniza- +tion bandwidth linearly. This is illustrated in Fig. 1(a) +where the synchronization bandwidth is plotted as a con- +tour against ¯β ≡ βr2 and F/r. The critical value of ¯β is +indicated by the red dashed line, where the bandwidth +is equal to its corresponding value at ¯β = 0. However, +this enhancement does not occur for all values of the +vdP nonlinearity. In Fig. 1(b) we see that an increase +of ¯λ ≡ λr2 from its value in Fig. 1(a) can ruin the gain +in synchronization bandwidth due to ¯β. Noting that the +dissipative terms in L are all proportional to λ, this ef- +fect can be qualitatively attributed to the phase diffusion +due to quantum noise, which is known to inhibit synchro- +nization [23, 24, 27]. We can develop some understanding +of the quantum DvdP by examining its classical analog. +Using the method of harmonic balance on x, we are able +to derive the conditions for nonlinearity-enhanced syn- +chronization analytically for the classical DvdP oscilla- +tor, given by [50] +¯β > +¯λ2 +3(1 − ¯λ2) , +0 < ¯λ < 1 . +(6) +This shows clearly the existence of a critical value of ¯β, +and a finite interval of ¯λ over which the synchronization +enhancement occurs. These results are consistent with +Fig. 1 except for the fact that quantum noise makes the +range of λ for synchronization enhancement in the quan- +tum DvdP oscillator smaller compared to the classical +range as seen in Fig. 1(b). +We have also shown in Ref. [50] that increasing the +vdP nonlinearity in the quantum model can only reduce +the synchronization bandwidth. One can thus be certain +that the synchronization enhancement seen in our model +is induced by the Duffing nonlinearity. We shall see next +that the vdP nonlinearity can induce synchronization in +coupled oscillators from a state of amplitude death. +Nonlinearity-induced synchronization and amplitude +death.—Two dissipatively coupled vdP oscillators (i.e. no +Duffing nonlinearity) can be described by the Lindbla- +dian +L = L1 + L2 − i ∆[ˆa† +2ˆa2,·] + η D[ˆa1 − ˆa2] , +(7) +where Lk (k = 1, 2) is the Lindbladian for oscillator k, +defined by setting ˆa to ˆak and β = F = 0 in (4) and (5). +We have assumed the oscillators to be identical (i.e. same +λ and r) except for an initial detuning of ∆, and denoted +their coupling strength by η. +0.50 +0.75 +0.25 +0.0 +1.0 +0.1 +0.2 +0.3 +0.4 +0.0 +0.4 +0.6 +0.8 +1.0 +0.2 +0.0 +0.0 +0.05 +0.1 +0.15 +0.2 (a) +(b) +AB6nicjVDLSgNBEOyNrxhfUY9eBoPgKe5KiB6DgniMaB +6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cfI4qChY0FBUdPdFSCa+O6H05uaXldS2/XtjY3NreKe7uNXWcKoYNFotYtQOqUXCJDcONwHaikEaBwFYwupz6rXtUmsfyzowT9CM6kDzkjBor3V6dqF6x5JXdGcjfpAQL1HvF924/ZmE0jBte54bmL8jCrDm +cBJoZtqTCgb0QF2LJU0Qu1ns1Mn5MgqfRLGypY0ZKZ+nchopPU4CmxnRM1Q/Sm4m9eJzXhuZ9xmaQGJZsvClNBTEymf5M+V8iMGFtCmeL2VsKGVFmbDqF/4XQPC171XLlplKqXSziyMBHMIxeHAGNbiGOjSAwQAe4AmeHeE8Oi/O67w15yxm9uEbnLdP4Fy +Niw=F/r +AB/HicjVDLSsNAFJ3UV62vaJduBovgKqRi1Y1QdOygn +1AE8rNZNIOnUzCzEQIof6KGxeKuPVD3Pk3Th8LFQUPXDicw/3coKUM6Vd98MqLS2vrK6V1ysbm1vbO/buXkclmS0TRKeyF4AinImaFszWkvlRTigNuML6a+t07KhVLxK3OU+rHMBQsYgS0kQZ21QtAFh43iRAm+AK7TmNg1+qOwP+m9TQAq2B/e6FCcliKj +ThoFS/7qbaL0BqRjidVLxM0RTIGIa0b6iAmCq/mD0/wYdGCXGUSDNC45n6NVFArFQeB2YzBj1SP72p+JvXz3R07hdMpJmgswPRnHOsHTJnDIJCWa54YAkcz8iskIJBt+qr8r4TOsVM/dRo3J7Xm5aKOMtpHB+gI1dEZaqJr1EJtRFCOHtATerburUfrxXqdr5 +asRaKvsF6+wQlyJPM¯� = 0.5 +AB/HicjVDLSsNAFJ34rPUV7dLNYBFchUR8bYSiG5cV7A +OaUG4mk3boZBJmJkI9VfcuFDErR/izr9x+lioKHjgwuGce7iXE2acKe26H9bC4tLymplrbq+sbm1be/stlWaS0JbJOWp7IagKGeCtjTnHYzSEJOe2Eo6uJ37mjUrFU3Ooio0ECA8FiRkAbqW/X/Bk6XOTiGCML7DreH27jnuFPhvUkdzNPv2ux+lJE+o0I +SDUj3PzXRQgtSMcDqu+rmiGZARDGjPUAEJVUE5fX6MD4wS4TiVZoTGU/VroREqSIJzWYCeqh+ehPxN6+X6/g8KJnIck0FmR2Kc451idN4IhJSjQvDAEimfkVkyFINr0Vf1fCe0jxzt1Tm6O643LeR0VtIf20SHy0BlqoGvURC1EUIEe0BN6tu6tR+vFep2tLl +jzTA19g/X2CR+4k8g=¯� = 0.1 +AB8nicjVDJSgNBEO2JW4xb1KOXwSB4C +jPidgx68RjBLDAzhJ5OTdKkp3vorhHCkM/w4kERr36N/GznJQUfBweO9KqrqxZngBj3vwyktLa+srpXKxubW9s71d29tlG5ZtBiSijdjakBwSW0kKOAbqaBprGATjy6nvqde9CGK3mH4wyilA4kTzijaKUgjKkuwhiQTn +rVml/3ZnD/JjWyQLNXfQ/7iuUpSGSCGhP4XoZRQTVyJmBSCXMDGWUjOoDAUklTMFExO3niHlml7yZK25LoztSvEwVNjRmnse1MKQ7NT28q/uYFOSaXUcFliNINl+U5MJF5U7/d/tcA0MxtoQyze2tLhtSTRnalCr/C6F9Uv +fP62e3p7XG1SKOMjkgh+SY+OSCNMgNaZIWYUSRB/JEnh10Hp0X53XeWnIWM/vkG5y3T5MakXY= ¯� +FIG. 1. Contour plot of the synchronization bandwidth for +the quantum DvdP oscillator as a function of ¯β ≡ βr2 (ver- +tical axis) and F/r (horizontal axis) with unit limit-cycle ra- +dius, i.e. r = 1. In this case ¯β = β and ¯λ = λ. The axes +are also indicated in subplot (b). (a) Illustration of synchro- +nization enhancement for ¯λ = 0.1. Above a critical value of +¯β, indicated by a red dashed line (obtained numerically), the +synchronization bandwidth is enlarged as the Duffing nonlin- +earity is increased. (b) Synchronization enhancement disap- +pears if we increase the vdP nonlinearity from ¯λ = 0.1 to +¯λ = 0.5, demonstrating the finite range of ¯λ over which the +enhancement is effective. +In this case, frequency locking occurs when the ob- +served frequencies of the two oscillators become identi- +cal at steady state. As before, we define an oscillator’s +observed frequency by the location of its spectral peak, +except now we must use the reduced state derived by +a partial trace over the two-oscillator steady state. For +a fixed η, we define the synchronization bandwidth to +be the range of ∆ for which the two oscillators lock fre- +quencies. It will also be interesting to look at position +correlations in the two oscillators at steady state, defined +by +Σ = +⟨ˆx1ˆx2⟩ − ⟨ˆx1⟩ ⟨ˆx2⟩ +�� +⟨ˆx2 +1⟩ − ⟨ˆx1⟩2 � � +⟨ˆx2 +2⟩ − ⟨ˆx2⟩2 � , +(8) +where ˆxk = ˆak + ˆa† +k. Note that frequency locking implies +a nonzero Σ, but not vice versa. +In addition to frequency locking, dissipatively coupled +oscillators can also cease to oscillate. If the oscillators +are classical, then this may happen for a range of η pro- +vided that ∆ is sufficiently large. And if both oscillators +stabilize to the same phase-space point, which may be +taken to be the origin without loss of generality, then +the effect is termed amplitude death [49, 56]. To define +amplitude death in quantum oscillators we generalize the +notion of P-bifurcations from classical stochastic systems +to the steady-state Wigner function of a reduced state +(see Ref. [57] and other references therein). In this case, +amplitude death is said to occur if the single-oscillator +Wigner functions peak at the origin in quantum phase +space. This approach is consistent with previous stud- +ies on amplitude death in coupled quantum oscillators +[32–35]. +In Fig. 2 we work out regions of frequency locking and + +4 +AMPLITUDE +DEATH +FREQUENCY +LOCKING +AMPLITUDE +DEATH +AB7XicbVBNS8NAEN3Ur1q/qh69LBb +BU0lE1GNRDx4r2A9oQ9lsJ+3aTbsToQS+h+8eFDEq/Hm/GbZuDtj4 +YeLw3w8y8IJHCoOt+O4WV1bX1jeJmaWt7Z3evH/QNCrVHBpcSaXbAT +MgRQwNFCihnWhgUSChFYxupn7rCbQRKn7AcQJ+xAaxCAVnaKVm9xYks +l654lbdGegy8XJSITnqvfJXt694GkGMXDJjOp6boJ8xjYJLmJS6qYGE8 +REbQMfSmEVg/Gx27YSeWKVPQ6VtxUhn6u+JjEXGjKPAdkYMh2bRm4r/ +eZ0Uwys/E3GSIsR8vihMJUVFp6/TvtDAUY4tYVwLeyvlQ6YZRxtQyYb +gLb68TJpnVe+ien5/Xqld53EUyRE5JqfEI5ekRu5InTQIJ4/kmbySN0c +5L8678zFvLTj5zCH5A+fzB2N6jwU=� +AB63icbVBNS8NAEJ3Ur1q/qh69LBb +BU0lE1GPRi8cK9gPaUDbTbt0dxN2J0IJ/QtePCji1T/kzX9j0uagrQ8 +GHu/NMDMviKWw6LrfTmltfWNzq7xd2dnd2z+oHh61bZQYxlskpHpBt +RyKTRvoUDJu7HhVAWSd4LJXe53nrixItKPOI25r+hIi1AwirnU50gH1 +Zpbd+cgq8QrSA0KNAfVr/4wYoniGpmk1vY8N0Y/pQYFk3xW6SeWx5RN6 +Ij3Mqp4tZP57fOyFmDEkYmaw0krn6eyKlytqpCrJORXFsl71c/M/r +JRje+KnQcYJcs8WiMJEI5I/TobCcIZymhHKjMhuJWxMDWYxVPJQvC +WX14l7Yu6d1W/fLisNW6LOMpwAqdwDh5cQwPuoQktYDCGZ3iFN0c5L86 +787FoLTnFzDH8gfP5AwvQjkE=⌘ +0.1 +0.6 +0.3 +0.4 +0.5 +0.0 +0.2 +0.1 +0.6 +0.3 +0.4 +0.5 +0.0 +0.2 +0.1 +0.3 +0.4 +0.5 +0.2 +0.0 0.2 0.4 +0.6 0.8 1.0 +1.2 1.4 +0.0 +1.5 +2.0 +2.5 +0.5 +1.0 +8 +4 +0 +2 +6 +10 +AB6nicbVBNS8NAEJ3Ur1q/qh69LBb +BU0lE1ItQ9OKxov2ANpTNdtMu3WzC7kQoT/BiwdFvPqLvPlv3LY5aOu +Dgcd7M8zMCxIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmvEGi2Ws2w +E1XArFGyhQ8naiOY0CyVvB6Hbqt564NiJWjzhOuB/RgRKhYBSt9KCv +V654lbdGcgy8XJSgRz1Xvmr249ZGnGFTFJjOp6boJ9RjYJPil1U8MTy +kZ0wDuWKhpx42ezUyfkxCp9EsbalkIyU39PZDQyZhwFtjOiODSL3lT8 +z+ukGF75mVBJilyx+aIwlQRjMv2b9IXmDOXYEsq0sLcSNqSaMrTplGw +I3uLy6R5VvUuquf35XaTR5HEY7gGE7Bg0uowR3UoQEMBvAMr/DmSOf +FeXc+5q0FJ585hD9wPn8A0cGNgQ=r = 1 +AB7HicbVBNS8NAEJ3Ur1q/qh69LBb +BU0i0qBeh6MVjBdMW2lA2027dHcTdjdCf0NXjwo4tUf5M1/47bNQas +PBh7vzTAzL0o508bzvpzSyura+kZ5s7K1vbO7V90/aOkU4QGJOGJ6k +RYU84kDQwznHZSRbGIOG1H49uZ36kSrNEPphJSkOBh5LFjGBjpUBde ++5v1rzXG8O9Jf4BalBgWa/+tkbJCQTVBrCsdZd30tNmGNlGOF0Wulm +qaYjPGQdi2VWFAd5vNjp+jEKgMUJ8qWNGiu/pzIsdB6IiLbKbAZ6WVv +Jv7ndTMTX4U5k2lmqCSLRXHGkUnQ7HM0YIoSwyeWYKYvRWREVaYGJt +PxYbgL7/8l7TOXP/Crd/Xa42bIo4yHMExnIPl9CAO2hCAQYPMELvDr +SeXbenPdFa8kpZg7hF5yPb68OjfU=r = 0.3 +(a) +(b) +(c) +(d) +0.0 +0.2 +0.4 +0.1 +0.3 +0.4 +0.5 +0.2 +0.0 +0.0 +0.25 +0.5 +0.75 +1.0 +0.2 +0.6 +0.8 +1.0 +0.4 +0.0 +AB ++3icdVDLSgMxFM3UV62vsS7dBIvgasi0Tq0LoejGZQX7gLaUTCbThmYeJBmxDPMrblwo4tYfcefmGkrqOiBwOGce7g3x405kwqhD6Owsrq2vl +HcLG1t7+zumfvljowSQWibRDwSPRdLylI24opTnuxoDhwOe2606vc795RIVkU3qpZTIcBHofMZwQrLY3M8sDFIh1wnfBwdoGsWn1kVpCF6k7t3 +IHIcpDdqOak6tgI1aBtoTkqYInWyHwfeBFJAhoqwrGUfRvFaphioRjhNCsNEkljTKZ4TPuahjigcpjOb8/gsVY86EdCv1DBufo9keJAylng6skA +q4n87eXiX14/UX5jmLIwThQNyWKRn3CoIpgXAT0mKF8pgkmgulbIZlgYnSdZV0CV8/hf+TtWy65Zzc1pXi7rKIJDcAROgA3OQBNcgxZoAw +LuwQN4As9GZjwaL8brYrRgLDMH4AeMt087t5Pv¯� = 0.36 +AB ++3icdVDLSsNAFJ34rPUV69LNYBFchaT46EYounFZwT6gCeVmMmHTh7MTMQS8ituXCji1h9x5984fQj1deDC4Zx7uJfjp5xJZdsfxtLyuraem +mjvLm1vbNr7lXaMskEoS2S8ER0fZCUs5i2FOcdlNBIfI57fijq4nfuaNCsiS+VeOUehEMYhYyAkpLfbPi+iByl+tEAMWFbdXqfbPqWPYU2P5Fv +qwqmqPZN9/dICFZRGNFOEjZc+xUeTkIxQinRdnNJE2BjGBAe5rGEFHp5dPfC3yklQCHidATKzxVFxM5RFKOI19vRqCG8qc3Ef/yepkK617O4jRT +NCazQ2HGsUrwpAgcMEGJ4mNgAimf8VkCAKI0nWVF0v4n7RrlnNmnd6cVBuX8zpK6AdomPkoHPUQNeoiVqIoHv0gJ7Qs1EYj8aL8TpbXTLmX +30DcbJ+dNk7U=¯� = 0.28 +AB ++3icdVDLSgMxFM3UV62vsS7dBIvgasgUrdOFUHTjsoJ9QGcomUzahmYeJBmxDPMrblwo4tYfcefmGkrqOiBwOGce7g3x084kwqhD6O0srq2vl +HerGxt7+zumfvVroxTQWiHxDwWfR9LylEO4opTvuJoDj0Oe3506vC791RIVkc3apZQr0QjyM2YgQrLQ3NqutjkblcJwKcXyDLdoZmDVlIo9GAB +bEdZGvSbDr1ehPacwuhGliPTf3SAmaUgjRTiWcmCjRHkZFoRTvOKm0qaYDLFYzrQNMIhlV42vz2Hx1oJ4CgW+kUKztXviQyHUs5CX0+GWE3k +b68Q/IGqRo5XsaiJFU0IotFo5RDFcOiCBgwQYniM0wEUzfCskEC0yUrquiS/j6KfyfdOuW3bDObk5rctlHWVwCI7ACbDBOWiBa9AGHUDAPX +gAT+DZyI1H48V4XYyWjGXmAPyA8fYJO5yT8A=¯� = 0.18 +AB ++nicdVDLSgMxFM3UV62vqS7dBIvgasi0TlsXQtGNywr2AW0pmTRtQzOZIckoZeynuHGhiFu/xJ1/Y6atoKIHAodz7uHeHD/iTGmEPqzMyura+k +Z2M7e1vbO7Z+f3myqMJaENEvJQtn2sKGeCNjTnLYjSXHgc9ryJ5ep37qlUrFQ3OhpRHsBHgk2ZARrI/XtfNfHMulykxjg2TlyKn27gBxU9kpnH +kSOh9xqMSVFz0WoBF0HzVEAS9T79nt3EJI4oEITjpXquCjSvQRLzQins1w3VjTCZIJHtGOowAFVvWR+gweG2UAh6E0T2g4V78nEhwoNQ18Mxlg +PVa/vVT8y+vEeljtJUxEsaCLBYNYw51CNMe4IBJSjSfGoKJZOZWSMZYqJNWzlTwtdP4f+kWXTcsuNdnxZqF8s6suAQHIET4IKqIErUAcNQM +AdeABP4Nm6tx6tF+t1MZqxlpkD8APW2yfEqZOz¯� = 0.7 +AB+nicdVDLSsNAFJ34rPWV6tLNYBFchUSsuhGKblxWsA9oQ7mZTNqhk0m +YmSgl9lPcuFDErV/izr9x+hDq68CFwzn3cC8nSDlT2nU/rIXFpeWV1cJacX1jc2vbLu0VJQusk4YlsBaAoZ4LWNdOctlJIQ4bQaDy7HfvKVSsUTc6GFK/Rh6gkWMgDZS1y51ApB5h5tECKNz16l07bLnuBNg9xf5spohlrXfu+ECcliKjThoFTbc1Pt5yA1I5yOip1M0RTIAHq0baiAmCo/ +n7w+wgdGCXGUSDNC4k6n8ghVmoYB2YzBt1XP72x+JfXznR05udMpJmgkwPRnHOsHjHnDIJCWaDw0BIpn5FZM+SCDatFWcL+F/0jhyvBOncn1crl7M6igPbSPDpGHTlEVXaEaqiOC7tADekLP1r31aL1Yr9PVBWuW2UXfYL19AmvDk3Y=¯� = 0.5 +AB ++nicdVBLSwMxGMzWV62vVo9egkXwtGRr17YHoejFYwX7gLaUbDbhmazS5JVytqf4sWDIl79Jd78N6YPQUHApOZb8iX8WLOlEbow8qsrK6tb2 +Q3c1vbO7t7+cJ+S0WJLRJIh7JjocV5UzQpma04sKQ49Ttve+HLmt2+pVCwSN3oS036Ih4IFjGBtpEG+0POwTHvcJHw8PUd2aZAvIhvVkHtah +ubqlC1YkitUim7DnRsNEcRLNEY5N97fkSkApNOFaq6BY91MsNSOcTnO9RNEYkzEe0q6hAodU9dP56lN4bBQfBpE0R2g4V78nUhwqNQk9Mxli +PVK/vZn4l9dNdFDtp0zEiaCLB4KEg51BGc9QJ9JSjSfGIKJZGZXSEZYqJNWzlTwtdP4f+kVbKdM9u9LhfrF8s6suAQHIET4IAKqIMr0ABNQM +AdeABP4Nm6tx6tF+t1MZqxlpkD8APW2yfWM5PA¯� = 0.2 +AB+nicbVDLSsNAFL2pr1pfqS7dBIv +gKiTF10YounFZwT6gCWUymbRDJ5MwM1FK7Ke4caGIW7/EnX/jtM1CWw8 +MHM65h3vnBCmjUjnOt1FaWV1b3yhvVra2d3b3zOp+WyaZwKSFE5aIbo +AkYZSTlqKkW4qCIoDRjrB6Gbqdx6IkDTh92qcEj9GA04jipHSUt+se +gESucd0IkSTK8eu982aYzszWMvELUgNCjT75pcXJjiLCVeYISl7rpMqP +0dCUczIpOJlkqQIj9CA9DTlKCbSz2enT6xjrYRWlAj9uLJm6u9EjmIp +x3GgJ2OkhnLRm4r/eb1MRZd+TnmaKcLxfFGUMUsl1rQHK6SCYMXGmiA +sqL7VwkMkEFa6rYouwV38jJp123D67O601ros6ynAIR3ACLlxA26 +hCS3A8AjP8ApvxpPxYrwbH/PRklFkDuAPjM8fZbeTcg=¯� = 0.2 +AB+3icbVDLSsNAFL2pr1pfsS7dBIv +gKiTioxuh6MZlBfuAJpTJZNIOnUzCzEQsIb/ixoUibv0Rd/6N0zYLrR4 +YOJxzD/fOCVJGpXKcL6Oysrq2vlHdrG1t7+zumfv1rkwygUkHJywR/Q +BJwignHUVI/1UEBQHjPSCyc3M7z0QIWnC79U0JX6MRpxGFCOlpaFZ9 +wIkco/pRIiK8d2m0Oz4djOHNZf4pakASXaQ/PTCxOcxYQrzJCUA9dJl +Z8joShmpKh5mSQpwhM0IgNOYqJ9P57YV1rJXQihKhH1fWXP2ZyFEs +5TQO9GSM1FguezPxP2+Qqajp5SnmSIcLxZFGbNUYs2KsEIqCFZsqgn +CgupbLTxGAmGl6rpEtzlL/8l3VPbvbDP784areuyjiocwhGcgAuX0IJ +baEMHMDzCE7zAq1EYz8ab8b4YrRhl5gB+wfj4BuRIk7M=¯� = 0.18 +FIG. 2. Regions of frequency locking and amplitude death +(as defined in the text by the power spectrum and Wigner +function) for two dissipatively coupled vdP oscillators [sub- +plots (a)–(d)] along with contours of Σ [subplots (a) and (c)]. +All subplots have ∆ on the vertical axis, and η on horizon- +tal axis which we also indicate in subplot (c). (a) Large r +(semiclassical regime). Note the region on the left does not +correspond to any identifiable effect and is demarcated us- +ing a solid line while the boundary between frequency locking +and amplitude death is a Hopf bifurcation, which we denote +by a dash-dotted line. As r is increased, the classical bound- +ary is recovered. (b) Effect of varying λ on synchronization +for r = 1. +Boundaries of the frequency-locking region and +its corresponding λ are shown (i.e. the frequency-locking re- +gion is the area underneath each curve). +Increasing λ can +enlarge the synchronization bandwidth and induce frequency +locking from a state of amplitude death. (c) Small r (deep +quantum regime). At small r, amplitude death occurs even +at zero initial detuning, which is a quantum effect. Note that +¯λ in (a) and (c) are approximately equal, leaving the dif- +ferences between the two plots only as a result of quantum +effects. (d) Shifts in amplitude-death boundary as ¯λ is in- +creased (quantum-to-semiclassical transition). +amplitude death in the (η, ∆) parameter space for (7), +along with Σ, shown as a contour in Fig. 2(a) and (c). +Two especially interesting scenarios are—when the limit +cycles are relatively large compared to quantum noise +[Fig. 2(a)]; and when they become small, being more sus- +ceptible to quantum noise [Fig. 2(c)]. +In Fig. 2(a) we have indicated the boundary between +frequency locking and amplitude death by a dash-dotted +line, while no identifiable phenomenon occurs to the left +of the solid line. Note the dash-dotted line is a Hopf- +bifurcation curve because the transition from amplitude +death to frequency locking is facilitated by a Hopf bifur- +cation. Clearly, Σ is larger inside the frequency-locking +region. +Especially significant here is the effect of the +vdP nonlinearity on synchronization. +Whereas in the +single-oscillator case the vdP nonlinearity had only detri- +mental effects, it now has a constructive role by enlarg- +(a) +AB8nicbVDLSgMxFM3UV62vqks3wSK4KjNi1Y1Q1IXLCvYB06Fk0kwbmkmG5I +5Qhn6GxeKuPVr3Pk3pu0stPVAyOGce7n3njAR3IDrfjuFldW19Y3iZmlre2d3r7x/0DIq1ZQ1qRJKd0JimOCSNYGDYJ1EMxKHgrXD0e3Ubz8xbiSjzBOWBCTgeQRpwSs5HfvmABy7VbdWq9csd8MeJl4OamgHI1e+avbVzSNmQqiDG+5yYQZEQDp4JNSt3UsITQERkw31JYmaCbLbyBJ9YpY8jpe2TgG +fq746MxMaM49BWxgSGZtGbiv95fgrRVZBxmaTAJ0PilKBQeHp/bjPNaMgxpYQqrndFdMh0YSCTalkQ/AWT14mrbOqd1GtPZxX6jd5HEV0hI7RKfLQJaqje9RATUSRQs/oFb054Lw4787HvLTg5D2H6A+czx+uEpA4� = 0.05 +AB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0j8vghFLx4rmLbQhrLZbtqlu5uwux +FK6G/w4kERr/4gb/4bt20OWn0w8Hhvhpl5UcqZNp735ZSWldW18rlY3Nre2d6u5eUyeZIjQgCU9UO8KaciZpYJjhtJ0qikXEaSsa3U791iNVmiXywYxTGgo8kCxmBsrBerac0971ZrnejOgv8QvSA0KNHrVz24/IZmg0hCOte74XmrCHCvDCKeTSjfTNMVkhAe0Y6nEguownx07QUdW6aM4UbakQTP150 +SOhdZjEdlOgc1QL3pT8T+vk5n4KsyZTDNDJZkvijOTIKmn6M+U5QYPrYE8XsrYgMscLE2HwqNgR/8eW/pHni+hfu+f1ZrX5TxFGAziEY/DhEupwBw0IgACDJ3iBV0c6z86b8z5vLTnFzD78gvPxDa9gjfY=r = 0.3 +AB7nicbVDLSsNAFL2pr1pfVZduBovgqiTia1l0 +47KCtYU2lMlk0g6dTMLMjVBCP8KNC0Xc+j3u/BunbRbaemDgcM65zL0nSKUw6LrfTmldW19o7xZ2dre2d2r7h8miTjLdYIhPdCajhUijeQoGSd1LNaRxI3g5Gt1O/cS1EYl6wHK/ZgOlIgEo2ildk/aEj71Zpbd2cgy8QrSA0KNPvVr1 +6YsCzmCpmkxnQ9N0U/pxoFk3xS6WGp5SN6IB3LVU05sbPZ+tOyIlVQhIl2j6FZKb+nshpbMw4Dmwypjg0i95U/M/rZhd+7lQaYZcsflHUSYJmR6OwmF5gzl2BLKtLC7EjakmjK0DVsCd7iycvk8azuXdYv7s9rjZuijIcwTGcgdX0IA7 +aEILGIzgGV7hzUmdF+fd+ZhHS04xcwh/4Hz+AEAFj4c=� +AB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx6 +8ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN/GSbIHTSxoKq6e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVB71S2a24M5Bl4uWkDlqvdJXtx+zNE +JpmKBadzw3MX5GleFM4KTYTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGlCmuL2VsCFVlBmbTdG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1g +gPAMr/DmPDovzrvzMW9dcfKZI/gD5/MHz7WM9Q=g +AB7HicdVDLSgNBEOyNrxhfUY9eBoPgaZldNCYHIejFYwTzgGQJs5PZMjs7DIzK4SQb/DiQRGvfpA3/8bJQ1DRgoaiqpvurjAVXBuMP5zcyura+kZ+s7C1vbO7V9w/aOokU5Q1aCIS1Q6JZoJL1jDcCNZOFSNxKFgrHF3P/NY9U5on +8s6MUxbEZCB5xCkxVmoMLrHr94ol7GKLchnNiFfBniXVasX3q8ibWxiXYIl6r/je7Sc0i5k0VBCtOx5OTAhynAq2LTQzTRLCR2RAetYKknMdDCZHztFJ1bpoyhRtqRBc/X7xITEWo/j0HbGxAz1b28m/uV1MhNVgmXaWaYpItFUSaQSdDsc9TnilEjxpYQqri9FdEhUYQam0/BhvD1KfqfNH3XK7vnt2el2tUyjwcwTGcgcXUIMbqEMDKHB4gCd4dqTz6Lw4r4vWnLOcOYQfcN4+AfRJjic=g = 0.2 +AB7HicdVDJSgNBEK2OW4xb1KOXxiB4GmYkLhch6MVjBCcJEPo6fQkTXp6hu4eIQz5Bi8eFPHqB3nzb+wsQtweFDzeq6KqXpgKro3rfqDC0vLK6lpxvbSxubW9U97da+gkU5T5NBGJaoVEM8El8w03grVSxUgcCtYMh9cTv3nPlOaJ +vDOjlAUx6UsecUqMlfz+petUu+WK57hTYPcX+bIqMEe9W37v9BKaxUwaKojWbc9NTZATZTgVbFzqZJqlhA5Jn7UtlSRmOsinx47xkV6OEqULWnwVF2cyEms9SgObWdMzED/9CbiX147M9FkHOZoZJOlsUZQKbBE8+xz2uGDViZAmhitbMR0QRaix+ZQWQ/ifNE4c78w5va1WalfzOIpwAIdwDB6cQw1uoA4+UODwAE/wjCR6RC/odZaQPOZfgG9PYJoYyN7Q=g = 0.4 +AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU9i0ptaDUPTisYJpC20om+2mXbrZhN +2NUEJ/gxcPinj1B3nz37hpK6jog4HezPMzAsSzpRG6MqrKyurW8UN0tb2zu7e+X9g7aKU0moR2Iey26AFeVMUE8zWk3kRHAaedYHKd+517KhWLxZ2eJtSP8EiwkBGsjeSNLpHdGJQryEZ1t3bhQmS7yGlUc1J1HYRq0LHRHBWwRGtQfu8PY5JGVGjCsVI9ByXaz7DUjHA6K/VTRNMJnhEe4YKHFHlZ/ +NjZ/DEKEMYxtKU0HCufp/IcKTUNApMZ4T1WP32cvEvr5fqsOFnTCSpoIsFoUphzqG+edwyCQlmk8NwUQycyskYywx0Safkgnh61P4P2lXbadu7dnlebVMo4iOALH4BQ4Bw0wQ1oAQ8QwMADeALPlrAerRfrdFasJYzh+AHrLdP/XqOLA=g = 0.8 +(b) +AB7XicbVDJSgNBEK1xjXGLevTSGARPYUbcjkEv +HiOaBZIh9HR6kja9DN09QhjyD148KOLV/Hm39hJ5qCJDwoe71VRVS9KODPW97+9peWV1bX1wkZxc2t7Z7e0t98wKtWE1oniSrcibChnktYts5y2Ek2xiDhtRsObid98otowJR/sKGhwH3JYkawdVKjc8/6AndLZb/iT4EWSZCTMuSodUtfnZ +4iqaDSEo6NaQd+YsMa8sIp+NiJzU0wWSI+7TtqMSCmjCbXjtGx07poVhpV9Kiqfp7IsPCmJGIXKfAdmDmvYn4n9dObXwVZkwmqaWSzBbFKUdWocnrqMc0JZaPHMFEM3crIgOsMbEuoKILIZh/eZE0TivBReX87qxcvc7jKMAhHMEJBHAJVbiF +GtSBwCM8wyu8ecp78d69j1nrkpfPHMAfeJ8/bp+PDQ=⌃ +AB7nicbVDLSsNAFL2pr1pfVZduBovgqiTia1l047KCtYU2lMlk0g6dTMLMjVBCP8KNC0Xc+j3u/BunbRbaemDgcM65zL0nSKUw6LrfTmldW19o7xZ2dre2d2r7h8miT +TjLdYIhPdCajhUijeQoGSd1LNaRxI3g5Gt1O/cS1EYl6wHK/ZgOlIgEo2ildk/aEj71Zpbd2cgy8QrSA0KNPvVr16YsCzmCpmkxnQ9N0U/pxoFk3xS6WGp5SN6IB3LVU05sbPZ+tOyIlVQhIl2j6FZKb+nshpbMw4Dmwypjg0i95U/M/rZhd+7lQaYZcsflHUSYJmR6OwmF5gzl2BLKtLC7EjakmjK0DVsCd7iycvk8azuXdYv7s9rjZuijIcwTGcgdX0IA7aEILGIzgGV7hzUmdF+fd+ZhHS04xcwh/4Hz+AEAFj4c=� +0.1 +0.3 +0.4 +0.5 +0.2 +0.0 +0.1 +0.3 +0.4 +0.2 +0.0 +FIG. 3. Positional correlation (8) for two reactively coupled +vdP oscillators. (a) Contour of Σ as a function of λ and g for +r = 0.3 and ∆ = 0.05. The bottom gap of zero correlation +agrees with the SL limit. +(b) Correlations along the three +vertical dashed lines at g = 0.2, 0.4, and 0.8 in subplot (a). +ing the (mutual) synchronization bandwidth. We illus- +trate this in Fig. 2(b) where additional frequency-locking +boundaries for different values of vdP nonlinearity λ are +plotted. +It can be seen that increasing λ enlarges the +frequency-locking region (see also Ref. [50] for some clas- +sical analysis). This means that two oscillators in a state +of amplitude death with an (η, ∆) lying above the Hopf- +bifurcation curve in Fig. 2(a) will transit suddenly to a +state of synchronized oscillations when λ is sufficiently +increased. This may be appropriately called nonlinearity- +induced mutual synchronization. +Turning now to the case of a small limit cycle (r ≪ 1) +in Fig. 2(c), we find frequency locking to be absent while +position correlations become negligible. The blue solid +line delineates the boundary of amplitude death. Most +striking is the persistence of amplitude death at zero de- +tuning (i.e. ∆ = 0). +Classically, some frequency mis- +match between the two oscillators must be present in +order for amplitude death to occur [49]. This signifies +a clear distinction between classical and quantum dy- +namics. A loss of amplitude death at ∆ = 0 may then +be expected if we increased either r or λ (while hold- +ing the other constant). +This is indeed what we find, +as illustrated in Fig. 2(d), where such a quantum-to- +semiclassical transition is captured by increasing ¯λ ≡ +λr2. +Since we have focused exclusively on the vdP nonlin- +earity here, we note that incorporating a Duffing non- +linearity into our model will not in fact change the +frequency-locking boundary. +We can understand this +from the classical coupled equations of motion, where +we see that β appears only in the phase dynamics of the +oscillators, and that such terms vanish at steady state for +identical oscillators (equal limit cycle radii) [50, 58]. +Nonlinearity-induced correlations.—It is known that +two reactively coupled SL oscillators cannot synchronize +nor share position correlations. This is true even in the +quantum case [50]. +However, we show here that posi- +tional correlations between two reactively coupled quan- +tum vdP oscillators do develop. Here we must use the + +5 +exact vdP Lindbladian [41, 50], because under reactive +coupling, the approximate model does not produce any +off-diagonal elements in the steady state, and hence can- +not generate correlations in the two oscillators. As with +the dissipatively coupled system, two reactively coupled +vdP oscillators can be modelled by considering two un- +coupled vdP oscillators with annihilation operators ˆa1 +and ˆa2, coupled by the Hamiltonian g(ˆa1ˆa† +2+ˆa† +1ˆa2), where +g is the reactive coupling strength. As before, we assume +that both oscillators have the same nonlinearity λ. +In Fig. 3(a), we generate a contour plot of Σ as a func- +tion of λ and g at r = 0.3 and ∆ = 0.05. From this we see +that for a fixed g, increasing the oscillator nonlinearity +beyond the SL regime leads to stronger correlations. We +illustrate this more clearly in Fig. 3(b) by showing how Σ +varies as a function of λ for g = 0.2, 0.4, and 0.8, which +are marked in Fig. 3(a) by vertical dashed lines. Note this +also shows the existence of an optimal λ which maximizes +Σ. Such nonlinearity-induced correlations are absent in +the corresponding classical model [50]. At a given cou- +pling strength, the position correlation in two reactively +coupled classical vdP oscillators decreases monotonically +as λ increases [50]. +Conclusion.— Our work goes beyond the well-studied +paradigm of weak nonlinearity in quantum synchroniza- +tion, and provides the first systematic study of quan- +tum synchronization effects for strong nonlinearity. We +introduced a new quantum oscillator model which cap- +tures intriguing effects induced by two strong nonlineari- +ties. We showed that a strong Duffing nonlinearity leads +to a linear enhancement of the synchronization band- +width in driven oscillators. +We also reported genuine +quantum synchronization effects exclusive to strong non- +linearity which are not observed previously: Increasing +the vdP nonlinearity enhances the synchronization band- +width, and revives synchronization between dissipatively- +coupled oscillators in amplitude death. For reactively- +coupled vdP oscillators on the other hand, we find that +strong nonlinearity induces position correlations which +are impossible in the weakly-nonlinear limit. 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We also provide +analytical and numerical results for the synchronization of quantum Stuart-Landau oscillators in the deep quantum +limit where the nonlinear damping dominates over the pumping. +CONTENTS +I. Exact quantum model +1 +II. A single forced classical oscillator +2 +A. Poincar´e–Lindstedt method +2 +B. Krylov–Bogoliubov averaging +3 +C. Synchronization analysis +3 +III. Two classical oscillators with dissipative coupling +6 +A. Synchronization analysis +6 +B. Amplitude death +7 +C. Effect of the Duffing nonlinearity +7 +D. Synchronization bandwidth +8 +IV. Two classical oscillators with reactive coupling +9 +V. Quantum synchronization in the deep quantum limit +9 +A. Dissipatively-coupled oscillators +9 +B. Reactively-coupled oscillators +11 +References +11 +I. +EXACT QUANTUM MODEL +Here, we describe how the quantum master equation for the exact Duffing-van der Pol model is obtained, following +the approach in [1]. Starting from the classical equation (2) in the main text, and defining the complex amplitude +α = (˜x + i˜y)/2, we get the equation of motion +α′ = i F +2 cos(ωdt) − i α − i β +2 (α + α∗)3 − λ +2 (α3 + |α|2α − |α|2α∗ − α∗3 − r2α + r2α∗) . +(1) +Quantum theory defines the state of a dynamical system by a density operator ρ satisfying ρ′ = Lρ, where L is +a linear superoperator. By regarding (ˆa, ˆa†) as the analog of (α, α∗), where [ˆa, ˆa†] = ˆ1, we can quantize a system +prescribed by α′ = h(α, α∗) by searching for an L such that in the Schr¨odinger picture, ⟨ˆa⟩′ = Tr[ˆa Lρ] = ⟨:h(ˆa, ˆa†):⟩. +Note that we have chosen to normally order h(ˆa, ˆa†), denoted by colons [e.g. if h(ˆa, ˆa†) = ˆaˆa†, then :h(ˆa, ˆa†): = ˆa†ˆa]. +Such an L corresponding to (1) can be found in Lindblad form [2–4], and which we refer to as a Lindbladian: +L = −i [ ˆH,·] + λD[ˆa†ˆa − ˆa†2/2] + λr2D[ˆa†] + 3λ +4 D[ˆa2] , +where +ˆH = ˆa†ˆa − F +2 cos(ωdt)(ˆa + ˆa†) + 3β +4 ˆa†2ˆa2 + β +2 (ˆa†ˆa3 + ˆa†3ˆa) + β +8 (ˆa4 + ˆa†4) +− iλ +2 (ˆa2 − ˆa†2) − iλ +4 (ˆa†ˆa3 − ˆa†3ˆa) − iλ +8 (ˆa4 − ˆa†4) . +(2) +arXiv:2301.02948v1 [quant-ph] 8 Jan 2023 + +2 +We remark that the choice of the master equation is not unique, since we only demand that the classical equation of +motion is recovered in the mean-field limit. Different choices of the master equation will in general lead to different +quantum noise effects, which are most pronounced at low excitations (sometimes referred to as the deep quantum +limit [5]). +II. +A SINGLE FORCED CLASSICAL OSCILLATOR +A. +Poincar´e–Lindstedt method +The Poincar´e–Lindstedt method is a perturbative technique to find approximate periodic solutions [6]. Regular +perturbation theory fails at long time due to the existence secular terms. Taking λ to be the perturbative parameter, +such secular terms in the van der Pol (vdP) oscillator are of the form t cos t, which are unbounded in t. The Poincar´e– +Lindstedt method overcomes this problem by removing secular terms explicitly. +Let us consider the undriven vdP oscillator (setting F = 0). Defining a rescaled time τ ≡ ωt where ω is the +oscillation frequency, we have the equation +ω2 x′′(τ) + λ(x2 − 1) ω x′(τ) + x(τ) = 0 . +(3) +where a prime denotes differentiation with respect to the argument. Now, we perform the perturbative expansions +x(τ) = ˘x0(τ) + λ ˘x1(τ) + λ2 ˘x2(τ) + O(λ3) , +ω = 1 + λ ˘ω1 + λ2 ˘ω2 + O(λ3) . +(4) +Collecting terms of the same order in λ, the zeroth-order equation reads +˘x′′ +0 + ˘x0 = 0 , +(5) +which just describes simple harmonic oscillations given by ˘x0(τ) = a cos τ, where a = ˘x0(0). The first-order equation +reads +˘x′′ +1 + ˘x1 = − 2 ˘ω1˘x′′ +0 − (˘x2 +0 − 1)˘x′′ +0 += 2 ˘ω1a cos τ − a +� +1 − a2 +4 +� +sin τ +� +�� +� +resonant forcing ++a3 +4 sin(3τ) . +(6) +The resonant forcing gives rise to secular terms such as τ cos τ and τ sin τ which are unwanted. Hence, setting the +resonant forcing to zero yields ˘ω1 = 0, a = 2 which gives ˘x0(τ) = 2 cos τ, ω = 1 + O(λ2). This is the Stuart–Landau +(SL), i.e. weakly nonlinear limit of the vdP which describes quasiharmonic oscillations. Solving the resulting first- +order equation, we get the leading-order correction for x(τ): x1(τ) = sin3 τ. To obtain the leading-order correction +for ω, we look at the second-order equation +˘x′′ +2 + ˘x2 = −(˘ω2 +1 + 2 ˘ω2) ˘x′′ +0 − 2 ˘ω1 ˘x′′ +1 − (˘x2 +0 − 1) (˘x′ +1 + ˘ω1 ˘x′ +0) − 2 ˘x0 ˘x1 ˘x′ +0 += +� +4 ˘ω2 + 1 +4 +� +cos τ + higher harmonics . +(7) +Again, eliminating the resonant forcing, we get ˘ω2 = −1/16. In terms of t, we then have +x(t) = 2 cos(ωt) + λ sin3(ωt) + O(λ2) , +ω = 1 − λ2 +16 + O(λ3) . +(8) +Thus, the effect of small nonlinearity on the limit cycle is a decrease in frequency and a distortion without a change +in the limit cycle amplitude (from the first-order approximation of x(t) we can see that for λ < 4/3 the amplitude of +the oscillation is constant at 2. Hence, the amplitude is unchanged to order λ.) A numerical simulation of x(t) and +observed frequency ω as given by as given by (8) are plotted in Fig. 1. Good agreement between the analytic and +numeric simulation can be seen for λ < 1. + +3 +(a) +(b) +AB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAe +UCyhNnJbDJmdnaZ6RXCki/w4kERr36SN/GSbIHTSxoKq6e4KEikMu63s7K6t +r6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR3dRvPXFtR +KwecJxwP6IDJULBKFqpjr1S2a24M5Bl4uWkDlqvdJXtx+zNOIKmaTGdDw3QT+jG +gWTfFLspoYnlI3ogHcsVTixs9mh07IqVX6JIy1LYVkpv6eyGhkzDgKbGdEcWgWva +n4n9dJMbzxM6GSFLli80VhKgnGZPo16QvNGcqxJZRpYW8lbEg1ZWizKdoQvMWXl0 +nzvOJdVS7rF+XqbR5HAY7hBM7Ag2uowj3UoAEMODzDK7w5j86L8+58zFtXnHzmCP +7A+fwB42mNAg=t +AB7XicbVDJS +gNBEK2JW4xb1KOXxiB4CjPidgx68RjBLJAMoafTk7TpZejuEcKQf/DiQRGv/o83/8ZOMgdNfFDweK+KqnpRwpmxv/tFVZW19Y3ipulre2d3b3y/kHTqFQT2iCK92Os +KGcSdqwzHLaTjTFIuK0FY1up37riWrDlHyw4SGAg8kixnB1knNrhJ0gHvlil/1Z0DLJMhJBXLUe+Wvbl+RVFBpCcfGdAI/sWGtWE0mpmxqaYDLCA9pxVGJBTZjNrp +2gE6f0Uay0K2nRTP09kWFhzFhErlNgOzSL3lT8z+ukNr4OMyaT1FJ5ovilCOr0PR1GeaEsvHjmCimbsVkSHWmFgXUMmFECy+vEyaZ9Xgsnpxf16p3eRxFOEIjuEUAr +iCGtxBHRpA4BGe4RXePOW9eO/ex7y14OUzh/AH3ucPk3GPJQ=! +AB7nicbVDLS +sNAFL2pr1pfVZduBovgqiTia1l047KCtYU2lMlk0g6dTMLMjVBCP8KNC0Xc+j3u/BunbRbaemDgcM65zL0nSKUw6LrfTmldW19o7xZ2dre2d2r7h8miTjLdYIhPdC +ajhUijeQoGSd1LNaRxI3g5Gt1O/cS1EYl6wHK/ZgOlIgEo2ildk/aEj71Zpbd2cgy8QrSA0KNPvVr16YsCzmCpmkxnQ9N0U/pxoFk3xS6WGp5SN6IB3LVU05sbPZ+ +tOyIlVQhIl2j6FZKb+nshpbMw4Dmwypjg0i95U/M/rZhd+7lQaYZcsflHUSYJmR6OwmF5gzl2BLKtLC7EjakmjK0DVsCd7iycvk8azuXdYv7s9rjZuijIcwTGcg +dX0IA7aEILGIzgGV7hzUmdF+fd+ZhHS04xcwh/4Hz+AEAFj4c=� +AB6HicbVDLTgJBEOzF+IL9ehlI +jHxRHaNryPRi0dI5JHAhswODYzMzm5mZ +o1kwxd48aAxXv0kb/6NA+xBwUo6qVR1p +7sriAXxnW/ndzK6tr6Rn6zsLW9s7tX3 +D9o6ChRDOsEpFqBVSj4BLrhuBrVghD +QOBzWB0O/Wbj6g0j+S9Gcfoh3QgeZ8za +qxUe+oWS27ZnYEsEy8jJchQ7Ra/Or2IJ +SFKwTVu25sfFTqgxnAieFTqIxpmxEB +9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3RE +pDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLOD +Eo2XxRPxHERGT6NelxhcyIsSWUKW5vJW +xIFWXGZlOwIXiLy+TxlnZuyxf1M5LlZ +sjwcwTGcgdXUIE7qEIdGCA8wyu8OQ +/Oi/PufMxbc042cwh/4Hz+AOl5jQY=x +(a) +(b) +FIG. 1. Dynamics of the undriven vdP oscillator. Results from an exact numerical simulation are shown as a blue solid line, +while results from (8) are plotted as an orange dashed curve. (a) Long-time limit of x(t) for λ = 0.5 (transient dynamics have +been discarded). (b) Observed frequency ω. The Poincar´e–Lindstedt method remains to be a good approximation up till about +λ ≈ 1. +B. +Krylov–Bogoliubov averaging +The Krylov–Bogoliubov averaging method essentially assumes that the amplitude is slowly varying and can be +treated as a constant when averaging the dynamics over one period, which produced the time-averaged equations of +motion. Introducing the complex amplitude, +α(t) = 1 +2 [ x(t) + i y(t) ] , +(9) +where y = x′, we can rewrite the vdP equation as a complex equation of motion +α′ = −i α + i F +2 cos(ωdt) − λ +2 (α3 − α∗3 + |α|2α − |α|2α∗ − α + α∗) . +(10) +Performing the Krylov–Bogoliubov time averaging to first-order in λ, we obtain +α′ = −i α + λ +2 (1 − |α|2) α , +(11) +which is the well-known SL equation representing the normal form of a supercritical Hopf bifurcation. The stable +oscillations are described by α(t) = exp(−it). This gives x(t) = 2 cos t which is consistent with the Poincar´e–Lindstedt +method up to zeroth order in λ. +The second-order averaging yields [7] +α′ = −i α + λ +2 (1 − |α|2) α + i λ2 +8 +� +1 − 6 |α|2 + 11 +2 |α|4 +� +α , +(12) +which predicts a limit cycle amplitude of ˘x0 = 2|α| = 2 and frequency ω = 1 − λ2/16, again consistent with the +zeroth-order Poincar´e–Lindstedt method. +C. +Synchronization analysis +Here, we use the harmonic-balance method to analyze the synchronization behavior for the Duffing–van der Pol +(DvdP) equation in nondimensionalized form, given by +x′′ + λ (x2 − r2)x′ + x + βx3 = F cos(ωdt) . +(13) + +4 +Note that for ease of writing we have omitted tildes for x and all dimensionless parameters (e.g. time). We then +assume a synchronized solution of the form x(t) = A cos(ωdt − φ), where A is the amplitude of the motion and φ is a +constant phase shift from the driving force. Substituting the ansatz for x into (13), we get +− ω2 +dA cos(ωdt − φ) − λ ωdA3 cos2(ωdt − φ) sin(ωdt − φ) + λ ωdr2A sin(ωdt − φ) + A cos(ωdt − φ) = F cos(ωdt) . (14) +The second term on the left-hand side may be written as +cos2(ωdt − φ) sin(ωdt − φ) = 1 +4 sin(ωdt − φ) + higher harmonics . +(15) +Neglecting the higher harmonics and then collecting the coefficients of cos(ωdt) and sin(ωdt), we obtain +(1 − ω2 +d)A cos φ + 1 +4λ ωdA3 sin φ − λ ωdAr2 sin φ + 3 +4βA3 cos φ = F , +(1 − ω2 +d)A sin φ − 1 +4λ ωdA3 cos φ + λ ωdAr2 cos φ + 3 +4βA3 sin φ = 0 . +(16) +These may be expressed compactly as a single complex equation as +� +1 − ω2 +d +� +A − iλ ωdA +�A2 +4 − r2 +� ++ 3 +4 βA3 = Fe−iφ . +(17) +A = A⋆ + p δA , +ωd = ω⋆ + p δω . +(18) +Substituting (18) into (17) then gives the first-order equation in p, +δA +� +6βr2 − 2i +� +1 + 3βr2 λr2� +− 4r +� +1 + 3βr2 δω = λ e−iφ . +(19) +The modulus of the left-hand side must be λ, hence we have +� +6βr2δA − 4r +� +1 + 3βr2 δω +�2 ++ +� +2λr2� +1 + 3βr2 δA +�2 += λ2 . +(20) +This can be solved to obtain the relationship between δA and δω. In order to derive the synchronization frequency +range, we first notice that (20) is a quadratic equation in δA. For synchronization to exist, δA must be real, hence +the discriminant must be non-negative, i.e., +� +48βr3� +1 + 3βr2 δω +�2 +− 4 +� +4r4 � +9β2 + λ2(1 + 3βr2) +�� � +16 r2(1 + 3βr2) δω2 − λ2� +≥ 0 , +(21) +which yields +δω2 ≤ 9β2 + λ2(1 + 3βr2) +16r2(1 + 3βr2)2 +≡ ω2 +c . +(22) +Hence, the vdP oscillator is synchronized if the driving frequency is within the critical interval +ω0 − F +λ ωc ≤ ωd ≤ ω0 + F +λ ωc . +(23) +The synchronization bandwidth is thus +2F +λ ωc = +(F/r) +2λr2(1 + 3βr2) +� +(λr2)2(1 + 3βr2) + 9(βr2)2 . +(24) +Evidently, by fixing F/r, λr2, and βr2, the synchronization bandwidth becomes independent of the scale parameter +r. As shown in the main text, this does not hold true in the quantum case due to quantum noise. Defining +¯F = F/r , +¯λ = λr2 , +¯β = βr2 , +(25) + +5 +(a) +(b) +Synchronization bandwidth +Enhancement factor +Numerics +AB7nicbVDLSgNBEJyNrxhfUY9eBoPgKeyKr4sQ9OIxgnlAsoTZSW8yZHZ2mekVw +pKP8OJBEa9+jzf/xkmyB0saCiqunuChIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmkODxzLW7Y +AZkEJBAwVKaCcaWBRIaAWju6nfegJtRKwecZyAH7GBEqHgDK3U6gaA7MbtlStu1Z2BLhMvJxWSo94r +f3X7MU8jUMglM6bjuQn6GdMouIRJqZsaSBgfsQF0LFUsAuNns3Mn9MQqfRrG2pZCOlN/T2QsMmYcB +bYzYjg0i95U/M/rpBhe+5lQSYqg+HxRmEqKMZ3+TvtCA0c5toRxLeytlA+ZhxtQiUbgrf48jJpnlW +9y+rFw3mldpvHUSRH5JicEo9ckRq5J3XSIJyMyDN5JW9O4rw4787HvLXg5DOH5A+czx+6YI8v� = 0 +AB7nicbVDLSgNBEJyNrxhfUY9eBoPgKeyKr4sQ9OIxgnlAsoTZSW8yZHZ2mekVw +pKP8OJBEa9+jzf/xkmyB0saCiqunuChIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmkODxzLW7Y +AZkEJBAwVKaCcaWBRIaAWju6nfegJtRKwecZyAH7GBEqHgDK3U6gaA7MbrlStu1Z2BLhMvJxWSo94r +f3X7MU8jUMglM6bjuQn6GdMouIRJqZsaSBgfsQF0LFUsAuNns3Mn9MQqfRrG2pZCOlN/T2QsMmYcB +bYzYjg0i95U/M/rpBhe+5lQSYqg+HxRmEqKMZ3+TvtCA0c5toRxLeytlA+ZhxtQiUbgrf48jJpnlW +9y+rFw3mldpvHUSRH5JicEo9ckRq5J3XSIJyMyDN5JW9O4rw4787HvLXg5DOH5A+czx+75I8w� = 1 +AB6HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2NQEI8JmAckS5id +dJIxs7PLzKwQlnyBFw+KePWTvPk3TpI9aGJBQ1HVTXdXEAujet+O7mV1bX1jfxmYWt7Z3 +evuH/Q0FGiGNZJCLVCqhGwSXWDTcCW7FCGgYCm8Hoduo3n1BpHskHM47RD+lA8j5n1Fi +pdtctltyOwNZJl5GSpCh2i1+dXoRS0KUhgmqdtzY+OnVBnOBE4KnURjTNmIDrBtqaQh +aj+dHTohJ1bpkX6kbElDZurviZSGWo/DwHaG1Az1ojcV/Paielf+ymXcWJQsvmifiKIi +cj0a9LjCpkRY0soU9zeStiQKsqMzaZgQ/AWX14mjbOyd1m+qJ2XKjdZHk4gmM4BQ+uoAL +3UIU6MEB4hld4cx6dF+fd+Zi35pxs5hD+wPn8AZ2xjNQ=F +AB7HicbVBNS8NAEJ +3Ur1q/qh69LBbBU0nEr2PRi8cKxhbaUDbSbt0swm7G6GE/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTAXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmGPosEYlqh1Sj4BJ9w43AdqQ +xqHAVji6nfqtJ1SaJ/LBjFMYjqQPOKMGiv53RAN7Vrbt2dgSwTryA1KNDsVb+6/YRlMUrDBNW647mpCXKqDGcCJ5VupjGlbEQH2LFU0h1kM+OnZATq/RJlChb0pCZ+nsip7HW4zi +0nTE1Q73oTcX/vE5mousg5zLNDEo2XxRlgpiETD8nfa6QGTG2hDLF7a2EDamizNh8KjYEb/HlZfJ4Vvcu6xf357XGTRFHGY7gGE7BgytowB0wQcGHJ7hFd4c6bw4787HvLXkFDOH8 +AfO5w/HFo6u� +(a) +(b) +FIG. 2. (a) Synchronization bandwidth against driving force F for λ = 0.5. Adding β enhances the synchronization bandwidth +for weak to moderate forcing (compared to λ). The analytical predictions given by the dashed lines) are accurate up to F ∼ λ. +(b) Enhancement factor defined as the ratio between the bandwidth and its baseline value F/2. A value greater than one +indicates enhancement of the bandwidth. λ = 0.5, F = 0.2, and the predicted minimum β for enhancement is 1/9. +the bandwidth can be rewritten as +2 ¯F +¯λ ωc = +¯F +2¯λ(1 + 3¯β) +� +¯λ2(1 + 3¯β) + 9¯β2 ≈ +� ¯F/2¯λ , +¯β −→ ∞ +¯F/2 , +¯β −→ 0 +(26) +where in the last step we have taken the limit of large and small ¯β. For the usual vdP oscillator (¯β = 0), the synchro- +nization bandwidth reduces to ¯F/2 which is independent of λ. In order to obtain nonlinearity-induced enhancement +in the bandwidth, we require +¯F +2¯λ(1 + 3¯β) +� +¯λ2(1 + 3¯β) + 9¯β2 > +¯F +2 . +(27) +This results in the conditions +¯β > +¯λ2 +3(1 − ¯λ2) , +0 < ¯λ < 1 . +(28) +Without loss of generality, we can set r = 1 in the following discussion. In this case ¯λ = λ, ¯β = β, and ¯F = F. +Interestingly, the nonlinearity-induced enhancement only acts for a finite range of λ and requires a minimum β. In the +limit of large β, the bandwidth tends to F/2λ. However, the method of harmonic balance assumes λ and β to be weak. +In this regime, we can interpret the existence of a minimum β as a competition between the effects of λ and β. Note +that the oscillator frequency is ω ≈ 1 + 3β/2 − λ2/16 from our previous analysis. The threshold for synchronization +enhancement occurs when these two opposite effects on the frequency are of the same scale, i.e. β ∼ λ2. +Figure 2 shows the effect of adding the Duffing nonlinearity on the synchronization bandwidth for r = 1. Comparing +β = 0 and β = 1 in Fig. 2(a), we can see that the synchronization bandwidth is enhanced for the β = 1 case for weak to +moderate forcing F (compared to λ). The analytical prediction of the bandwidth agrees with the numerical simulations +up to F ∼ λ, beyond which the numerical bandwidth exceeds the predicted value indicating the suppression of natural +dynamics at strong forcing. Using λ = 0.5, we also predict that the synchronization enhancement occurs when β > 1/9. +Indeed, by plotting in Fig. 2(b) the enhancement factor, defined as the ratio between the bandwidth and its baseline +value F/2, we see a good agreement between the numerical results and the analytical calculations. Fluctuations in the +numerics can be attributed to a few reasons, such as the time resolution dt in the simulation, the number of samples +used for x(t), the frequency resolution dω used in determining the bandwidth, and the threshold observed detuning +used to determine the frequency-locked regions (set to 0.001). The numerical results also appear to deviate slightly +from the prediction at large β, which is likely due to the breakdown in making the ansatz x(t) = A cos(ωd t − φ) used +in the derivation. Including higher harmonics in the ansatz might result in more accurate predictions at larger β. + +6 +III. +TWO CLASSICAL OSCILLATORS WITH DISSIPATIVE COUPLING +The equations of motion for two dissipatively coupled vdP oscillators are often written as +x′′ +1 + λ(x2 +1 − 1) x′ +1 + x1 = η +2 (x′ +2 − x′ +1) , +x′′ +2 + λ(x2 +2 − 1) x′ +2 + (1 + ∆)x2 = η +2 (x′ +1 − x′ +2) , +(29) +where η is the coupling strength and ∆ is the detuning between the two oscillators. The system has four phase-space +dimensions, two for each oscillator, given by (x1, y1) = (x1, x′ +1) and (x2, y2) = (x2, x′ +2). We can express the coupled +system more simply using complex amplitudes as we did in (9), except now +αk(t) = 1 +2 [ xk(t) + i yk(t) ] , +k = 1, 2 . +(30) +We will analyze the following first-order averaged equations for the coupled system assuming the nonlinearity to be +weak, +α′ +1 = −i α1 + λ +2 (1 − |α1|2)α1 + η +2 (α2 − α1) , +α′ +2 = −i (1 + ∆)α2 + λ +2 (1 − |α2|2)α2 + η +2 (α1 − α2) . +(31) +A. +Synchronization analysis +In fact, only three real variables are needed in polar coordinates. If we write +αk = Rk exp (−iφk) , +k = 1, 2, +(32) +then we may express the coupled dynamics in terms of only R1, R2, and the phase difference ϕ ≡ φ2 − φ1: +R′ +1 = λ +2 R1 (1 − R2 +1) + η +2 (R2 cos ϕ − R1) , +R′ +2 = λ +2 R2 (1 − R2 +2) + η +2 (R1 cos ϕ − R2) , +ϕ′ = ∆ − η +2 +�R1 +R2 ++ R2 +R1 +� +sin ϕ . +(33) +By symmetry, if the two oscillators synchronize, we must have R1 = R2 = R. In this case the above equations simplify +further to +R′ = λ +2 R(1 − R2) + η +2 R(cos ϕ − 1) , +ϕ′ = ∆ − η sin ϕ . +(34) +In the synchronized state, R′ = ϕ′ = 0. Solving the phase equation, we get two solutions, ϕ(1) +⋆ += sin−1(∆/η) and +ϕ(2) +⋆ += π −sin−1(∆/η). To determine the stable phase, we use linear stability analysis. Substituting ϕ(t) = ϕ(k) +⋆ ++ϵ(t) +(k = 1, 2) into the phase equation of motion and keeping only first-order terms in ϵ, we get +ϵ′ = ∆ − η sin +� +ϕ(k) +⋆ ++ ϵ +� += ∆ − η sin +� +sin−1 +�∆ +η +� +− (−1)kϵ +� +≈ (−1)kϵ η +� +1 − ∆2 +η2 . +(35) +From this we see that ϕ(1) +⋆ +is stable while ϕ(2) +⋆ +is unstable. We also find that |∆| < η to be a necessary condition for +synchronization. Substituting ϕ(1) +⋆ +into the radial equation R′ = 0 then gives +λ +2 R (1 − R2) + η +2 R +�� +1 − ∆2 +η2 − 1 +� += 0 . +(36) + +7 +This has the solution +R2 +⋆ = 1 + η +λ +�� +1 − ∆2 +η2 − 1 +� +. +(37) +The zero-amplitude solution corresponds to the case of amplitude death and will be analyzed next. Performing linear +stability analysis, we find that r⋆ is stable for 0 < |∆| ≤ λ and |∆| < η, or when, |∆| > λ and |∆| < +� +λ(2η − λ). +Combining the phase and amplitude solutions, the conditions for synchronization is then +|∆| < η , +for η ≤ λ , +(38) +|∆| < +� +λ(2η − λ) , +for η > λ . +(39) +B. +Amplitude death +It is possible for coupled oscillators to achieve amplitude death, where the limit cycle oscillations are suppressed +due to the mutual coupling. To obtain the conditions for amplitude death, we require the solution α1 = α2 = 0 +to be stable. Since the phase is undefined in this case, we will analyze in Cartesian coordinates instead. Denoting +αk = ℜ[ϵk] + i ℑ[ϵk] (k = 1, 2) for small ϵk, we can linearize the coupled equations around the origin, giving, +� +� +� +ℜ[ϵ1] +ℑ[ϵ1] +ℜ[ϵ2] +ℑ[ϵ2] +� +� +� +′ += +� +� +� +(λ − η)/2 +ω1 +η/2 +0 +−ω1 +(λ − η)/2 +0 +η/2 +η/2 +0 +(λ − η)/2 +ω2 +0 +η/2 +−ω2 +(λ − η)/2 +� +� +� +� +� +� +ℜ[ϵ1] +ℑ[ϵ1] +ℜ[ϵ2] +ℑ[ϵ2] +� +� +� , +(40) +where ω1 = 1 and ω2 = 1 + ∆. Let h be the eigenvalues of the stability matrix in (40), and p = h − (λ − η)/2. The +characteristic polynomial for h then reads +p4 + p2 +� +ω2 +1 + ω2 +2 − η2 +2 +� ++ +�η2 +4 + ω1 ω2 +�2 += 0 +=⇒ +� +p2 − +�η2 +4 + ω1 ω2 +��2 += −(ω1 + ω2)2 p2 . +(41) +The eigenvalues h are therefore +h = 1 +2 +� +λ − η ± +� +η2 − ∆2 +� +± i (ω1 + ω2) +2 +. +(42) +For α1 = α2 = 0 to be stable, the real part of h must be negative. This gives us the the following necessary and +sufficient condition for amplitude death +η > λ , +|∆| > +� +λ(2η − λ) . +(43) +The curve |∆| = +� +λ (2η − λ) thus separates the regions of synchronization and amplitude death, also known as the +Hopf bifurcation curve. +C. +Effect of the Duffing nonlinearity +Recall from (13) that a Duffing–van der Pol (DvdP) oscillator has a nonlinear frequency term βx3 in its second-order +equation of motion for x. If we were to define the DvdP by its equation of motion for its complex amplitude α, then +this term amounts to adding −i3β|α|2α to α′ under first-order time averaging with respect to the Duffing nonlinearity. +We show here that such a modification due to the Duffing nonlinearity in two dissipatively coupled DvdP oscillators +does not change their mutual synchronization from the β = 0 case. The classical time-averaged equations of motion +for two dissipatively-coupled DvdP oscillators are +α′ +1 = −i ω1 α1 + λ +2 (r2 − |α1|2)α1 + iλ2 +8 +� +r4 − 6r2|α1|2 + 11 +2 |α1|4 +� +α1 − i3β +2 |α1|2α1 + η +2 (α2 − α1) , +(44) +α′ +2 = −i ω2 α2 + λ +2 (r2 − |α2|2)α2 + iλ2 +8 +� +r4 − 6r2|α2|2 + 11 +2 |α2|4 +� +α2 − i3β +2 |α2|2α2 + η +2(α1 − α2) , +(45) + +8 +Δ +! +AB8HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIVTdC0Y +3LCvYh7VAymUwbmSGJCOUoV/hxoUibv0cd/6NaTsLbT0QOJxzLrn3BAln2rjut1NYWV1b3yhulra2d3b3yvsHLR2nitAmiXmsOgHWlDNJm4YZTjuJolgEnLaD0e3Ubz9RpVksH8w4ob7A8kiRrCx0mOP2iIr71+ueJW3RnQMvFyUoEcjX75qxfG +JBVUGsKx1l3PTYyfYWUY4XRS6qWaJpiM8IB2LZVYUO1ns4Un6MQqIYpiZ80aKb+nsiw0HosApsU2Az1ojcV/O6qYmu/IzJDVUkvlHUcqRidH0ehQyRYnhY0swUczuisgQK0yM7ahkS/AWT14mrbOqd1Gt3Z9X6jd5HU4gmM4BQ8uoQ530IAmEB +DwDK/w5ijnxXl3PubRgpPHMIfOJ8/NmWQCQ=� = 1 +AB8nicbVDLSgMxFL1TX7W+qi7dBIvgapgRq26Eoh +uXFewDpkPJZDJtaCYZkoxQSj/DjQtF3Po17vwb03YWj0QOJxzLrn3RBln2njel1NaWV1b3yhvVra2d3b3qvsHbS1zRWiLSC5VN8KaciZoyzDaTdTFKcRp51odDvzO49UaSbFgxlnNEzxQLCEWysFPS4jcb42nPr/WrNc7050F/iF6QGBZr96mcv +liRPqTCEY60D38tMOMHKMLptNLNc0wGeEBDSwVOKU6nMxXnqITq8Qokco+YdBc/TkxwanW4zSyRSboV72ZuJ/XpCb5CqcMJHlhgqy+CjJOTISze5HMVOUGD62BPF7K6IDLHCxNiWKrYEf/nkv6R95voXbv3+vNa4KeowxEcwyn4cAkNuIMmtI +CAhCd4gVfHOM/Om/O+iJacYuYQfsH5+AYbaJB/� = 0.5 +AB8nicbVDLSgMxFL3js9ZX1aWbYBFcDTPF10Youn +FZwT5gOpRMJtOGZpIhyQil9DPcuFDErV/jzr8xbWehrQcCh3POJfeKONMG8/7dlZW19Y3Nktb5e2d3b39ysFhS8tcEdokvVibCmnAnaNMxw2skUxWnEaTsa3k39hNVmknxaEYZDVPcFyxhBsrBV1uozG+8dxar1L1XG8GtEz8glShQKNX+erG +kuQpFYZwrHXge5kJx1gZRjidlLu5phkmQ9yngaUCp1SH49nKE3RqlRglUtknDJqpvyfGONV6lEY2mWIz0IveVPzPC3KTXIdjJrLcUEHmHyU5R0ai6f0oZoSw0eWYKY3RWRAVaYGNtS2ZbgL568TFo1790Lx7Oq/Xbo4SHMJnIEPV1CHe2hAEw +hIeIZXeHOM8+K8Ox/z6IpTzBzBHzifPxbckHw=� = 0.2 +AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQzqwWME84BkCbOT3 +mTM7OwyMyuEJf/gxYMiXv0fb/6Nk2QPmljQUFR1090VJIJr47rfztLyuraemGjuLm1vbN +b2tv6DhVDOsFrFqBVSj4BLrhuBrUQhjQKBzWB4M/GbT6g0j+WDGSXoR7QvecgZNVZqd +G5RGNotld2KOwVZJF5OypCj1i19dXoxSyOUhgmqdtzE+NnVBnOBI6LnVRjQtmQ9rFtqaQ +Raj+bXjsmx1bpkTBWtqQhU/X3REYjrUdRYDsjagZ63puI/3nt1IRXfsZlkhqUbLYoTAUxM +Zm8TnpcITNiZAlitbCRtQRZmxARVtCN78y4ukcVrxLirn92fl6nUeRwEO4QhOwINLqM +Id1KAODB7hGV7hzYmdF+fd+Zi1Ljn5zAH8gfP5A2PMjwY=� +AB63icbVBNS8NAEN +3Ur1q/qh69BIvgqSTi17HoxWMFawtKJvtpF26uwm7E6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvTAQ36HnfTmldW19o7xZ2dre2d2r7h8mjVDFosFrHuhNSA4ApayFAJ9FAZS +igHY5vc7/9BNrwWD3gJIFA0qHiEWcUc6kHSPvVmlf3ZnCXiV+QGinQ7Fe/eoOYpRIUMkGN6fpegkFGNXImYFrpQYSysZ0CF1LFZVgmx269Q9scrAjWJtS6E7U39PZFQaM5Gh7ZQUR2 +bRy8X/vG6K0XWQcZWkCIrNF0WpcDF28fdAdfAUEwsoUxze6vLRlRThjaeig3BX3x5mTye1f3L+sX9ea1xU8RJkfkmJwSn1yRBrkjTdIijIzIM3klb450Xpx352PeWnKmUPyB87nD +wijkI=⌘ +FIG. 3. Synchronization boundaries for different values of λ. The synchronization region lies below the curve in the all cases. +The synchronization bandwidth is enlarged as λ is increased for a fixed η. +where ω1 = 1 and ω2 = 1 + ∆. Note these equations assume second-order time averaging with respect to the vdP +nonlinearity [see (12)]. As in the case of β = 0 in Sec. III A, mutual synchronization is more easily analyzed using +polar coordinates, where αk = Rk exp(−iφk) (k = 1, 2). Equations (44) and (45) are then equivalent to +R′ +1 = λ +2 +� +r − R2 +1 +� +R1 + η +2 +� +R2 cos ϕ − R1 +� +, +R′ +2 = λ +2 +� +r − R2 +2 +� +R2 + η +2 +� +R1 cos ϕ − R2 +� +, +ϕ′ = ∆ − λ2 +8 +� +6r2� +R2 +1 − R2 +2 +� +− 11 +2 +� +R4 +1 − R4 +2 +�� +− 3β +2 +� +R2 +1 − R2 +2 +� +− η +2 +�R1 +R2 ++ R2 +R1 +� +sin ϕ . +(46) +Recall from Sec. III A that we have defined ϕ ≡ φ2 − φ1. By symmetry, when the two oscillators synchronize, we +have R1 = R2. Consequently, the equation of motion for the phase difference ϕ becomes independent of R1 and R2. +Moreover, the β term vanishes, which suggests that the β nonlinearity has no effect on the mutual synchronization. +D. +Synchronization bandwidth +For a fixed η, we define from Eq. (39) the synchronization bandwidth to be the range of initial detunings ∆ for +which the two oscillators synchronize. This is given by the piecewise function +Bandwidth = +� +2 +� +λ(2η − λ) +λ ≤ η +2 η +λ > η +(47) +where the factor of 2 arises because ∆ can be either positive or negative (or zero). Increasing λ from 0, we find +that the bandwidth increases monotonically until λ = η, after which the bandwidth saturates at 2η. Thus, the vdP +nonlinearity enlarges the synchronization bandwidth. This is shown by the synchronization boundaries in Fig. 3 +for various values of λ, where synchronization regions lie below the boundary curves. Fixing η and increasing the +nonlinearity parameter λ leads to a larger synchronization bandwidth. This also implies that fixing ∆ and η while +increasing λ can lead to nonlinearity-induced mutual synchronization whereby the two oscillators transition suddenly +from amplitude death to synchronized oscillations. +As a measure for synchronization enhancement, we can calculate the ‘total bandwidth’ by integrating the syn- +chronization bandwidth from η = 0 to some η = ηmax, which can then be evaluated asymptotically for large ηmax +(compared to λ). This gives (assuming ηmax > λ) +B(λ, ηmax) = +� λ +0 +dη η + +� ηmax +λ +dη +� +λ(2η − λ) = 1 +6λ2 + 1 +3 λ1/2 (2ηmax − λ)3/2 ≈ 2 +√ +2 +3 +λ1/2 η3/2 +max , +(48) +which increases as λ1/2. Since B has the geometrical interpretation of the area covered by the synchronization region +in the parameter space (λ, η), this means that the synchronization region grows with the vdP nonlinearity λ. + +9 +IV. +TWO CLASSICAL OSCILLATORS WITH REACTIVE COUPLING +Let us first consider two classical SL oscillators which are reactively coupled with strength g. The coupled complex- +amplitude equations are +α′ +1 = −i α1 + λ +2 (1 − |α1|2) α1 + i g(α2 − α1) , +α′ +2 = −i (1 + ∆) α2 + λ +2 (1 − |α2|2) α2 + i g(α1 − α2) . +(49) +Similar to before, we work in polar coordinates, giving +R′ +1 = λ +2 R1(1 − R2 +1) + g R2 sin ϕ , +R′ +2 = λ +2 R2(1 − R2 +2) − g R1 sin ϕ , +ϕ′ = ∆ + g +�R2 +R1 +− R1 +R2 +� +cos ϕ . +(50) +At steady state, R1 = R2, and we see that ϕ′ = ∆. In other words, the rate at which the phase difference grows +is exactly the detuning between the two oscillators. Thus for reactive coupling, the two oscillators simply cannot +synchronize. This result generalizes to the case two quantum SL oscillators. +What is particularly interesting in the case of reactive coupling is how the oscillators behave beyond the λ −→ 0+ +limit. +In the main text we have shown that increasing the nonlinearity in two reactively coupled quantum vdP +oscillators increases their position correlation before a plateau is reached. +This result is particularly interesting +because the analogous classical system fails to produce the same effect. Here we show this explicitly by computing the +steady-state correlation coefficient for the positions of two classical vdP oscillators as a function of their nonlinearities +(assumed identical, given by λ) and their coupling strength. The position correlation coefficient in this case is given +by +Σ = +�M +m=1 +� +x(m) +1 +− ¯x1 +�� +x(m) +2 +− ¯x2 +� +��M +m=1 +� +x(m) +1 +− ¯x1 +�2��M +m=1 +� +x(m) +2 +− ¯x2 +�2 , +(51) +where +�� +x(1) +1 , x(1) +2 +� +, +� +x(2) +1 , x(2) +2 +� +, . . . , +� +x(M) +1 +, x(M) +2 +�� +are pairwise samples of +� +x1(t), x2(t) +� +. Note that since we are in- +terested in how x1(t) and x2(t) are correlated in the long-time time limit, each pairwise sample must be taken from +x1(t) and x2(t) after all transience has died out. We have also defined +¯xk = 1 +M +M +� +m=1 +x(m) +k +. +(52) +For the two classical vdP oscillators we may simply take +� +x(m) +1 +, x(m) +2 +) +� += +� +x1(m δt), x2(m δt) +� +where we have introduced +a small time increment δt. For a sufficiently large sample size M, (51) measures how correlated x1(t) and x2(t) are +in the long-time limit (or steady state). The result of such a computation is shown in Fig. 4 as a function of λ and +g. As λ −→ 0, we can see that the correlation vanishes, as expected from the SL model. But more importantly, +and omitting the degenerate λ = 0 case, we find that Σ appears to decrease monotonically with λ, and decreases +sharply to zero when λ exceeds some critical value. It is simply impossible to increase Σ by increasing λ for two +reactively-coupled classical vdP oscillators at a fixed g. This is in stark contrast to the same calculation performed +for two such quantum vdP oscillators for which Σ can increase if λ is increased for a given g. +V. +QUANTUM SYNCHRONIZATION IN THE DEEP QUANTUM LIMIT +A. +Dissipatively-coupled oscillators +We derive here a sufficient condition for amplitude death in two dissipatively coupled SL oscillators in the deep +quantum regime. The density operator for two dissipatively-coupled oscillators satisfies ρ′ = Lρ where L given by +L = −i ∆ +� +ˆa† +1ˆa1,· +� ++ κ +� +D[ˆa† +1] + D[ˆa† +2] +� ++ γ +� +D[ˆa2 +1] + D[ˆa2 +2] +� ++ η D[ˆa1 − ˆa2] , +(53) + +10 +Σ +AB6HicbVDLSgNBEO +z1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN/GSbIHTSxoKq6e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQR +oHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVB71S2a24M5Bl4uWkDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOi +ZqgXvan4n9dJTXjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGlCmuL2VsCFVlBmbTdG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1gPAMr/DmPDovzrvzMW9dcfKZI/gD5/M +Hz7WM9Q=g +AB7nicbVDLSsNAFL2pr1pfVZduBovgqiTia1l047KCtYU2lMlk0 +g6dTMLMjVBCP8KNC0Xc+j3u/BunbRbaemDgcM65zL0nSKUw6LrfTmldW19o7xZ2dre2d +2r7h8miTjLdYIhPdCajhUijeQoGSd1LNaRxI3g5Gt1O/cS1EYl6wHK/ZgOlIgEo2il +dk/aEj71Zpbd2cgy8QrSA0KNPvVr16YsCzmCpmkxnQ9N0U/pxoFk3xS6WGp5SN6IB3LV +U05sbPZ+tOyIlVQhIl2j6FZKb+nshpbMw4Dmwypjg0i95U/M/rZhd+7lQaYZcsflHUSY +JmR6OwmF5gzl2BLKtLC7EjakmjK0DVsCd7iycvk8azuXdYv7s9rjZuijIcwTGcgdX0 +IA7aEILGIzgGV7hzUmdF+fd+ZhHS04xcwh/4Hz+AEAFj4c=� +AB7XicbVDJSgNBEK +1xjXGLevTSGARPYUbcjkEvHiOaBZIh9HR6kja9DN09QhjyD148KOLV/Hm39hJ5qCJDwoe71VRVS9KODPW97+9peWV1bX1wkZxc2t7Z7e0t98wKtWE1oniSrcibChnktYts5y2Ek2xi +DhtRsObid98otowJR/sKGhwH3JYkawdVKjc8/6AndLZb/iT4EWSZCTMuSodUtfnZ4iqaDSEo6NaQd+YsMa8sIp+NiJzU0wWSI+7TtqMSCmjCbXjtGx07poVhpV9Kiqfp7IsPCmJGI +XKfAdmDmvYn4n9dObXwVZkwmqaWSzBbFKUdWocnrqMc0JZaPHMFEM3crIgOsMbEuoKILIZh/eZE0TivBReX87qxcvc7jKMAhHMEJBHAJVbiFGtSBwCM8wyu8ecp78d69j1nrkpfPHMA +feJ8/bp+PDQ=⌃ +FIG. 4. Correlation coefficient of two reactively-coupled vdP oscillators for ∆ = 0.1, r = 1. +where ∆ is the detuning between the two oscillators and η is the dissipative coupling strength. Note that we have +assumed equal single-photon amplification rates κ, and equal two-photon dissipation rates γ for the two oscillators. +In the deep quantum limit, i.e. when γ/κ −→ ∞, the process of two-photon loss dominates and this confines each +oscillator to the zero- and one-photon subspace. This allows us to treat each oscillator effectively as a two-level system. +After adiabatically eliminating the higher excited states [8]) we have +¯L = −i ¯∆ [ˆσ+ +1 ˆσ− +1 ,· ] + D[ˆσ+ +1 ] + D[ˆσ+ +2 ] + 2 D[ˆσ− +1 ] + 2 D[ˆσ− +2 ] + ¯η D[ˆσ1 − ˆσ− +2 ] +(54) +where ¯∆ ≡ ∆/κ and ¯η ≡ η/κ. We have also denoted the creation and annihilation operators for the jth oscillator +in the {|0⟩, |1⟩} subspace by ˆσ+ +j and ˆσ− +j . With this simplification, we can solve for the steady-state density matrix +exactly, defined by ¯Lϱ = 0. In the basis {|00⟩ , |01⟩ , |10⟩ , |11⟩} we find +ϱ = +� +� +� +ϱ11 +0 +0 +0 +0 +ϱ22 ϱ23 +0 +0 +ϱ32 ϱ33 +0 +0 +0 +0 +ϱ44 +� +� +� . +(55) +The matrix elements are given explicitly by +ϱ11 = +� +6 ¯η3 + (¯η + 2)2 ¯∆2 + 34 ¯η2 + 60 ¯η + 36 +� +/ν , +ϱ22 = ϱ33 = +� +¯η3 + (¯η + 2)2 ¯∆2 + 8¯η2 + 21¯η + 18 +� +/ν , +ϱ44 = +� +¯η2 + ¯∆2 + 6 ¯η + 9 +� +/ν , +ϱ23 = ϱ∗ +32 = +� +¯η(¯η + 1)(¯η + 3) + i ¯η(¯η + 1) ¯∆ +� +/ν , +(56) +where for ease of writing we have introduced the factor +ν = 8 ¯η3 + (¯η + 3)2 ¯∆2 + 51 ¯η2 + 108 ¯η + 81 . +(57) +It is straightforward to evaluate the position correlation coefficient +Σ = +⟨ˆx1ˆx2⟩ − ⟨ˆx1⟩ ⟨ˆx2⟩ +�� +⟨ˆx2 +1⟩ − ⟨ˆx1⟩2 �� +⟨ˆx2 +2⟩ − ⟨ˆx2⟩2 � +=ρ23 + ρ32 +Tr[ϱ] += +2 ¯η (¯η + 1) +8 ¯η2 + 27 ¯η + (¯η + 3) ¯∆2 + 27 +(58) +where ˆxj = ˆσ+ +j + ˆσ− +j (j = 1, 2). The maximum correlation Σ −→ 1/4 is achieved in the limit ¯∆ −→ 0 and ¯η −→ ∞. +To study amplitude death, we trace out the second oscillator, giving the single-oscillator density matrix +ϱ1 = Tr2[ϱ] = +� +ϱ11 + ϱ22 +0 +0 +ϱ33 + ϱ44 +� +. +(59) + +11 +Its Wigner distribution, taken as a function of the radial coordinate r, can then be calculated explicitly as +W1(r) = (ϱ11 + ϱ22) e−r2 + (ϱ33 + ϱ44)(2 r2 − 1) e−r2 . +(60) +We define amplitude death to be the regime where the Wigner distribution possesses a single peak at the origin instead +of being ring like. A sufficient and necessary condition for this is when ϱ11 + ϱ22 ≥ 3/4, or equivalently, +¯∆2 ≥ 27 − 15 ¯η2 − 4 ¯η3 +(¯η − 1)(¯η − 2) +. +(61) +Interestingly, when the two oscillators are on resonance ( ¯∆ = ∆ = 0), amplitude death can still occur as long as +¯η ≳ 1.17. This is due to the quantum noise which is significant in the deep quantum limit, rather than coming from +the frequency mismatch between the oscillators. +B. +Reactively-coupled oscillators +We show here that two identical reactively-coupled quantum SL oscillators cannot share any position correlations. +This permits calling the correlations observed in two similarly coupled vdP oscillators to be nonlinearity induced. For +two reactively-coupled quantum SL oscillators, its Lindbladian (in units where κ = 1, ∆ = 0) +L = −ig +� +ˆa† +1ˆa2 + ˆa† +2ˆa1,· +� ++ D +� +ˆa† +1] + D +� +ˆa† +2 +� ++ γ +� +D +� +ˆa2 +1] + D +� +ˆa2 +2 +�� +. +(62) +To order 1/γ, we can truncate the Hilbert space of each oscillator to the first three levels, and solve for the steady-state +density matrix ϱ (again defined by Lϱ = 0), +ϱ = +�4 +9 + 4(7g2 − 6) +81γ +� +|00⟩ ⟨00| + +�2 +9 − 4(g2 + 3) +81γ +� +(|01⟩ ⟨01| + |10⟩ ⟨10|) + 2 +9γ |02⟩ ⟨02| + 2 +9γ (|02⟩ ⟨02| + ⟨20| ⟨20|) ++ +�1 +9 − 2(10g2 + 3) +81γ +� +|11⟩ ⟨11| + 1 +9γ +� +|12⟩ ⟨12| + |21⟩ ⟨21| +� ++ i +√ +2g +9γ +� +|11⟩ ⟨20| + |11⟩ ⟨02| − |02⟩ ⟨11| − |20⟩ ⟨11| +� +. +(63) +It can then be checked explicitly from this expression for ϱ that Σ is always zero. +[1] A. Chia, L. C. Kwek, and C. Noh. Relaxation oscillations and frequency entrainment in quantum mechanics. Phys. Rev. +E, 102:042213, Oct 2020. +[2] G. Lindblad. On the generators of quantum dynamical semigroups. Commun. Math. Phys, 48(2):119–130, Jun 1976. +[3] Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan. +Completely positive dynamical semigroups of n-level +systems. J. Math. Phys, 17(5):821–825, 1976. +[4] H.-P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. +[5] W.-K. Mok, L.-C. Kwek, and H. Heimonen. Synchronization boost with single-photon dissipation in the deep quantum +regime. Phys. Rev. Research, 2:033422, Sep 2020. +[6] Steven H Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC +press, 2018. +[7] Jan A Sanders, Ferdinand Verhulst, and James Murdock. Averaging methods in nonlinear dynamical systems, volume 59. +Springer, 2007. +[8] Tony E. Lee, Ching-Kit Chan, and Shenshen Wang. Entanglement tongue and quantum synchronization of disordered +oscillators. Phys. Rev. E, 89:022913, Feb 2014. + diff --git a/WtE1T4oBgHgl3EQfJAP_/content/tmp_files/load_file.txt b/WtE1T4oBgHgl3EQfJAP_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8303bad42d9a0b6010b3eeac8305bdc68e72f6c --- /dev/null +++ b/WtE1T4oBgHgl3EQfJAP_/content/tmp_files/load_file.txt @@ -0,0 +1,1317 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf,len=1316 +page_content='Quantum synchronization effects induced by strong nonlinearities Yuan Shen,1 Wai-Keong Mok,2, 3 Changsuk Noh,4 Ai Qun Liu,1, ∗ Leong-Chuan Kwek,2, 5, 6, 7, † Weijun Fan,1, ‡ and Andy Chia2 1School of Electrical and Electronic Engineering, Nanyang Technological University, Block S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 50 Nanyang Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Singapore 639798 2Centre for Quantum Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' National University of Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Singapore 3California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' CA 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' USA 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Kyungpook National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Daegu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' South Korea 5MajuLab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' CNRS-UNS-NUS-NTU International Joint Research Unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Singapore UMI 3654,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Singapore 6National Institute of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Nanyang Technological University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Singapore 637616,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Singapore 7Quantum Science and Engineering Centre (QSec),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Nanyang Technological University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Singapore A paradigm for quantum synchronization is the quantum analog of the Stuart–Landau oscillator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' which corresponds to a van der Pol oscillator in the limit of weak (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' vanishingly small) nonlin- earity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Due to this limitation, the quantum Stuart–Landau oscillator fails to capture interesting nonlinearity-induced phenomena such as relaxation oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' To overcome this deficiency we propose an alternative model which approximates the van der Pol oscillator to finitely large non- linearities while remaining numerically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This allows us to uncover interesting phenomena in the deep-quantum strongly-nonlinear regime with no classical analog, such as the persistence of amplitude death on resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We also report nonlinearity-induced position correlations in reac- tively coupled quantum oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Such coupled oscillations become more and more correlated with increasing nonlinearity before reaching some maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Again, this behavior is absent classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We also show how strong nonlinearity can enlarge the synchronization bandwidth in both single and coupled oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This effect can be harnessed to induce mutual synchronization between two oscillators initially in amplitude death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='—Mathematical modelling has shown us how the immense variety and beauty of nature can be governed by nonlinear differential equations [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Such equations, owing to their nonlinearity, are difficult to analyze and their application to physical processes has come to be known as nonlinear science [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In physics, interest in nonlinear phenomena has spread to quantum-mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Effects such as chaos [7– 10], stochastic resonance [11–14], and coherence reso- nance [12, 15], are some of the better known exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Besides fundamental research, there are also several promising applications of nonlinear dissipation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' sta- bilizing bosonic qubits for fault-tolerant quantum com- puting [16–18], and enhancing the sensitivity of quantum sensors [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' A relative newcomer to the study of nonlinear effects in quantum systems is synchronization [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Its most ele- mentary form consists of applying a sinusoidal force, say with amplitude f, and frequency Ωd, to a self-sustained oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Synchronization is then the modification of the oscillator frequency to Ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' A prototypical model is the driven van der Pol (vdP) oscillator [22], defined by phase-space coordinates (x, y) satisfying x′ = y , y′ = f cos(Ωdt) − ω2 0 x − µ (x2 − q2) y , (1) where primes denote differentiation with respect to the argument (in this case t, representing time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In the ab- sence of forcing (f = 0) the oscillator is characterized by ω0, and a nonlinearity parameter µ which controls how much the oscillator is damped towards an ampli- tude of order |q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' An important feature of the undriven vdP system is the existence of a supercritical Hopf bifur- cation at µ = 0, via which a stable limit cycle appears for µ > 0 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' At µ = 0, (1) is entirely linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This motivates one to consider the quasilinear limit of (1), defined by µ −→ 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In this limit the vdP oscillator is well approximated by the Stuart–Landau (SL) oscillator, the steady state of which is rotationally symmetric in phase space (a circular limit cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This makes the SL oscillator much simpler to analyze, and has thus served as a starting point in the literature on quantum synchronization for continuous- variable systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' [23–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The trade-off of course, is that effects taking place at finite values of µ are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' A prominent example is relaxation oscillations in the undriven vdP oscillator1 [3, 22, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' More effects start to appear if driving is included, such as quasiperi- odicity and chaos [44–46], both of which are absent in the driven SL oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In this work, we investigate the effects of nonlinear- ity in quantum oscillators by considering a more general model based on the classical Duffing–van der Pol (DvdP) oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This adds ζ x3 to y′ where ζ is another nonlin- earity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' To overcome the inadequacy of the SL model we propose a quantum DvdP oscillator in which 1 In fact, vdP intended for (1) to model relaxation oscillations in an electrical circuit [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' To observe relaxation oscillations in quantum theory one needs to quantize the exact vdP model, and it is only relatively recently that such efforts have been made [40– 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='02948v1 [quant-ph] 8 Jan 2023 2 the vdP and Duffing nonlinearities (respectively µ and ζ) are nonvanishing, but also not arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Our model is accurate up to order (µ/ω0)2, at which the dis- tinct signatures of strong nonlinearity appear, such as relaxation oscillations [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Our approach has the bene- fit of capturing novel nonlinear effects while evading the large computational cost of simulating quantum systems with very strong nonlinear dissipations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We show that for a single oscillator with periodic forc- ing there exists a critical Duffing nonlinearity, above which further increases in ζ enlarges the synchroniza- tion bandwidth (the amount of detuning the forcing can tolerate from the oscillator and still entrain it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This re- sult is similar to the synchronization enhancement from the classical literature [47], but now generalized to quan- tum oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='2 In contrast, the vdP nonlinearity ac- tivates genuine quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Coupling two vdP os- cillators dissipatively may lead to either amplitude death (the cessation of oscillations), or mutual synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Classically, amplitude death occurs only when the two oscillators are sufficiently detuned [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Interestingly, we find this need not be the case for quantum oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We show that two quantum vdP oscillators possessing relatively small limit cycles and nonvanishing nonlinear- ities can exhibit amplitude death even with zero detun- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Larger limit cycles on the other hand can mutually synchronize from a state of amplitude death if their non- linearity is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We also consider reactively coupled oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Two such SL oscillators cannot develop positional correla- tions, and hence do not synchronize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This is true regard- less of whether the oscillators are classical or quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We show here that at finitely large nonlinearity, posi- tion correlations behave rather differently between the classical and quantum oscillators: Two reactively cou- pled quantum vdP oscillators can undergo nonlinearity- induced correlations whereby their position correlation increases as they become more nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In contrast, we find that making the analogous classical oscillators more nonlinear monotonically reduces their position correla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The nonlinearity-induced correlations in the quan- tum vdP oscillators are thus a consequence of both their quantum nature and strong nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='—For simplicity we consider here a dimension- less DvdP model in terms of the nonlinearity parameters λ ≡ µq2/ω0r2 and β ≡ ζq2/ω2 0r2 in which r is a dimen- sionless scale parameter: ˜x′ = ˜y , ˜y′ = F cos(ωd˜t ) − ˜x − λ(˜x − r2)˜x′ − β ˜x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (2) Note that ˜x ≡ xr/q is now a function of ˜t = ω0t, and 2 It is also worth mentioning that nonlinear oscillators are of inter- est to quantum information too, when they are coupled to qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In this context the Duffing nonlinearity has been shown to both increase and stabilize the oscillator-qubit entanglement [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' we have also included a dimensionless external force pa- rameterized by F = fr/ω2 0q and ωd = Ωd/ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' From the approximate analysis of (2), the leading contribution to the oscillator frequency is quadratic in λ, and linear in β [50], given by ω ≈ 1 + r2(3β/2 − λ2r2/16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This moti- vates a Bogoliubov–Krylov time-average of the equations of motion up to these orders, giving [50, 51] α′ =i F 2 cos(ωdt) − i α − i3β 2 |α|2α + λ 2 (r2 − |α|2)α + iλ2 8 � r4 − 6r2|α|2 + 11 2 |α|4 � α, (3) where α = (˜x + i ˜y)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' For F = 0, (3) predicts a limit- cycle amplitude of 2|α| = 2r with the expected frequency shifts due to λ and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Additionally, note the first-order averaging in λ for β = 0 yields the SL equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Our approximate model captures the effects of strong vdP nonlinearity of order λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We seek a quantum master equation ρ′ = Lρ such that ⟨ˆa⟩′ = Tr[ˆa Lρ] (with [ˆa, ˆa†] = ˆ1) agrees with (3) in the mean-field limit [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' It can then be shown that this is satisfied by the Lindbladian [50, 52– 54] L = −i [ ˆH,·] + λr2D[ˆa†] + λ 2 D[ˆa2] , (4) where ˆH = � 1 − λ2r4 8 � ˆa†ˆa + 3λ2r2 8 ˆa†2ˆa2 − 11λ2 48 ˆa†3ˆa3 + 3β 4 ˆa†2ˆa2 − F 2 cos(ωdt)(ˆa + ˆa†) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (5) We have also defined D[ˆc] ≡ ˆc· ˆc†−(ˆc†ˆc·+· ˆc†ˆc)/2 for any ˆc, and a dot denotes the position of ρ when acted upon by a superoperator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We remark that both the higher- order Kerr terms and the nonlinear two-photon dissipa- tion in our proposed model can be implemented in circuit QED [17, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The tunability of the limit cycle radius r allows us to access different parameter regimes of the quantum oscillator, in particular the quantum (r ≪ 1), and semiclassical (r ≈ 1) regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We have included the second-order contributions in λ in our model for its nonlinearity-tuning capability, since the terms linear in λ neither affect the limit-cycle amplitude nor phase dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The Duffing nonlinearity translates to a Kerr term in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This model can be considered as an alternative to the quantum SL oscillator, but with flexibility in tuning the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' All our numerical results for a given parameter set are obtained with a sufficiently large trun- cation of the Hilbert space by ensuring the corresponding steady-state power spectrum converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Nonlinearity-enhanced synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='—We study first the frequency locking of the approximate quantum DvdP oscillator to a periodic force [(4) and (5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The syn- chronization bandwidth is the range of ωd for which the oscillator frequency is locked to the driving frequency at 3 steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This is achieved when |ωd−˜ω| = 0, where ˜ω the observed frequency of the driven oscillator, obtained from the peak of its spectrum averaged over one period of the drive [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Here we find the Duffing nonlinearity to enhance quan- tum synchronization: For a range of λ and a fixed r, increasing β past a critical value widens the synchroniza- tion bandwidth linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 1(a) where the synchronization bandwidth is plotted as a con- tour against ¯β ≡ βr2 and F/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The critical value of ¯β is indicated by the red dashed line, where the bandwidth is equal to its corresponding value at ¯β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' However, this enhancement does not occur for all values of the vdP nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 1(b) we see that an increase of ¯λ ≡ λr2 from its value in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 1(a) can ruin the gain in synchronization bandwidth due to ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Noting that the dissipative terms in L are all proportional to λ, this ef- fect can be qualitatively attributed to the phase diffusion due to quantum noise, which is known to inhibit synchro- nization [23, 24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We can develop some understanding of the quantum DvdP by examining its classical analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Using the method of harmonic balance on x, we are able to derive the conditions for nonlinearity-enhanced syn- chronization analytically for the classical DvdP oscilla- tor, given by [50] ¯β > ¯λ2 3(1 − ¯λ2) , 0 < ¯λ < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (6) This shows clearly the existence of a critical value of ¯β, and a finite interval of ¯λ over which the synchronization enhancement occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' These results are consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 1 except for the fact that quantum noise makes the range of λ for synchronization enhancement in the quan- tum DvdP oscillator smaller compared to the classical range as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We have also shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' [50] that increasing the vdP nonlinearity in the quantum model can only reduce the synchronization bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' One can thus be certain that the synchronization enhancement seen in our model is induced by the Duffing nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We shall see next that the vdP nonlinearity can induce synchronization in coupled oscillators from a state of amplitude death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Nonlinearity-induced synchronization and amplitude death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='—Two dissipatively coupled vdP oscillators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' no Duffing nonlinearity) can be described by the Lindbla- dian L = L1 + L2 − i ∆[ˆa† 2ˆa2,·] + η D[ˆa1 − ˆa2] , (7) where Lk (k = 1, 2) is the Lindbladian for oscillator k, defined by setting ˆa to ˆak and β = F = 0 in (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We have assumed the oscillators to be identical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' same λ and r) except for an initial detuning of ∆, and denoted 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Contour plot of the synchronization bandwidth for the quantum DvdP oscillator as a function of ¯β ≡ βr2 (ver- tical axis) and F/r (horizontal axis) with unit limit-cycle ra- dius, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In this case ¯β = β and ¯λ = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The axes are also indicated in subplot (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (a) Illustration of synchro- nization enhancement for ¯λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Above a critical value of ¯β, indicated by a red dashed line (obtained numerically), the synchronization bandwidth is enlarged as the Duffing nonlin- earity is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (b) Synchronization enhancement disap- pears if we increase the vdP nonlinearity from ¯λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='1 to ¯λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='5, demonstrating the finite range of ¯λ over which the enhancement is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In this case, frequency locking occurs when the ob- served frequencies of the two oscillators become identi- cal at steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' As before, we define an oscillator’s observed frequency by the location of its spectral peak, except now we must use the reduced state derived by a partial trace over the two-oscillator steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' For a fixed η, we define the synchronization bandwidth to be the range of ∆ for which the two oscillators lock fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' It will also be interesting to look at position correlations in the two oscillators at steady state, defined by Σ = ⟨ˆx1ˆx2⟩ − ⟨ˆx1⟩ ⟨ˆx2⟩ �� ⟨ˆx2 1⟩ − ⟨ˆx1⟩2 � � ⟨ˆx2 2⟩ − ⟨ˆx2⟩2 � , (8) where ˆxk = ˆak + ˆa† k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Note that frequency locking implies a nonzero Σ, but not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In addition to frequency locking, dissipatively coupled oscillators can also cease to oscillate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' If the oscillators are classical, then this may happen for a range of η pro- vided that ∆ is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' And if both oscillators stabilize to the same phase-space point, which may be taken to be the origin without loss of generality, then the effect is termed amplitude death [49, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' To define amplitude death in quantum oscillators we generalize the notion of P-bifurcations from classical stochastic systems to the steady-state Wigner function of a reduced state (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' [57] and other references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In this case, amplitude death is said to occur if the single-oscillator Wigner functions peak at the origin in quantum phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This approach is consistent with previous stud- ies on amplitude death in coupled quantum oscillators [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2 we work out regions of frequency locking and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='AMPLITUDE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='DEATH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='FREQUENCY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='LOCKING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='AMPLITUDE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='DEATH ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Regions of frequency locking and amplitude death (as defined in the text by the power spectrum and Wigner function) for two dissipatively coupled vdP oscillators [sub- plots (a)–(d)] along with contours of Σ [subplots (a) and (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' All subplots have ∆ on the vertical axis, and η on horizon- tal axis which we also indicate in subplot (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (a) Large r (semiclassical regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Note the region on the left does not correspond to any identifiable effect and is demarcated us- ing a solid line while the boundary between frequency locking and amplitude death is a Hopf bifurcation, which we denote by a dash-dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' As r is increased, the classical bound- ary is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (b) Effect of varying λ on synchronization for r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Boundaries of the frequency-locking region and its corresponding λ are shown (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' the frequency-locking re- gion is the area underneath each curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Increasing λ can enlarge the synchronization bandwidth and induce frequency locking from a state of amplitude death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (c) Small r (deep quantum regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' At small r, amplitude death occurs even at zero initial detuning, which is a quantum effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Note that ¯λ in (a) and (c) are approximately equal, leaving the dif- ferences between the two plots only as a result of quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (d) Shifts in amplitude-death boundary as ¯λ is in- creased (quantum-to-semiclassical transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' amplitude death in the (η, ∆) parameter space for (7), along with Σ, shown as a contour in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2(a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Two especially interesting scenarios are—when the limit cycles are relatively large compared to quantum noise [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2(a)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' and when they become small, being more sus- ceptible to quantum noise [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2(a) we have indicated the boundary between frequency locking and amplitude death by a dash-dotted line, while no identifiable phenomenon occurs to the left of the solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Note the dash-dotted line is a Hopf- bifurcation curve because the transition from amplitude death to frequency locking is facilitated by a Hopf bifur- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Clearly, Σ is larger inside the frequency-locking region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Especially significant here is the effect of the vdP nonlinearity on synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Whereas in the single-oscillator case the vdP nonlinearity had only detri- mental effects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' it now has a constructive role by enlarg- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='AB8nicbVDLSgMxFM3UV62vqks3wSK4KjNi1Y1Q1IXLCvYB06Fk0kwbmkmG5I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='5Qhn6GxeKuPVr3Pk3pu0stPVAyOGce7n3njAR3IDrfjuFldW19Y3iZmlre2d3r7x/0DIq1ZQ1qRJKd0JimOCSNYGDYJ1EMxKHgrXD0e3Ubz8xbiSjzBOWBCTgeQRpwSs5HfvmABy7VbdWq9csd8MeJl4OamgHI1e+avbVzSNmQqiDG+5yYQZEQDp4JNSt3UsITQERkw31JYmaCbLbyBJ9YpY8jpe2TgG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Positional correlation (8) for two reactively coupled vdP oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (a) Contour of Σ as a function of λ and g for r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='3 and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The bottom gap of zero correlation agrees with the SL limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (b) Correlations along the three vertical dashed lines at g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='4, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='8 in subplot (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' ing the (mutual) synchronization bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We illus- trate this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2(b) where additional frequency-locking boundaries for different values of vdP nonlinearity λ are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' It can be seen that increasing λ enlarges the frequency-locking region (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' [50] for some clas- sical analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This means that two oscillators in a state of amplitude death with an (η, ∆) lying above the Hopf- bifurcation curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2(a) will transit suddenly to a state of synchronized oscillations when λ is sufficiently increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This may be appropriately called nonlinearity- induced mutual synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Turning now to the case of a small limit cycle (r ≪ 1) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2(c), we find frequency locking to be absent while position correlations become negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The blue solid line delineates the boundary of amplitude death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Most striking is the persistence of amplitude death at zero de- tuning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' ∆ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Classically, some frequency mis- match between the two oscillators must be present in order for amplitude death to occur [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This signifies a clear distinction between classical and quantum dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' A loss of amplitude death at ∆ = 0 may then be expected if we increased either r or λ (while hold- ing the other constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This is indeed what we find, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 2(d), where such a quantum-to- semiclassical transition is captured by increasing ¯λ ≡ λr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Since we have focused exclusively on the vdP nonlin- earity here, we note that incorporating a Duffing non- linearity into our model will not in fact change the frequency-locking boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We can understand this from the classical coupled equations of motion, where we see that β appears only in the phase dynamics of the oscillators, and that such terms vanish at steady state for identical oscillators (equal limit cycle radii) [50, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Nonlinearity-induced correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='—It is known that two reactively coupled SL oscillators cannot synchronize nor share position correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This is true even in the quantum case [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' However, we show here that posi- tional correlations between two reactively coupled quan- tum vdP oscillators do develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Here we must use the 5 exact vdP Lindbladian [41, 50], because under reactive coupling, the approximate model does not produce any off-diagonal elements in the steady state, and hence can- not generate correlations in the two oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' As with the dissipatively coupled system, two reactively coupled vdP oscillators can be modelled by considering two un- coupled vdP oscillators with annihilation operators ˆa1 and ˆa2, coupled by the Hamiltonian g(ˆa1ˆa† 2+ˆa† 1ˆa2), where g is the reactive coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' As before, we assume that both oscillators have the same nonlinearity λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 3(a), we generate a contour plot of Σ as a func- tion of λ and g at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='3 and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' From this we see that for a fixed g, increasing the oscillator nonlinearity beyond the SL regime leads to stronger correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We illustrate this more clearly in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 3(b) by showing how Σ varies as a function of λ for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='4, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='8, which are marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 3(a) by vertical dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Note this also shows the existence of an optimal λ which maximizes Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Such nonlinearity-induced correlations are absent in the corresponding classical model [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' At a given cou- pling strength, the position correlation in two reactively coupled classical vdP oscillators decreases monotonically as λ increases [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='— Our work goes beyond the well-studied paradigm of weak nonlinearity in quantum synchroniza- tion, and provides the first systematic study of quan- tum synchronization effects for strong nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We introduced a new quantum oscillator model which cap- tures intriguing effects induced by two strong nonlineari- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We showed that a strong Duffing nonlinearity leads to a linear enhancement of the synchronization band- width in driven oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We also reported genuine quantum synchronization effects exclusive to strong non- linearity which are not observed previously: Increasing the vdP nonlinearity enhances the synchronization band- width, and revives synchronization between dissipatively- coupled oscillators in amplitude death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' For reactively- coupled vdP oscillators on the other hand, we find that strong nonlinearity induces position correlations which are impossible in the weakly-nonlinear limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Our model provides a new paradigm for studying other strongly non- linear effects such as chaos [7, 22, 59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='—YS and WJF would like to thank the support from NRF-CRP19-2017-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' CN was sup- ported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2022R1F1A1063053).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' WKM, AC and LCK are grateful to the National Research Foundation, Singapore and the Ministry of Education, Singapore for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' ∗ EAQLiu@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='sg † cqtklc@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='sg ‡ EWJFan@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='edu.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Haake, Quantum Signatures of Chaos 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' ((Springer, Switzerland), 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' [60] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Braun, Dissipative Quantum Chaos and Decoherence ((Springer, Berlin, Heidelberg), 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Supplementary material for “Quantum synchronization effects induced by strong nonlinearities” In this supplementary note, we provide details about the quantum model for the Duffing-van der Pol oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We analyze the classical synchronization dynamics of the oscillator models discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' We also provide analytical and numerical results for the synchronization of quantum Stuart-Landau oscillators in the deep quantum limit where the nonlinear damping dominates over the pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Exact quantum model 1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' A single forced classical oscillator 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Poincar´e–Lindstedt method 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Krylov–Bogoliubov averaging 3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Synchronization analysis 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Two classical oscillators with dissipative coupling 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Synchronization analysis 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Amplitude death 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Effect of the Duffing nonlinearity 7 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Synchronization bandwidth 8 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Two classical oscillators with reactive coupling 9 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Quantum synchronization in the deep quantum limit 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Dissipatively-coupled oscillators 9 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Reactively-coupled oscillators 11 References 11 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' EXACT QUANTUM MODEL Here, we describe how the quantum master equation for the exact Duffing-van der Pol model is obtained, following the approach in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Starting from the classical equation (2) in the main text, and defining the complex amplitude α = (˜x + i˜y)/2, we get the equation of motion α′ = i F 2 cos(ωdt) − i α − i β 2 (α + α∗)3 − λ 2 (α3 + |α|2α − |α|2α∗ − α∗3 − r2α + r2α∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (1) Quantum theory defines the state of a dynamical system by a density operator ρ satisfying ρ′ = Lρ, where L is a linear superoperator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' By regarding (ˆa, ˆa†) as the analog of (α, α∗), where [ˆa, ˆa†] = ˆ1, we can quantize a system prescribed by α′ = h(α, α∗) by searching for an L such that in the Schr¨odinger picture, ⟨ˆa⟩′ = Tr[ˆa Lρ] = ⟨:h(ˆa, ˆa†):⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Note that we have chosen to normally order h(ˆa, ˆa†), denoted by colons [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' if h(ˆa, ˆa†) = ˆaˆa†, then :h(ˆa, ˆa†): = ˆa†ˆa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Such an L corresponding to (1) can be found in Lindblad form [2–4], and which we refer to as a Lindbladian: L = −i [ ˆH,·] + λD[ˆa†ˆa − ˆa†2/2] + λr2D[ˆa†] + 3λ 4 D[ˆa2] , where ˆH = ˆa†ˆa − F 2 cos(ωdt)(ˆa + ˆa†) + 3β 4 ˆa†2ˆa2 + β 2 (ˆa†ˆa3 + ˆa†3ˆa) + β 8 (ˆa4 + ˆa†4) − iλ 2 (ˆa2 − ˆa†2) − iλ 4 (ˆa†ˆa3 − ˆa†3ˆa) − iλ 8 (ˆa4 − ˆa†4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (2) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='02948v1 [quant-ph] 8 Jan 2023 2 We remark that the choice of the master equation is not unique, since we only demand that the classical equation of motion is recovered in the mean-field limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Different choices of the master equation will in general lead to different quantum noise effects, which are most pronounced at low excitations (sometimes referred to as the deep quantum limit [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' A SINGLE FORCED CLASSICAL OSCILLATOR A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Poincar´e–Lindstedt method The Poincar´e–Lindstedt method is a perturbative technique to find approximate periodic solutions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Regular perturbation theory fails at long time due to the existence secular terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Taking λ to be the perturbative parameter, such secular terms in the van der Pol (vdP) oscillator are of the form t cos t, which are unbounded in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The Poincar´e– Lindstedt method overcomes this problem by removing secular terms explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Let us consider the undriven vdP oscillator (setting F = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Defining a rescaled time τ ≡ ωt where ω is the oscillation frequency, we have the equation ω2 x′′(τ) + λ(x2 − 1) ω x′(τ) + x(τ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (3) where a prime denotes differentiation with respect to the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Now, we perform the perturbative expansions x(τ) = ˘x0(τ) + λ ˘x1(τ) + λ2 ˘x2(τ) + O(λ3) , ω = 1 + λ ˘ω1 + λ2 ˘ω2 + O(λ3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (4) Collecting terms of the same order in λ, the zeroth-order equation reads ˘x′′ 0 + ˘x0 = 0 , (5) which just describes simple harmonic oscillations given by ˘x0(τ) = a cos τ, where a = ˘x0(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' The first-order equation reads ˘x′′ 1 + ˘x1 = − 2 ˘ω1˘x′′ 0 − (˘x2 0 − 1)˘x′′ 0 = 2 ˘ω1a cos τ − a � 1 − a2 4 � sin τ � �� � resonant forcing +a3 4 sin(3τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (6) The resonant forcing gives rise to secular terms such as τ cos τ and τ sin τ which are unwanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Hence, setting the resonant forcing to zero yields ˘ω1 = 0, a = 2 which gives ˘x0(τ) = 2 cos τ, ω = 1 + O(λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' This is the Stuart–Landau (SL), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' weakly nonlinear limit of the vdP which describes quasiharmonic oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Solving the resulting first- order equation, we get the leading-order correction for x(τ): x1(τ) = sin3 τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' To obtain the leading-order correction for ω, we look at the second-order equation ˘x′′ 2 + ˘x2 = −(˘ω2 1 + 2 ˘ω2) ˘x′′ 0 − 2 ˘ω1 ˘x′′ 1 − (˘x2 0 − 1) (˘x′ 1 + ˘ω1 ˘x′ 0) − 2 ˘x0 ˘x1 ˘x′ 0 = � 4 ˘ω2 + 1 4 � cos τ + higher harmonics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (7) Again, eliminating the resonant forcing, we get ˘ω2 = −1/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' In terms of t, we then have x(t) = 2 cos(ωt) + λ sin3(ωt) + O(λ2) , ω = 1 − λ2 16 + O(λ3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' (8) Thus, the effect of small nonlinearity on the limit cycle is a decrease in frequency and a distortion without a change in the limit cycle amplitude (from the first-order approximation of x(t) we can see that for λ < 4/3 the amplitude of the oscillation is constant at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Hence, the amplitude is unchanged to order λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=') A numerical simulation of x(t) and observed frequency ω as given by as given by (8) are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' Good agreement between the analytic and numeric simulation can be seen for λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='(a) ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content='dX0IA7aEILGIzgGV7hzUmdF+fd+ZhHS04xcwh/4Hz+AEAFj4c=� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfJAP_/content/2301.02948v1.pdf'} +page_content=' 1 MK) +but their fast cooling, their sub-pixel multithermal structure, +and the width of the AIA response function, prevent us from +detecting significant time lags; +3. The observed events are the transition region (∼ 1MK) emis- +sion of long and hot (i.e ∼ 10 – 100 Mm, ∼ 3 MK ; Reale +2014) loops, which are heated impulsively. +Let us start with the interpretation given by the scenario 1. Look- +ing at the most intense bands of AIA (Figure 2), we understand +that a time lag zero arises when the plasma temperature does not +reach the peak of the 171 band. At the temperatures below this +peak, all the bands behave similarly, and so do the light curves. +Furthermore, the observational properties (low-lying, short +time lags) of these events resemble what is observed by +Winebarger et al. (2013) for the "Hi-C loops" (Te ∼ 105 K +and ne ∼ 1010 cm−3) in the inter moss loops areas. Their time +lag analysis on the AIA light curves also displayed near-zero +time lags, which brought them to conclude that the loops did not +reach one million degree. Their interpretation was the observa- +tion of impulsively, low-energy (nanoflares) heated loops which +cool rapidly due to their small length. Given the similarities of +the HRI brightenings to these events, we suggest that they may +have a similar physical origin, being the result of an impulsive +heating. +For such cold events to be visible in the AIA bands and in +HRIEUV, they should be quite dense. We did a first order estima- +tion of their density, using an average value of the background- +subtracted event intensity on AIA 171 and assuming an isother- +mal plasma. We obtained ne ∼ 109 cm−3 for Te = 1.3 × 106 K +and ne ∼ 1010 cm−3 for Te = 3 × 105 K. The latter supports the +result of Winebarger et al. (2013). +However, we must consider possible differences between the +Hi-C loops and our HRIEUV events. First, as mentioned, the ob- +served solar region is not the same. But small low-lying cool +loops (Te ≤ 0.5 MK) are observed in the QS (Hansteen et al. +2014), and are ubiquitous along the supergranular cell bound- +aries in the solar upper atmosphere (see for instance, Feldman +et al. 1999; Sánchez Almeida et al. 2007, and references therein). +And since there is no distinction between supergranular cells in +QS and AR, we expect to observe similar events in both regions. +Berghmans et al. (2021) showed with HRILyα observations that +the HRIEUV events are organised mostly around the supergranu- +lar network. +Another difference between the Hi-C and the HRIEUV events +are their estimated temperature, around Te ≈ 0.25 ± 0.06 MK +for the first case (Winebarger et al. 2013) and around 1.3 ± 0.1 +MK for the latter case (Berghmans et al. 2021). Again, if we are +looking at similar events in the two cases, we suggest that such +discrepancy may be due to the uncertainties in the data, the in- +version methods and associated assumptions applied to relatively +broad band instruments, as for these imagers. Indeed, the mea- +surement of the temperature of these events is very challenging. +For instance, Schonfeld & Klimchuk (2020) showed that often +the cool plasma emission dominates the bands, even though the +hot plasma is there. +Let us now assume that we are in the scenario 2. A time lag +close to zero for AIA bands has been predicted in the TR emis- +sion of active region coronal loops heated by nanoflares (Viall +& Klimchuk 2015, see also references therein). They showed +Article number, page 9 of 13 + +A&A proofs: manuscript no. arxiv_version +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = 1.3s +Event pixels +QS pixels +Random +| +Confidence levels: +80 % +90 % +95 % +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = 1.2 s +0.0 +0.2 +(a) 193 - 171 (background not subtracted) +0.00 +0.25 +0.0 +0.2 +(b) 193 - 171 (background subtracted) +0.00 +0.25 +100 +101 +102 +Event pixel counts +Fig. 7. Margin and 2D histograms of time lags and associated maximum correlation values for the couple 193-171. Sub-figure (a) in red shows +the original distribution for the event pixels, sub-figure (b) shows the background subtracted event pixels. The blue contours in the central panel +of sub-figure (a) are the [20, 40, 60, 80] % percentiles of the QS pixels distribution. The green colors in the main panels are the confidence levels, +and the distributions of the light curves used to compute them is plotted with the same color in the margin histograms. The margin histograms are +normalised by their total number of pixels. The parameter ν>95 is defined as in Fig. 5. +that the combination of the multi-temperature sensitivity of the +AIA bands, combined with the almost constant pressure prop- +erty of the TR and its variable extension along the loop during +the heating-cooling phases, result in a narrower time lag with +respect to the coronal emission part of the loop. To be empha- +sized here that the TR of a loop is defined as the region where +the thermal conduction acts as a plasma coolant, contrary to the +coronal region where it acts as a heater (e.g. Klimchuk et al. +2008). While the presence in the simulation of short time lags +for all the AIA couples corroborates our results, the loops mod- +eled by Viall & Klimchuk (2015) are much longer than what we +are dealing with here (L ≈ 30 – 50 Mm, with respect to 0.4 – 4 +Mm). Moreover, in those simulations, a clear different signature +in the time lag exists between TR and coronal emission, while +this is not visible in our data. This could be possibly explained +by the short cooling time from coronal temperatures of one of +these tiny loops. For instance, for the shorter loops (∼ 0.4 Mm) +detected by HRIEUV at a temperature of ∼ 1.3 MK and density +of ne = 1010cm−3 the cooling time is about 14s. +It is possible that our time lag method is not sensitive enough, +due to the AIA cadence of 12 s, to detect both TR and coronal +emission populations of short time lags. We propose to investi- +gate further this aspect in the future through numerical simula- +tions. The small asymmetries we have in our time lag distribu- +tions are below the cadence of our observation. We would need +a higher temporal resolution data to corroborate such result. The +cadence should be at last similar to the one of HRIEUV, where +the emission variation of the event is better captured. At present, +we verified that the measured time lags are independent of the +event’s duration. +Concerning the scenario 3, if such large loops exist in the QS, +they remain undetected by the AIA channels, meaning that they +would have a very low density. Without independent evidence +that this is the case, we exclude for now this possibility. +In conclusion, in the picture of impulsive heating phenomena +acting in the QS region, and considering the wide temperature +response of the AIA bands, our results appear also to be con- +sistent with predominantly fast cooling plasma from more than +1 MK, that is satisfying our scenario 2. Consistently with this +picture are the results from a 3D MHD simulations using MU- +RaM code by Chen et al. (2021). Here magnetic reconnections +in the coronal part of small QS loops produced events with prop- +erties similar to what observed in HRIEUV. They noticed that the +simulated HRIEUV emission only showed the apex of the heated +loop, where the lower density allows the available stored energy +to heat the plasma up to ≈ 1.3 MK, even though some hotter +temperatures could also be reached. +To summarize, our results are consistent with two possible +scenarios: either the events do not reach coronal temperatures, +or they do, but they cool faster than the AIA temporal resolu- +tion. It is possible that the two scenarios coexist, as the HRIEUV +catalog does not separate events produced by different physical +processes. The AIA cadence and the multithermal nature of the +bands do not allow separating the emissions from the possible +cool and hot plasma along the line of sight. +To solve the ambiguity on the temperature, we need to use +spectroscopic data. This has been done recently by using the +Spectral Imaging of the Coronal Environement (SPICE) instru- +ment on board Solar Orbiter (Huang et al. submitted to this is- +sue). They investigated a few HRIEUV events and came to the +conclusion that the studied events do not show significant emis- +sion at temperatures higher than that of Ne viii (0.63 MK). +Although such spectroscopic analysis needs to be extended +to a larger sample to better quantify the fraction of events not +Article number, page 10 of 13 + +Dolliou et al.: Temperature of EUI QS small scale brightenings: evidence for a cooler component +reaching high temperatures, we find it to support our conclusion +that quiet Sun small-scale EUI brightenings are in most cases +largely dominated by cool emission. +Further investigations are needed to confirm this idea. For +these reasons, we plan to extend our methodology to forward +modeling constrained by spectroscopic data. +Acknowledgements. The authors gratefully thank J.A. Klimchuk for the fruit- +ful discussions and suggestions. A.D. acknowledges the funding by CNES and +EDOM. S.P. acknowledges the funding by CNES through the MEDOC data and +operations center. G.P. was supported by a CNES postdoctoral allocation. P.A. +and D.M.L. are grateful to the Science Technology and Facilities Council for +the award of Ernest Rutherford Fellowships (ST/R004285/2 and ST/R003246/1, +respectively). The ROB team thanks the Belgian Federal Science Policy Of- +fice (BELSPO) for the provision of financial support in the framework of the +PRODEX Programme of the European Space Agency (ESA) under contract +numbers 4000134474 and 4000136424. This paper uses the Solar Orbiter/EUI +data release 1.0 https://doi.org/10.24414/WVJ6-NM32. Solar Orbiter is +a space mission of international collaboration between ESA and NASA, oper- +ated by ESA. The EUI instrument was built by CSL, IAS, MPS, MSSL/UCL, +PMOD/WRC, ROB, LCF/IO with funding from the Belgian Federal Science Pol- +icy Office (BELSPO/PRODEX PEA 4000134088, 4000112292, 4000117262, +and 400013447); the Centre National d’Etudes Spatiales (CNES); the UK Space +Agency (UKSA); the Bundesministerium für Wirtschaft und Energie (BMWi) +through the Deutsches Zentrum für Luft- und Raumfahrt (DLR); and the Swiss +Space Office (SSO). This work used data provided by the MEDOC data and op- +erations centre (CNES / CNRS / Univ. Paris-Saclay), http://medoc.ias.u-psud.fr/. +This research used version 0.6.4 (Barnes et al. 2021) of the aiapy open source +software package (Barnes et al. 2020). +References +Antolin, P., Pagano, P., Testa, P., Petralia, A., & Reale, F. 2021, Nature Astron- +omy, 5, 54 +Aschwanden, M. J. & Parnell, C. 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P., et al. 2020, A&A, 642, A3 +Article number, page 11 of 13 + +A&A proofs: manuscript no. arxiv_version +Appendix A: Computation of the confidence levels +The cross-correlation of two uncorrelated random time series has +a nonzero probability of resulting in a time lag with a nonzero +value for the maximum correlation. This is why the interpreta- +tion of our time lag results is challenging, especially for couples +involving low to medium SNR AIA channels, such as 131, 94 +and 335 which are noise dominated in several pixels. +For our goal we adopted a Monte-Carlo approach inspired by +Max-Moerbeck et al. (2014). We computed the time lags (corre- +sponding to the maximum cross correlation) between many un- +correlated simulated AIA light curves to estimate the probability +of chance occurrence of each time lag value. +The simulated light curves are built using the observational +results that the coronal emission has a temporal Power Spectral +Density (PSD) that can be modeled by a power law (Auchère +et al. 2014; Gupta 2014; Threlfall et al. 2017). Specifically, for +the QS, Ireland et al. (2014) fitted the exponents n = 1.72 ± 0.01 +for AIA 171 and n = 2.20 ± 0.01 for AIA 193. To keep the +empirical model simple, we adopted a power law with exponent +n = 2 for all the AIA channels. From this PSD, we generate 105 +random light curves of 4 min length and 12 s cadence using the +method described in (Timmer & Koenig 1995). +The obtained time series ˆI(t) (in arbitrary units) are converted +into Digital Number (DN) as the follow: +I(t) [DN] = +�ˆI(t) − µˆI +� σDN +σˆI ++ µDN +(A.1) +where µˆI and σˆI are, respectively, the mean and standard de- +viation of ˆI(t); µDN and σDN are the spatial mean intensity and +the standard deviation derived from the first image of the AIA +sequence (see Fig. 1 (d)). +Photon noise is then added by picking random values from +a Poisson distribution peaking at the average photons per image. +We assume it to be equal to the incident photons I(t). Negative in- +tensity values are set to zero. Next, we simulate the regular AIA +acquisition chain by re-converting the time series into DN. Read +noise is then added, in the form of a normal distribution of mean +zero and standard deviation σRN. Using the inverse of the camera +gain, I(t) is converted into photons. All the conversion constants +are taken from the initial AIA calibration (Boerner et al. 2012). +The resulting time series are now used for the time lag anal- +ysis applied to each of the AIA couples used in Sect. 4.1 and +4.3. +The time lags and maximum correlation distributions of +these random light curves are displayed for the couple 193-171 +in the margin histograms of Fig. 7. The confidence levels are +defined as the [80 %, 90 %, 95 %] percentiles of the maximum +correlation distribution. They are displayed as green dashed lines +in Fig. 7 and Fig. 5. According to our simulation, counts above +the 95 % confidence level are at most 5 % likely to be caused by +chance. +Appendix B: Event-based time lag analysis +The main work we have presented is based on the single pixel +analysis. Here we summarize the results from the full-event in- +vestigation in order to verify if the resulting thermal behavior +reflects the one deduced with the single-pixel analysis. +Both the pixel-based and the full-event approaches have their +advantages. The full-event approach increases the SNR of the +light curves, as it is represented by the averaged intensity over +the selected event area, but does not separate the "cold" and the +"hot" pixels populations. This is because inside an "event sur- +face", one pixel might reach a higher temperature compared with +the others. The high temperature pixel and the lower tempera- +ture ones appear as separate counts in the resulting figures of +the pixel-based approach (Fig. 5). On the contrary, the temper- +ature associated with the average intensity will be something in +between the hottest and cooler pixels, so reducing the tempera- +ture excursion over time. Each event area is a single count in the +statistical analysis (Fig. B.1). +To build the single-event light curves, we proceeded by spa- +tially averaging the light curves within each event mask. The +time lag analysis is then applied to these new time sequences +in the same way as it was done for the pixel-based approach +(Sect. 3.2). +The results of the analysis are displayed in Fig. B.1. The +time lags are centered around short values (>12s), above the 95 +% confidence levels. There is no noticeable difference with the +pixel-based approach (Fig. 5), apart from small variations in the +asymmetries ν95, which remains close to the exposure time. The +variations are probably caused by the lower number of counts +above the 95 % confidence level, compared with the pixel-based +approach. It decreases the statistical significance of the asymme- +try, and the events should be studied individually. +Article number, page 12 of 13 + +Dolliou et al.: Temperature of EUI QS small scale brightenings: evidence for a cooler component +100 +101 +Event pixel counts +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = 0.4 s +193-171 +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = 3.3 s +211-171 +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = -0.1 s +193-131 +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = 1.2 s +171-131 +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = 2.8 s +335-171 +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = 0.1 s +211-131 +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = -3.8 s +94-171 +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = 4.2 s +335-211 +1 +0 +1 +Time lag [min] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Maximum correlation +95 = -0.7 s +94-335 +Confidence levels: +80 % +90 % +95 % +Fig. B.1. Same as Fig. 5, but with a full-event approach, as opposed to a pixel-based one. For every 1314 events, the light curves are spatially +averaged over each of their respective event surface. Then, time lag extraction is performed similarly as the pixel-based approach (Sect. 3.2). The +estimated background has been previously subtracted on event pixels with the "inpainting" algorithm. +Article number, page 13 of 13 + diff --git a/ZtA0T4oBgHgl3EQfF_9p/content/tmp_files/load_file.txt b/ZtA0T4oBgHgl3EQfF_9p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..874ac1f1252ba038c11d25404673aafe4e9c541a --- /dev/null +++ b/ZtA0T4oBgHgl3EQfF_9p/content/tmp_files/load_file.txt @@ -0,0 +1,1329 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf,len=1328 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' arxiv_version ©ESO 2023 January 6, 2023 Temperature of Solar Orbiter/EUI quiet Sun small scale brightenings: evidence for a cooler component A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Verbeeck3, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Gissot3, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Aznar Cuadrado7, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Buchlin1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Mierla3, 8, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Teriaca7, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Zhukov3, 9 1 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405, Orsay, France 2 Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle Upon Tyne, NE1 8ST, UK 3 Solar-Terrestrial Centre of Excellence – SIDC, Royal Observatory of Belgium, Ringlaan -3- Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Circulaire, 1180 Brussels, Belgium 4 Physikalisch-Meteorologisches Observatorium Davos, World Radiation Center, 7260, Davos Dorf, Switzerland 5 ETH-Zürich, Wolfgang-Pauli-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 27, 8093 Zürich, Switzerland 6 UCL-Mullard Space Science Laboratory, Holmbury St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Mary, Dorking, Surrey, RH5 6NT, UK 7 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany 8 Institute of Geodynamics of the Romanian Academy, Bucharest, Romania 9 Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia e-mail: antoine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='dolliou@universite-paris-saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='fr Received 7 September 2022 / Accepted 4 January 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' On 2020 May 30, small and short-lived EUV brightenings were observed in the Quiet Sun (QS) during a four minutes sequence by EUI/HRIEUV on board Solar Orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Their physical origin and possible impact on coronal or Transition Region (TR) heating are still to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Our aim is to derive the statistical thermal evolution of these events in order to establish their coronal or TR origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Our thermal analysis takes advantage of the multithermal sensitivity of the Atmospheric Imaging Assembly (AIA) imager on board the Solar Dynamics Observatory (SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We first identified these HRIEUV events in the six coronal bands of AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We then performed a statistical time lag analysis, which quantifies the delays between the light curves from different bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These time lags can give significant insights into the temperature evolution of these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The analysis is performed taking into account the possible contribution to the results from the background and foreground emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The events are characterized by time lags inferior to the AIA cadence of 12 s, for all nine couples of AIA bands analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Our interpretation is the possible co-presence of events which reach or do not reach coronal temperatures (≈ 1 MK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We believe that the cool population dominates the events analyzed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Sun: corona – Sun: transition region – Sun: UV radiation – Instrumentation: high angular resolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Introduction Decades of investigation suggest that the solar corona is formed and maintained through small scale processes, even though the mechanisms at the origin of such processes are only partially understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Waves dissipation and magnetic field reconnection are present in the solar atmosphere and are the main candidates processes for the plasma heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' See for instance Reale (2014) and Viall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) for a review on the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The observations suggest that the dissipation of magnetic en- ergy leading to coronal heating must happen at unresolved spa- tial scales, and while many dissipation mechanisms are impul- sive in nature, it is unclear whether the dissipation has a more continuous or bursty character on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The properties of these heating events, such as their amplitude, the duration and the occurrence frequency, are still a matter of debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Parker (1988) proposed magnetic reconnection as the origin of these heating events (which became known as nanoflares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' His theory is based on the shuffling and intermixing of the photo- spheric footpoints of magnetic flux tubes, which would produce reconnection with subsequent formation of tiny current sheets in which the energy is dissipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This idea has been generalized in more recent years, particularly for active region heating, where other processes (waves propagation) than reconnection may also be at the origin of the nanoflares energy (Van Doorsselaere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Viall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For instance, small scale energy dissi- pation can occur through turbulent cascade created by nonlinear waves interaction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Buchlin & Velli 2007), or through shock heating from nonlinear mode conversion (Moriyasu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Studies addressing the heating of the Quiet Sun (QS) indi- cate that waves and reconnections are also present (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' McIn- tosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Hahn & Savin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Upendran & Tripathi 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Observations of the corona from the hard X-rays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Crosby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Shimizu 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Hannah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2010) to the UV bands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Harra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' As- chwanden & Parnell 2002) also suggest that small scale impul- sive heating may play a role here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These observations reveal that unresolved small bright transient events increase in number ev- erywhere in the corona, any time we increase the spatial and temporal resolutions of our instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Examples of small and fast phenomena in the corona have been observed during the High-Resolution Coronal Imager (Hi- Article number, page 1 of 13 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='02040v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='SR] 5 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' arxiv_version C) sounding rocket flights (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2014), during which images were recorded in a band centered on 193 Å (including the Fe XII 195 Å line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These observations were made with a spatial resolution of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3′′ (≈ 220 km, Winebarger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The Hi-C instrument resolved small cool loops (Winebarger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2013) and EUV bright dots with characteristic lengths of 680 km, durations of 25 s and temperatures ranging between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 MK (Régnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The Interface Region Imaging Spectrograph (IRIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' De Pon- tieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2014) reaches a resolution of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='33′′ – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4′′ (≈ 240 – 290 km in the corona), but is mostly sensitive to transition region (TR) and chromospheric temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' With IRIS and SDO/AIA (Solar Dynamics Observatory, Pesnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2012) it was possi- ble, for instance, to observe tiny, short-lived and multithermal "nanojets" (size 1000 – 2000 km, ∼15 s, with chromospheric to coronal temperatures, Antolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Sukarmadji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2022) in large cool loops, interpreted as the transverse motion of field lines reconnecting at small angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Larger jet-like struc- tures (Innes & Teriaca 2013) were detected with the Solar Ultra- violet Measurements of Emitted Radiation (SUMER) spectrom- eter (Wilhelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1995) on board the Solar and Heliospheric Observatory (SOHO), along with Ultraviolet (UV) (Peter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2014) and Extreme-UV (EUV) (Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2018) bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' IRIS has also observed ‘unresolved fine structures’ (UFS) in TR lines, which has been associated with short (≈ 4 – 12 Mm) loops or part of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' They were seen at the limb in QS regions, and showed to be highly variable (few minutes), with strong Doppler shift dynamics (up to 100 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Besides the aforementioned sporadic and short duration Hi-C rocket flights, the Atmospheric Imaging Assembly (AIA, Lemen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2012), onboard the Solar Dynamics Observatory (SDO, Pesnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2012) obtains full Sun images with a resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5′′, corresponding to ≈ 1100 km in the corona).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Using the AIA 171 and 193 Å channels, Raouafi & Stenborg (2014) detected small jets (“jetlets") at the footpoint of coronal plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' More recently, Chitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) characterised the statistical proper- ties of small EUV bursts detected in AIA 171, 193 and 211 Å sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The Solar Orbiter mission (Müller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Zouganelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2020) carries, as part of the remote-sensing pay- load (Auchère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2020), the Extreme Ultraviolet Imager (EUI) suite (Rochus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The High-Resolution Imager (HRIEUV) and the Full Sun Imager (FSI) 174 channels are dom- inated by emission from lines of Fe ix and Fe x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' They image the plasma emission of the high TR and corona, which is the region of interest for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' At the closest, Solar Orbiter approaches the Sun down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='28 AU, allowing a two pixels spatial resolution of ≈ 200 km on the corona, along with a maximal cadence of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 s, thus provid- ing the highest spatial and temporal resolution images to date at these wavelengths, for extended periods of time and on a variety of targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' On May 30, 2020, when Solar Orbiter was at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='556 AU, HRIEUV made its first observation of the QS corona at high resolution (400 km) and cadence (5 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' During this 4 minutes sequence, 1467 small EUV brightenings of variable size (400 to 4000 km) and lifetime (10 to 200 s) have been detected and called "campfires" (Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The HRIEUV field of view was also visible by SDO/AIA and part of the events de- tected by HRIEUV were also visible in at least one of the AIA coronal bands, because of the lower spatial and temporal resolu- tions of AIA (about 1100 km and 12 s, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) used the AIA observations to infer their temperature applying the Differential Emission Measure (DEM) diagnostic method of Hannah & Kontar (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The resulting distribution was centered around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These features have yet to be better characterized, but the first investigations suggest that their origin is linked to photo- spheric magnetic cancellation (Panesar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021) or magnetic reconnection close to the TR or the chromospheric part of the loops (Kahil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Zhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) found that these EUV brightenings are low-lying (1 Mm to 5 Mm), which indi- cates that they could be chromospheric or transition region fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The authors noticed that the estimated heights of the fea- tures are larger than their apparent lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' If these events are loops, this implies that HRIEUV does not see their full extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Therefore, if they reach 1 MK, they do so only at their apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Winebarger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2013) used Hi-C and SDO/AIA data to estimate the temperature of small inter-moss loops to be about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 × 105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These had a projected length between about 5 and 7 Mm and their light curves, from the different AIA bands, peaked at the same time, suggesting the absence of cooling from coronal temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These loops are larger than the ones ob- served by Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) and Zhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021), furthermore they are observed in active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' However, it is possible that they share similar physical mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These results motivated our work to further investigate the thermal properties of the HRIEUV events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We perform a statisti- cal study over more than the 1000 detected events, and the rest of the QS used as a reference (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Our analysis is based on the time lag method (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3) applied to the AIA light curves from several pairs of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This method has been extensively used in active regions to study loops submitted to Thermal Non Equilibrium (TNE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Froment et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Froment et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2017, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Froment 2016), and to test the nanoflares the- ory (Viall & Klimchuk 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Viall & Klimchuk 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Viall & Klimchuk 2015, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The novelty of the present work relies on the application of this technique to QS region data and over short time lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4, we show that there is no or little sign of lag between all the chosen AIA bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The implications of these results will be discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Observations and data reduction On 2020 May 30, while the Solar Orbiter mission was still per- forming commissioning activities, HRIEUV observed a QS re- gion at 5 seconds cadence for 4 minutes, from 14:54:00 UT to 14:58:05 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The field of view of HRIEUV (blue square) is vis- ible in a full Sun image taken in the FSI 174 channel (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1 (c) shows the corresponding field of view on a full Sun image of AIA 171, as seen by SDO, which has a similar temperature response, peaking at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='9 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The different apparent position of the HRIEUV field of view between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1 (a) and (c) is caused by the separation angle, equal to 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 between the Solar Orbiter line of sight and the Sun-Earth line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Detection of the EUV brightenings by HRIEUV The HRIEUV data used for the present work1 was taken at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='556 AU from the Sun, resulting in a spatial resolution of ∼ 400 km in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In this sequence, Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) automatically detected and cataloged 1467 brightening events, nicknamed campfires, and called events from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The detection was performed after remapping the images on a 1 EUI Data Release 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='24414/wvj6-nm32 Article number, page 2 of 13 Dolliou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' : Temperature of EUI QS small scale brightenings: evidence for a cooler component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Solar-X [103 arcsec] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Solar-Y [103 arcsec] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='(a) FSI 00:55:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Intensity [DN/s] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='260 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='270 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='280 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Longitude [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Latitude [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='(b) HRIEUV 14:54:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Intensity [DN/s] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='500" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='500" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='500" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='500" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1000" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Solar-X [103 arcsec] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Solar-Y [103 arcsec] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='(c) AIA 171 14:57:45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Intensity [DN/s] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='260 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='270 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='280 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Longitude [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Latitude [°] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='(d) AIA 171 14:57:45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Intensity [DN/s] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Images captured on May 30, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Upper row : field of view observed by FSI 174 (a), and the first image of the HRIEUV sequence (b) in Carrington coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The FSI image is the closest available to the HRIEUV sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Lower row : AIA 171 image (c) and remapped on the same grid as HRIEUV (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Blue rectangles in the left column correspond to the field of view on the right column, and the blue dots in the right column are the positions of the 1467 detected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' regular 2400 × 2400 Carrington grid spanning from 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='9° to 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='9° in longitude and −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5° to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5° in latitude (correspond- ing to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='016 25° pitch, 198 km on the sphere) with a projection radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='004 R⊙ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The spacecraft jitter being docu- mented in the FITS headers, it is compensated by the Carrington remapping, and the absolute pointing values were determined by cross-correlation with AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The automated detection scheme (appendix B of Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021) defines the events as the pixels whose intensity is larger than an arbitrarily defined threshold of 5 times the lo- cal noise level, in the first two smaller scales of a spatial à trous wavelet transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Events overlapping between successive frames were merged to produce the final set of spatio-temporal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Their surfaces range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='04 Mm2 (the HRIEUV spatial resolution) to 5 Mm2, the upper limit being partly a consequence of the chosen maximum wavelet scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' No restriction was im- posed on their duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We note that the number of detected events, as well as their properties (surface, lifetime), highly de- pends upon the detection parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For consistency, we used the (Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021) cataloged as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We however re- moved the events present in the first or last image of the HRIEUV observation, as their lifetime might not have been fully captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1 (b) shows the location of the 1314 selected events on the first HRIEUV image of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Multichannel observations with AIA A major limitation of HRIEUV is its single passband, which makes it impossible to derive information on the plasma temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For this purpose, we used data from 6 channels (94, 131, 171, 193, 211, and 335 Å) of the AIA instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We did not in- clude the 304 band because the He ii 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 nm spectral line is op- Article number, page 3 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' arxiv_version 105 106 107 Temperature [K] 10 28 10 27 10 26 10 25 10 24 Response [DN s ¹ px ¹ cm ] HRIEUV & AIA temperature response HRI 171 94 131 193 211 335 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' HRIEUV and AIA temperature response functions computed with CHIANTI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1 (Dere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Del Zanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021), assuming an electron number density ne = 109 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' tically thick and the interpretation of its intensity is not straight- forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The selected bands cover a wide range of plasma tem- peratures (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 MK to 8 MK, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2), but have only less than half the temporal resolution (12 s) of HRIEUV (5 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For our work, we need to take into account the lower spa- tial and temporal resolutions of AIA, compared with HRIEUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Therefore, small and short-lived events detected by HRIEUV can be unresolved when observed with AIA 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In addition, events might not be sufficiently bright in some of the AIA bands to be detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The HRIEUV and AIA images have been paired tak- ing into account the 229 s difference in light travel time to Solar Orbiter and to the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The AIA images have been remapped onto the same Carrington grid as the HRIEUV data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' On this common grid, the HRIEUV images are re-sampled with at least 1 grid point per pixel, and the AIA images with at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Method In order to characterize the evolution of the thermal structure of these events, we used the time lags method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Because the AIA bands peak at different temperatures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2), time lags between them are a signature of plasma cooling (or heating) over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For example, the response functions of the AIA 193 and 171 bands peak respectively at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 MK and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='9 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The intensity in the 171 band peaking after the 193 one can be interpreted as a hot plasma cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The opposite behavior, can be a signature of plasma heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We discuss the various possible scenarios in detail in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We describe below the computation and classification of the AIA light curves (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1), and the computation of the time lags (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The analysis is performed pixel-by-pixel to take into account the spatial and the temporal information contained in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Several events are spatially resolved in the AIA data, so that the thermal behavior in individual pixels of each event will be independently characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This avoids the assumption that the event has no thermal sub-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This method may involve the use of low SNR for some of the pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This could be avoided by performing the analysis over the integrated intensity from the whole spatial extension of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' However, the latter choice would impose the above-mentioned assumption, which we pre- fer to avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We verified in Appendix B that a same time lag analysis, performed over whole events, yields the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In the following, we call "background" the total of back- ground and foreground emission superimposed on the events along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Background emission can represent a large fraction of the total emission (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3) and has the same properties as the QS emission observed outside events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Since we want to measure the time lags of the events themselves, it is nec- essary to check the influence of the background (as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The background intensity is estimated for each pixel and time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Light curves For our analysis, we classify the pixels in two categories: the "event" pixels, that is those containing at least one event from Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) during the sequence, and the "non- event" pixels, which we call Quiet Sun (QS) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The QS pixels are used as a reference, and their statistics will be com- pared to that of the event pixels (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' While the AIA and HRIEUV data have been re-projected to the same Carrington grid, the location of each event can be dif- ferent in the two data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Indeed, the separation angle between the two vantage points induces a parallax shift for those events located above or below the projection sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The contour of each event detected in HRIEUV was shifted by the amount mea- sured by Zhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) to obtain the corresponding con- tour in AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In the case of spatially overlapping events, this can cause the classification mask (the union of the contours at each time step) to have a different shape in AIA than in HRIEUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This is the case for the area shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3 in which two successive events, peaking at 14:54:30 UT and 14:55:04 UT, are overlap- ping and do not have the same height, and thus not the same parallax shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We estimate the background emission at each pixel using the open-cv implementation of the inpainting method of Bertalmio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This method estimates the intensity inside the mask, by matching the intensity and intensity gradients at its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This operation is performed at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' When- ever this background subtraction is applied to the analysis, it will be mentioned explicitly in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Figure 3 (b) shows an example of the result from this treat- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We have selected a pixel inside the mask (pixel 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3 (a)) and we plotted the light curves as measured in the HRIEUV and AIA channels (dots), together with their calculated back- ground emission (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For display purposes, original and background subtracted light-curves are normalized to the stan- dard deviation of the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' To plot all the curves on the same panel, we separated vertically the curves from a given channel by an arbitrary value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The error bars are the root mean square of the photon shot noise (as computed in Appendix A) and read noise components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Boerner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2012) provides the read noise for all the AIA bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For HRIEUV, it is estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In this figure, the light curves of all channels but AIA 94 and 335 have a similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In the AIA 94 and 335 channels, the event in pixel 1 is not detected above the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The absence of signal in these two bands is caused by their low response (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2) and is common for most of the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Figure 3 (c) shows, for a comparison, the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3 (b) for a representative QS pixel (pixel 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We see ap- parently uncorrelated fluctuations of the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Article number, page 4 of 13 Dolliou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' : Temperature of EUI QS small scale brightenings: evidence for a cooler component 0 2 4 Time [min] 0 5 10 15 20 25 30 35 (c) Pixel 2 (quiet Sun) 0 2 4 Time [min] 5 0 5 10 15 20 25 30 Intensity [normalised] (b) Pixel 1 (event) HRI-EUV AIA : 171 193 211 131 94 335 1000 km 1 2 (a) HRIEUV(averaged over 4 minutes) 2 0 2 Time offset [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 (f) Pixel 2 (quiet Sun) 2 0 2 Time offset [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Cross-correlation (e) Pixel 1 (event) 193-171 211-171 211-131 335-171 94-171 1 2 1000 km (d) AIA 171 (averaged over 4 minutes) 350 400 450 500 550 600 650 700 Intensity [DN/s] 80 90 100 110 120 130 Intensity [DN/s] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Images of HRIEUV (a) (14:54:00 UT to 14:58:05 UT) and AIA 171 ˚A (d) (14:57:45 UT to 15:01:57 UT) averaged in time over their respective sequence on 2020 May 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Both images are centered around Carrington coordinates (275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='00, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='07)°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The white contours represent the masks that isolate the event pixels from the QS ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Pixels 1 and 2 are selected, respectively, as example for event pixel and QS pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (b) Light curves in pixel 1 for HRIEUV and the AIA channels original data (dots) and background data estimated with "inpainting" algorithm (solid curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For each channel, both curves are normalized over the standard deviation over time of the original data (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (c) light curves in pixel 2 for the same channels of (b), normalized to their standard deviation over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Different couples are separated by an arbitrary value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The error bars in sub-figures (b) and (c) are computed from the shot and read noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (e) and (f) show the cross-correlation as a function of the time offset between the AIA light curves for, respectively, pixels 1 (b) and 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Time lags In the following, we describe the computation of time lags be- tween couples of AIA light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The time lags are defined as the temporal offset between the two light curves that yields the maximum Pearson’s cross-correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' By design, the six channels’ images are not co-temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For this reason, we resample the light curves on the timeline of the 171 band using linear interpolation before applying the cross- correlation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The latter is performed on a range of tem- poral offsets of ±2 minutes, with 12 s steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' A finer estimate of the time lag is obtained by parabolic interpolation around the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Figures 3 (e) and (f) show the results of this analysis for the AIA pixels 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We plot the values of the correlation as a function of the time offset in time applied between the two light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For the event pixel, we chose three couples with the high SNR (193 – 171, 211 – 171, 211 – 131).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' They have a strong correlation peak at near-zero offsets: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 s for 193-171, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 s for 211 – 171, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 s for 211 – 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The other two curves (335 – 171 and 94 – 171) involve low SNR bands, and have a maximum of correlation at a time offset different from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' They are positive for 335 – 171 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 s) and neg- ative for 94 – 171 (−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 s), with a maximum correlation below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The SNR is low in the 335 and 94 bands and the peak cor- relation is of the order of that found in the QS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3 (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We will discuss the significance the cross-correlations involving low SNR bands in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Figure 3 (f) shows the results for the selected QS pixel: clearly there is no strong correlation at any time offset and for any pair of AIA channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Figure 4 displays the maps of AIA intensity averaged over the sequence (upper row), time lag (middle row) and maximum Article number, page 5 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' arxiv_version AIA 193 AIA 211 AIA 131 AIA 335 AIA 94 30 40 50 Intensity [DN/s] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 Intensity [DN/s] 2 3 4 Intensity [DN/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 Intensity [DN/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 Intensity [DN/s] 193-171 211-171 211-131 335-171 94-171 193-171 211-171 211-131 335-171 94-171 2 1 0 1 2 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Top row: intensity maps for five AIA bands (averaged over the temporal sequence) showing the event of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3 (a) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The "event" region is identified by the black contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Middle and bottom rows: time lag and associated maximum correlation maps for five couples of AIA bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These are the result of the pixel-by-pixel cross-correlation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The maximum correlations of the events decreases as the intensities of the involved AIA channels decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' of cross-correlation (lower row) for the area shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We notice that the emission is not co-spatial in all bands: The in- tensity maps show a displacement of emission peak for AIA 211 and 94 (even though the signal is very low for AIA 94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Since the AIA channels are all co-aligned, this could be due to the thermal structure of the observed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' These observations show the importance of analyzing the plasma evolution pixel by pixel, as opposed to averaging the intensity over the event surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' While doing the latter might increase the SNR, it will mix light curves of regions at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The bands in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4 are ordered by decreas- ing mean intensity and thus decreasing SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In the bottom row, within the mask we see correspondingly decreasing correlation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Higher correlation values are associated with spatially coherent near-zero time lags, whereas lower correlations show an apparently random distribution of the time lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Results In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1 we present the statistical analysis over the whole field of view of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 discusses the effect of the SNR on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 estimates the effect of the back- ground on the time lag analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Zero time lags For the event pixels, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5 displays the time lag and the maxi- mum correlation 2D histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We choose nine representative AIA couples, covering a wide range of temperature sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Here, the estimated background has been subtracted from the event pixel intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The green dashed lines are the 80, 90, and 95% confidence levels, as computed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The counts above the 95 % level are at most 5 % likely to occur by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For most of the couples, a significant number of pixels is centered about short time lags (below the twelve seconds cadence), and are above the 95 % confidence level in cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This part of the distribution is therefore statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' On the contrary, 94 – 335 shows no significant pixel counts above the 95 % con- fidence level, which matches the contour of the 2D histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Given that these bands are largely affected by noise, this vali- Article number, page 6 of 13 Dolliou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' : Temperature of EUI QS small scale brightenings: evidence for a cooler component 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 s 193-171 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='7 s 211-171 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 s 193-131 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 s 171-131 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 s 335-171 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 s 211-131 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 s 94-171 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 s 335-211 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 s 94-335 100 101 102 Event pixel counts Confidence levels: 80 % 90 % 95 % Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2D histograms (shades of red) of time lags and maximum correlations for nine couples of AIA channels, for the 4451 event pixels of the HRIEUV field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The estimated background has been subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The green dashed lines are the confidence levels, derived in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The ν95 parameter quantifies the asymmetry of the time lag distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' It is the average of the event time lags above the 95 % confidence level, weighted by their respective maximum correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' dates a posteriori the principle of computing confidence levels from uncorrelated light-curves (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' While the time lags are near zero, the distributions are slightly asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This can be quantified by the parameter ν95, which represents the average of the time lag values above the 95 % confidence level, weighted by their maximum correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Apart from the 335 – 211 couple, all the asymmetries are below the exposure time of 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For 335 – 211, the positive asymmetry is above the exposure time but below the temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Influence of the signal level The main panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 6 display the 2D histograms of the av- erage intensity over the time sequence versus the maximum cor- relations for the two AIA couples: 193 – 171 (left column, high Article number, page 7 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' arxiv_version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 50 100 150 200 250 300 350 AIA 171 intensity [DN/s] AIA 171 intensity [DN/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1 193 - 171 (a) 94 - 171 (b) Event pixels QS pixels 0.' metadata={'source': 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+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='00 AIA 94 intensity [DN/s] 100 101 102 Event pixel counts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation (193 - 171) Maximum correlation (193 - 171) 50 100 150 200 250 AIA 193 intensity [DN/s] (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Main panels: histograms of the time-averaged intensity, as a function of the maximum cross-correlation, in the whole HRIEUV field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The left and right columns show the results for, respectively, the 193-171 and the 94-171 couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The 2D orange histograms are the counts of event pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The 2D red and blue contours correspond to the [20, 40, 60, 80] percentiles of the events and the QS pixels distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The margin histograms are normalised by their total number of counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The right margin histogram of (a) is not displayed, as it is a repetition of the one of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Similarly, top margin histograms of (c) and (d) are respectively the ones of (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' – high SNR) and 94 – 171 (right column, low – high SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The bottom (respectively top) row displays the intensity of the first (respectively second) band of the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The orange distributions and red contours refer to the event pixels, and the blue contours to the QS pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For the 193 – 171 couple (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 6 (a) and (c)), the event pixels distribution shows a wide range of possible cor- relation values, as opposed to the QS pixels one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The latter is more compact, and centered around lower maximum correlation and intensity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' On the contrary, the 171 – 94 case shows both events and QS pixels histograms sharing a similar compact shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This is mostly due to the lower intensity, and thus lower SNR, in the 94 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The intensity distributions, which are displayed in the right margin histograms of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 6, peak at higher values for the event pixels than for the QS, for every channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This implies that, on average, the HRIEUV events are also visible in the AIA channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The most significant difference between the two AIA couples shown in the figure is their maximum correlation distributions, displayed in the top margin histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Indeed, while the event pixels distribution peaks at higher correlation values than the QS ones for 193 – 171, both distributions share a similar shape for 94 – 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' As shown in the intensity distributions of the right margin histograms, the signal in 94 band is much lower than in the other two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Given an exposure time of 3s, the 94 band inten- sity distributions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 6 (d)) are close to the read noise value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='14 DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The SNR of the median intensity over the QS in the field of view is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='7, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='7 for the 171, 193 and 94 bands respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Thus in the 94 band, the noise dominates and the events, if present in the band, remain undetected for most of the cases (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3, b as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This is why, in the 94 – 171 case, the maximum correlation distributions of the events and the QS pixels share the same statistical behavior: most of the signal in this band originates from the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5, it explains the low number of significant time lags for the couples 94 – 171 and 94 – 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Article number, page 8 of 13 Dolliou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' : Temperature of EUI QS small scale brightenings: evidence for a cooler component 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Influence of the background subtraction According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 6, the AIA 171 intensities distribution of the events peaks only about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 times higher values than the QS ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Therefore, the background largely contributes to the over- all signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This is why it is necessary to evaluate its influence on the cross-correlations, which we illustrate using the couple 193– 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Figure 7 displays the time lag and the maximum correla- tion distributions without (left) and with (right) the subtraction of the estimated background component (as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The 2D histogram of event pixels is represented in shades of red, while the distribution of QS pixels is visualized by blue contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In the margins, the 1D event pixels distributions are displayed in red and the QS ones in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The green histograms correspond to the uncorrelated light curves used to compute the confidence levels (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5, the events distributions peak at high correlation values, and are concentrated around short time lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The impact of the background intensity on the events distributions is visi- ble when comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 7 (a) with (b): the time-lags and their asymmetries are mostly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' However, when removing the background, the distribution of the event pixels is flattened (most visible in the margin his- togram) and the counts are redistributed in the low correlation, random time lag wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This has two causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' First, the noise from the QS is propagated to the background by the inpaint- ing (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1), and in turn to the background-subtracted light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Thus, the correlations are lowered in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Second, the QS signal is partly correlated around zero time lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This forms the high-correlation tail visible in the blue contours of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Subtracting the background removes this correlated signal, which also lowers the correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' To conclude, remov- ing the background isolates the contribution of the events to the time lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Thus, the time lags in Figures 7 (b) and 5 are a prop- erty of the events and not of the QS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Discussion In this work, we have presented the results from the statisti- cal analysis of the time lags measured in the AIA data for the small scales EUV brightening (the "campfires") cataloged by Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This catalog has the unique property of collecting the tiniest and most rapid brightening ever observed, which are the manifestation of physical processes probably al- ready known, but now observed over shorter temporal and spa- tial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For this reason, we preferred to use the general name of "EUV events".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Our observational work points to the following result: the events are characterized by short time lags (within ±12 s) and high correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We verified that these results are statistically significant, and are not caused by background variations alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In comparison, the QS mostly exhibits random time lags with lower correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' It is possible that the timescales of thermal changes between events and the surrounding areas are different, the latter being much longer than the maximum time lags con- sidered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' To our knowledge, this is the first time that the time lags as- sociated with small scale EUV brightenings and their surround- ings have been statistically characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Earlier works, as men- tioned in the introduction, and which used this technique, re- ported zero time lags in the QS surrounding active region loops, without taking into account the possible presence of small scale brightenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Concerning the interpretation of the short time lags, there are three possible scenarios that can be raised, and which we are going to discuss in the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The observed events do not reach the peak temperatures of the response function (∼ one million degree);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The observed events reach coronal temperatures (> 1 MK) but their fast cooling, their sub-pixel multithermal structure, and the width of the AIA response function, prevent us from detecting significant time lags;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The observed events are the transition region (∼ 1MK) emis- sion of long and hot (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='e ∼ 10 – 100 Mm, ∼ 3 MK ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Reale 2014) loops, which are heated impulsively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Let us start with the interpretation given by the scenario 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Look- ing at the most intense bands of AIA (Figure 2), we understand that a time lag zero arises when the plasma temperature does not reach the peak of the 171 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' At the temperatures below this peak, all the bands behave similarly, and so do the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Furthermore, the observational properties (low-lying, short time lags) of these events resemble what is observed by Winebarger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2013) for the "Hi-C loops" (Te ∼ 105 K and ne ∼ 1010 cm−3) in the inter moss loops areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Their time lag analysis on the AIA light curves also displayed near-zero time lags, which brought them to conclude that the loops did not reach one million degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Their interpretation was the observa- tion of impulsively, low-energy (nanoflares) heated loops which cool rapidly due to their small length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Given the similarities of the HRI brightenings to these events, we suggest that they may have a similar physical origin, being the result of an impulsive heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For such cold events to be visible in the AIA bands and in HRIEUV, they should be quite dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We did a first order estima- tion of their density, using an average value of the background- subtracted event intensity on AIA 171 and assuming an isother- mal plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We obtained ne ∼ 109 cm−3 for Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 × 106 K and ne ∼ 1010 cm−3 for Te = 3 × 105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The latter supports the result of Winebarger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' However, we must consider possible differences between the Hi-C loops and our HRIEUV events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' First, as mentioned, the ob- served solar region is not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' But small low-lying cool loops (Te ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='5 MK) are observed in the QS (Hansteen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2014), and are ubiquitous along the supergranular cell bound- aries in the solar upper atmosphere (see for instance, Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Sánchez Almeida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2007, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' And since there is no distinction between supergranular cells in QS and AR, we expect to observe similar events in both regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021) showed with HRILyα observations that the HRIEUV events are organised mostly around the supergranu- lar network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Another difference between the Hi-C and the HRIEUV events are their estimated temperature, around Te ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='06 MK for the first case (Winebarger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2013) and around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1 MK for the latter case (Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Again, if we are looking at similar events in the two cases, we suggest that such discrepancy may be due to the uncertainties in the data, the in- version methods and associated assumptions applied to relatively broad band instruments, as for these imagers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Indeed, the mea- surement of the temperature of these events is very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For instance, Schonfeld & Klimchuk (2020) showed that often the cool plasma emission dominates the bands, even though the hot plasma is there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Let us now assume that we are in the scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' A time lag close to zero for AIA bands has been predicted in the TR emis- sion of active region coronal loops heated by nanoflares (Viall & Klimchuk 2015, see also references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' They showed Article number, page 9 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' arxiv_version 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3s Event pixels QS pixels Random | Confidence levels: 80 % 90 % 95 % 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 (a) 193 - 171 (background not subtracted) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 (b) 193 - 171 (background subtracted) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='25 100 101 102 Event pixel counts Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Margin and 2D histograms of time lags and associated maximum correlation values for the couple 193-171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Sub-figure (a) in red shows the original distribution for the event pixels, sub-figure (b) shows the background subtracted event pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The blue contours in the central panel of sub-figure (a) are the [20, 40, 60, 80] % percentiles of the QS pixels distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The green colors in the main panels are the confidence levels, and the distributions of the light curves used to compute them is plotted with the same color in the margin histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The margin histograms are normalised by their total number of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The parameter ν>95 is defined as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' that the combination of the multi-temperature sensitivity of the AIA bands, combined with the almost constant pressure prop- erty of the TR and its variable extension along the loop during the heating-cooling phases, result in a narrower time lag with respect to the coronal emission part of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' To be empha- sized here that the TR of a loop is defined as the region where the thermal conduction acts as a plasma coolant, contrary to the coronal region where it acts as a heater (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Klimchuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' While the presence in the simulation of short time lags for all the AIA couples corroborates our results, the loops mod- eled by Viall & Klimchuk (2015) are much longer than what we are dealing with here (L ≈ 30 – 50 Mm, with respect to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 – 4 Mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Moreover, in those simulations, a clear different signature in the time lag exists between TR and coronal emission, while this is not visible in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This could be possibly explained by the short cooling time from coronal temperatures of one of these tiny loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For instance, for the shorter loops (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 Mm) detected by HRIEUV at a temperature of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 MK and density of ne = 1010cm−3 the cooling time is about 14s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' It is possible that our time lag method is not sensitive enough, due to the AIA cadence of 12 s, to detect both TR and coronal emission populations of short time lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We propose to investi- gate further this aspect in the future through numerical simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The small asymmetries we have in our time lag distribu- tions are below the cadence of our observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We would need a higher temporal resolution data to corroborate such result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The cadence should be at last similar to the one of HRIEUV, where the emission variation of the event is better captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' At present, we verified that the measured time lags are independent of the event’s duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Concerning the scenario 3, if such large loops exist in the QS, they remain undetected by the AIA channels, meaning that they would have a very low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Without independent evidence that this is the case, we exclude for now this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' In conclusion, in the picture of impulsive heating phenomena acting in the QS region, and considering the wide temperature response of the AIA bands, our results appear also to be con- sistent with predominantly fast cooling plasma from more than 1 MK, that is satisfying our scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Consistently with this picture are the results from a 3D MHD simulations using MU- RaM code by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Here magnetic reconnections in the coronal part of small QS loops produced events with prop- erties similar to what observed in HRIEUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' They noticed that the simulated HRIEUV emission only showed the apex of the heated loop, where the lower density allows the available stored energy to heat the plasma up to ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 MK, even though some hotter temperatures could also be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' To summarize, our results are consistent with two possible scenarios: either the events do not reach coronal temperatures, or they do, but they cool faster than the AIA temporal resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' It is possible that the two scenarios coexist, as the HRIEUV catalog does not separate events produced by different physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The AIA cadence and the multithermal nature of the bands do not allow separating the emissions from the possible cool and hot plasma along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' To solve the ambiguity on the temperature, we need to use spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This has been done recently by using the Spectral Imaging of the Coronal Environement (SPICE) instru- ment on board Solar Orbiter (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' submitted to this is- sue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' They investigated a few HRIEUV events and came to the conclusion that the studied events do not show significant emis- sion at temperatures higher than that of Ne viii (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='63 MK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Although such spectroscopic analysis needs to be extended to a larger sample to better quantify the fraction of events not Article number, page 10 of 13 Dolliou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' : Temperature of EUI QS small scale brightenings: evidence for a cooler component reaching high temperatures, we find it to support our conclusion that quiet Sun small-scale EUI brightenings are in most cases largely dominated by cool emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Further investigations are needed to confirm this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For these reasons, we plan to extend our methodology to forward modeling constrained by spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The authors gratefully thank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Klimchuk for the fruit- ful discussions and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' acknowledges the funding by CNES and EDOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' acknowledges the funding by CNES through the MEDOC data and operations center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' was supported by a CNES postdoctoral allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' are grateful to the Science Technology and Facilities Council for the award of Ernest Rutherford Fellowships (ST/R004285/2 and ST/R003246/1, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The ROB team thanks the Belgian Federal Science Policy Of- fice (BELSPO) for the provision of financial support in the framework of the PRODEX Programme of the European Space Agency (ESA) under contract numbers 4000134474 and 4000136424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This paper uses the Solar Orbiter/EUI data release 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='24414/WVJ6-NM32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Solar Orbiter is a space mission of international collaboration between ESA and NASA, oper- ated by ESA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The EUI instrument was built by CSL, IAS, MPS, MSSL/UCL, PMOD/WRC, ROB, LCF/IO with funding from the Belgian Federal Science Pol- icy Office (BELSPO/PRODEX PEA 4000134088, 4000112292, 4000117262, and 400013447);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' the Centre National d’Etudes Spatiales (CNES);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' the UK Space Agency (UKSA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' the Bundesministerium für Wirtschaft und Energie (BMWi) through the Deutsches Zentrum für Luft- und Raumfahrt (DLR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' and the Swiss Space Office (SSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This work used data provided by the MEDOC data and op- erations centre (CNES / CNRS / Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Paris-Saclay), http://medoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='ias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='u-psud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='fr/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This research used version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2021) of the aiapy open source software package (Barnes et al.' metadata={'source': 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+page_content=', De Groof, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=', Walsh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2020, A&A, 642, A3 Article number, page 11 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' arxiv_version Appendix A: Computation of the confidence levels The cross-correlation of two uncorrelated random time series has a nonzero probability of resulting in a time lag with a nonzero value for the maximum correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This is why the interpreta- tion of our time lag results is challenging, especially for couples involving low to medium SNR AIA channels, such as 131, 94 and 335 which are noise dominated in several pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For our goal we adopted a Monte-Carlo approach inspired by Max-Moerbeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We computed the time lags (corre- sponding to the maximum cross correlation) between many un- correlated simulated AIA light curves to estimate the probability of chance occurrence of each time lag value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The simulated light curves are built using the observational results that the coronal emission has a temporal Power Spectral Density (PSD) that can be modeled by a power law (Auchère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Gupta 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Threlfall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Specifically, for the QS, Ireland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' (2014) fitted the exponents n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='01 for AIA 171 and n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='01 for AIA 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' To keep the empirical model simple, we adopted a power law with exponent n = 2 for all the AIA channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' From this PSD, we generate 105 random light curves of 4 min length and 12 s cadence using the method described in (Timmer & Koenig 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The obtained time series ˆI(t) (in arbitrary units) are converted into Digital Number (DN) as the follow: I(t) [DN] = �ˆI(t) − µˆI � σDN σˆI + µDN (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1) where µˆI and σˆI are, respectively, the mean and standard de- viation of ˆI(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' µDN and σDN are the spatial mean intensity and the standard deviation derived from the first image of the AIA sequence (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 1 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Photon noise is then added by picking random values from a Poisson distribution peaking at the average photons per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' We assume it to be equal to the incident photons I(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Negative in- tensity values are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Next, we simulate the regular AIA acquisition chain by re-converting the time series into DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Read noise is then added, in the form of a normal distribution of mean zero and standard deviation σRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Using the inverse of the camera gain, I(t) is converted into photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' All the conversion constants are taken from the initial AIA calibration (Boerner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The resulting time series are now used for the time lag anal- ysis applied to each of the AIA couples used in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The time lags and maximum correlation distributions of these random light curves are displayed for the couple 193-171 in the margin histograms of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The confidence levels are defined as the [80 %, 90 %, 95 %] percentiles of the maximum correlation distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' They are displayed as green dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' According to our simulation, counts above the 95 % confidence level are at most 5 % likely to be caused by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Appendix B: Event-based time lag analysis The main work we have presented is based on the single pixel analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Here we summarize the results from the full-event in- vestigation in order to verify if the resulting thermal behavior reflects the one deduced with the single-pixel analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Both the pixel-based and the full-event approaches have their advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The full-event approach increases the SNR of the light curves, as it is represented by the averaged intensity over the selected event area, but does not separate the "cold" and the "hot" pixels populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' This is because inside an "event sur- face", one pixel might reach a higher temperature compared with the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The high temperature pixel and the lower tempera- ture ones appear as separate counts in the resulting figures of the pixel-based approach (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' On the contrary, the temper- ature associated with the average intensity will be something in between the hottest and cooler pixels, so reducing the tempera- ture excursion over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Each event area is a single count in the statistical analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' To build the single-event light curves, we proceeded by spa- tially averaging the light curves within each event mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The time lag analysis is then applied to these new time sequences in the same way as it was done for the pixel-based approach (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The results of the analysis are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The time lags are centered around short values (>12s), above the 95 % confidence levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' There is no noticeable difference with the pixel-based approach (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5), apart from small variations in the asymmetries ν95, which remains close to the exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The variations are probably caused by the lower number of counts above the 95 % confidence level, compared with the pixel-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' It decreases the statistical significance of the asymme- try, and the events should be studied individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Article number, page 12 of 13 Dolliou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' : Temperature of EUI QS small scale brightenings: evidence for a cooler component 100 101 Event pixel counts 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 s 193-171 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='3 s 211-171 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1 s 193-131 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 s 171-131 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 s 335-171 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1 s 211-131 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 s 94-171 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 s 335-211 1 0 1 Time lag [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='0 Maximum correlation 95 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='7 s 94-335 Confidence levels: 80 % 90 % 95 % Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 5, but with a full-event approach, as opposed to a pixel-based one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' For every 1314 events, the light curves are spatially averaged over each of their respective event surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Then, time lag extraction is performed similarly as the pixel-based approach (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' The estimated background has been previously subtracted on event pixels with the "inpainting" algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} +page_content=' Article number, page 13 of 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtA0T4oBgHgl3EQfF_9p/content/2301.02040v1.pdf'} diff --git a/ZtE3T4oBgHgl3EQfcwpV/content/tmp_files/2301.04528v1.pdf.txt b/ZtE3T4oBgHgl3EQfcwpV/content/tmp_files/2301.04528v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e594fc6c39e765b8a68d89b6b70a41efa94166b5 --- /dev/null +++ b/ZtE3T4oBgHgl3EQfcwpV/content/tmp_files/2301.04528v1.pdf.txt @@ -0,0 +1,1933 @@ +The Role of Interactive Visualization in Explaining (Large) NLP Models: +from Data to Inference +Richard Brath +Uncharted Software Inc +rbrath@uncharted.software +Daniel Keim +University of Konstanz +Johannes Knittel +University of Stuttgart +Shimei Pan +University of Maryland, BC +Pia Sommerauer +Vrije Universiteit Amsterdam +Hendrik Strobelt +IBM Research Cambridge +hendrik.strobelt@ibm.com +Abstract +With a constant increase of learned parame- +ters, modern neural language models become +increasingly more powerful. Yet, explaining +these complex model’s behavior remains a +widely unsolved problem. In this paper, we +discuss the role interactive visualization can +play in explaining NLP models (XNLP). We +motivate the use of visualization in relation to +target users and common NLP pipelines. We +also present several use cases to provide con- +crete examples on XNLP with visualization. +Finally, we point out an extensive list of re- +search opportunities in this field. +1 +Motivation +In recent years, NLP systems powered by very large +neural network models, such as BERT and GPT- +3, have provided an unprecedented performance. +The latest models have billions of parameters and +need enormous amount of data and computing re- +sources for training. While results in general are +of high quality, there are numerous applications +where explainability is of high importance, such as +medical diagnosis or bias detection and mitigation, +e.g., (Zhang et al., 2021; Stevens et al., 2020). +In this position paper, we examine the role of +interactive visualization in explaining (large) NLP +models. The reflections and proposals in this work +are the result of intensive discussions and close +collaboration of experts in NLP and visualization. +We first distinguish different user groups with +varied technical and domain expertise. Each user +group has different explainability needs, which may +guide the design of interactive visual tools. Next, +we discuss the use of visualizations in explaining +typical NLP pipelines, especially those employ- +ing pre-trained large language models (LLM), with +questions ranging from when to use visualizations +and why, which visualizations to use and how to +use them. It is important to note that visualizations +can be used in very different ways and for very dif- +ferent purposes, and probably in even more ways +than they have been used in the past, e.g. (Belinkov +and Glass, 2019; Danilevsky et al., 2020). However, +there are also pitfalls such as a misunderstanding +of what can be inferred from visualizations. To +support our arguments, we include a few use cases +ranging from identifying social bias in NLP mod- +els, acquiring linguistic insight, debugging com- +plex models to labeling ground truth in the main +work and the appendix. Finally, we present an out- +look on research opportunities that may arise in +the same context. They cover all phases of model +development starting from visualizing the data and +their properties over the different stages of model +development to the evaluation and interpretation of +the models. +2 +User Groups for XNLP Visualization +Visualization methods for XNLP should enable +users with different expertise, to solve specific +tasks. As shown in Figure 1, domain experts may +have high expertise in a task and text corpus, but +may not be experienced in language models and +may use these models as blackboxes. Model archi- +tects and builders may have high degree of knowl- +edge on advanced modeling techniques but might +not be experts in the application domain. General +users, such as a casual user of Google translate, +may have low knowledge of both NLP models and +application domain. Data scientists and analysts +may be in the middle, as users of general analytical +and NLP toolkits, in order to perform analytical +tasks on some datasets of interest. +Explainability of NLP models through visualiza- +tion can help across all user types, although each +may have differing explainability needs. A model +builder may be more interested in locating bugs and +understanding model performance which may re- +quire fine-grain visualization of the model structure +and parameters. Domain experts may have a criti- +arXiv:2301.04528v1 [cs.CL] 11 Jan 2023 + +INPUT/TRAINING DATA +MODEL +ASSESS +TRAIN LARGE +LANGUAGE +MODEL +FINE-TUNING +MODEL +INFERENCE +OR +PROMPT +Raw +Data +Prepared +Data +Model +Evaluate +Challenge +Raw +Data +Annotated +Data +Model +Evaluate +Challenge +Input +(& Steering) +Model +Output +Prompt +• Observe artifacts +• De-bias training data +• Monitor +training +• Changes over +iterations +• Understand +linguistic +knowledge +• Inspect +quality +• Discover +boundaries +• Assess +updates +à +à +• Audit, +inspect +• Bias, +active +learning +• Find edge +cases +• Review +annotations +• Prompt +engineering +• User-based manipulation +to understand model +sensitivity +• visual +comparison +• relate back to +training data +EXPLORE +TRAINING SPACE +EXPLAIN +MODEL +ASSESS +OUTPUT +(A) MODULAR +NLP PIPELINE +(B) END-TO-END +MODELS +(C) LARGE LANGUAGE +MODELS +(c1) Fine-Tuned +Model +End-to-End +NLP model +Task (classify, Q&A, translation, summarization,…) +models for POS, +NER, sentiment, ... +Large Language +Model +(c2) Few Shot +Learning & +Prompting +LEGEND +Black Box Model +Other algorithms / models +a +b +c2 +c1 +REVISED 20220713 +MODEL +KNOWLEDGE +DATA KNOWLEDGE +general +user +domain +expert +model +architect +data +scientist +Figure 1: User types +cal need to understand how concepts are encoded +in the model and require visualization of what con- +cepts or linguistic units the model attends to. Gen- +eral users may wish to understand if the model is +biased and may desire to gain some understanding +between the training data and a biased output from +the model. Model architects and builders should +understand their models and their implications in +downstream applications +3 +NLP Models and Interpretability +Black-box neural network models are important +building blocks in state-of-the-art NLP pipelines as +depicted in Figure 2. Classical pipeline approaches +still have their place in small-data scenarios, which +are common in interaction systems, e.g. when in- +teractively selecting document subsets, NLP tech- +niques such as NER, POS, LDA, can provide on- +demand descriptive statistics, which in turn can be +visualized to characterize the selected subset, in +comparison to the training corpus. Furthermore, +they can be used to combine several neural systems. +Another advantage is that the explicit inputs and +outputs of individual models may help to interpret +the combined system. We therefore include them +in our considerations. +A lot of research interest is currently directed +toward large language models (LLM) and how they +can be utilized to solve common NLP tasks. In +contrast to end-to-end models that are typically +trained on specific tasks, these LLMs are trained +in a self-supervised or semi-supervised fashion on +very large training data and use many trainable pa- +rameters. In mid-2022, the largest language model +reported has 540 billion parameters (Chowdhery +et al., 2022). One of the intriguing aspects of LLMs +is that they seem to capture factual knowledge to +some extent (Petroni et al., 2019), which makes +them very powerful. +Two main methods have been used recently to +apply LLMs for specific NLP tasks: fine-tuning +INPUT/TRAINING DATA +MODEL +ASSESS +TRAIN LARGE +LANGUAGE +MODEL +FINE-TUNING +MODEL +INFERENCE +OR +PROMPT +Raw +Data +Prepared +Data +Model +Evaluate +Challenge +Raw +Data +Annotated +Data +Model +Evaluate +Challenge +Input +(& Steering) +Model +Output +Prompt +• Observe artifacts +• De-bias training data +• Monitor +training +• Changes over +iterations +• Understand +linguistic +knowledge +• Inspect +quality +• Discover +boundaries +• Assess +updates +à +à +• Audit, +inspect +• Bias, +active +learning +• Find edge +cases +• Review +annotations +• Prompt +engineering +• User-based manipulation +to understand model +sensitivity +• visual +comparison +• relate back to +training data +EXPLORE +TRAINING SPACE +EXPLAIN +MODEL +ASSESS +OUTPUT +(A) MODULAR +NLP PIPELINE +(B) END-TO-END +MODELS +(C) LARGE LANGUAGE +MODELS +(c1) Fine-Tuned +Model +End-to-End +NLP model +Task (classify, Q&A, translation, summarization,…) +models for POS, +NER, sentiment, ... +Large Language +Model +(c2) Few Shot +Learning & +Prompting +LEGEND +Black Box Model +Other algorithms / models +a +b +c2 +c1 +REVISED 20220713 +MODEL +KNOWLEDGE +DATA KNOWLEDGE +general +user +domain +expert +model +architect +data +scientist +Figure 2: The role of neural network (NN) models in +NLP tasks. Four modes are commonly used: (a) using +separate NN models to solve intermediate tasks in mod- +ular NLP pipelines, (b) training end-to-end models on +NLP tasks, (c1) fine-tuning with task-specific ground +truth with text encoding from LLMs, and (c2) few-shot +learning and prompt engineering for ad-hoc NLP tasks. +task-specific models using text encodings gener- +ated by LLMs (Devlin et al., 2018) or employing +LLMs as few shot learners (Brown et al., 2020) and +adapting them to different NLP tasks using a few +examples or prompts. +LMs have been shown to have superior perfor- +mance on many complex tasks. At the same time, +they come with important limitations: (1) LLMs +require large amounts of data and computing power +and are often created in such a way that the train- +ing process and the training data are not openly +accessible. (2) The training data often consist of +large volumes of texts published online, which can +reflect harmful views and biases, which are then +propagated to downstream applications. (3) The +extremely large size of the models, together with +the size of the training data render XNLP a very +challenging problem. +From a XNLP perspective, visualization can be +applied to different tasks in different NLP work- +flows. In addition, both the scope of data and users’ +expectations can have a significant impact on the +design of a visualization. +4 +Visualization for XNLP +Visualization is broadly applicable across explain- +ing LLMs as indicated in Figure 3. Interactive +visualizations can be used to explore the training +data, the training processes, the resulting models, +or the model output (columns in Figure 3); and can +be used during model training, model adaptation +(e.g. fine-tuning), and model inference (rows in Fig- + +ure 3). The training data (first column) defines the +world view of the model, a first and important step +to better assess the performance of and issues in the +model (e.g. domain specificity, biases, and general- +izability). Simple visualizations like line charts are +often used to communicate the progression of high- +level measures (e.g. accuracy or loss), but more +advanced approaches can allow for more thorough +analyses such as whether the training is qualita- +tively improving w.r.t. the task. From a model +perspective (middle column), visualization can aid +model-builders building LLMs; downstream mod- +elers fine-tuning or prompt engineering LLMs; and +end-users visually analyzing the output of models +(right column). Inspecting output is a more shallow +yet model-agnostic way of exploring NLP models +that can lead to relevant insights. For instance, it +may help model builders and domain experts to +evaluate models more comprehensively instead of +just relying on a small set of computed metrics on +benchmark datasets (Ribeiro et al., 2020). +INPUT/TRAINING DATA +MODEL +ASSESS +TRAIN LARGE +LANGUAGE +MODEL +FINE-TUNING +MODEL +INFERENCE +OR +PROMPT +Raw +Data +Prepared +Data +Model +Evaluate +Challenge +Raw +Data +Annotated +Data +Model +Evaluate +Challenge +Input +(& Steering) +Model +Output +Prompt +• Observe artifacts +• De-bias training data +• Monitor +training +• Changes over +iterations +• Understand +linguistic +knowledge +• Inspect +quality +• Discover +boundaries +• Assess +updates +à +à +• Audit, +inspect +• Bias, +active +learning +• Find edge +cases +• Review +annotations +• Prompt +engineering +• User-based manipulation +to understand model +sensitivity +• visual +comparison +• relate back to +training data +EXPLORE +TRAINING SPACE +EXPLAIN +MODEL +ASSESS +OUTPUT +(A) MODULAR +NLP PIPELINE +(B) END-TO-END +MODELS +(C) LARGE LANGUAGE +MODELS +(c1) Fine-Tuned +Model +End-to-End +NLP model +Task (classify, Q&A, translation, summarization,…) +models for POS, +NER, sentiment, ... +Large Language +Model +(c2) Few Shot +Learning & +Prompting +LEGEND +Black Box Model +Other algorithms / models +a +b +c2 +c1 +REVISED 20220713 +MODEL +KNOWLEDGE +DATA KNOWLEDGE +general +user +domain +expert +model +architect +data +scientist +Figure 3: Areas for visualization in LLM’s in red. +4.1 +Why and When (Not) to Use +Visualization +Interactive visualizations are well suited to tackle +problems that are generally difficult to formulate +as a mathematical problem (Keim et al., 2010). +Exploring the inner workings of models with mil- +lions or even billions of parameters is a complex, +open-ended task that cannot be solved solely in an +automated way. Upon interacting with the visual- +izations, analysts may gain a better understanding +of the model or dataset at hand, and they may come +up with more formal hypotheses that can be investi- +gated later on. In addition, visualizations often play +a major role in communicating results and insights +to different audiences. +It is important to note that for some tasks, how- +ever, there is little benefit in using interactive visu- +alizations. For instance, visualizations rarely play +a role in confirming formal hypotheses quantita- +tively. For well-defined goals (e.g., text search), +automatic methods or simple visualizations such as +text highlighting are better suited. +4.2 +Visualization and Interaction +While the variety of potential visualizations is large, +some visualization techniques have shown to be +the workhorses across many tasks: +Simple charts are often used, such as line charts +for loss over iterations; or bar charts indicating the +probability of words (at a position in a sentence), +or for model performance or class prediction (e.g. +Figure 4b). +Markup and heatmaps are frequently applied +to (partly) explain model behavior. For instance, +we can visualize extracted rules that approximate +predictions or local behavior with heatmaps; or we +can highlight salient inputs that are relevant for the +model (e.g. Figure 4c). This can be applied to +narrative text, or word lists (e.g. one node (Dalvi +et al., 2019)). One of the advantages is that we can +zoom-out, such that only the markup and structure +remain, i.e., pixel-level visualization. +High-dimensional visualization, such as dimen- +sional reduction visualization (e.g. PCA, t-SNE) +or hierarchical clustering, may be used to visualize +thousands to millions of data points by plotting sim- +ilar data points in similar locations. They are often +used to visualize the distribution of embeddings +and hidden states, for instance, to better understand +whether semantically similar input leads to similar +internal representations (e.g. Figure 4a). +Graph visualization can be used to show net- +work relationships. While they are perfectly suited +to show relationships between a few nodes at a +local scale, it is challenging to scale them up to +global structures. In addition, they can be used a) +to visualize alternative inputs or outputs (e.g. Sen- + +tenTree (Hu et al., 2017) to view alternative outputs +from one point forwards); b) to visualize grammat- +ical structures, such as parse trees or co-referents; +(c) model internals like attention relations (e.g. Fig- +ure 4d). +Text visualization techniques beyond markup are +important for exploring the underlying training data +set but they can also play an important role in visu- +alizing the inner workings of a model, for instance, +by showing visual summaries of the layers or nodes +the model seems to focus on. Word clouds are popu- +lar but controversial (in the traditional layout, word +position, color, orientation are non-meaningful), al- +though there are improvements, e.g. (Hearst et al., +2019; Knittel et al., 2021). Other techniques exist +for visualizing text with text, e.g. (Kucher and Ker- +ren, 2015; Brath, 2020; Lang and Nacenta, 2022). +On top of an effective visual encoding, some +classes of interaction techniques are critical when +exploring large scale datasets and models. Selec- +tions and tooltips help to explore subsets and access +detailed data on demand, without cluttering the pri- +mary representation. Filters and facets allow to +reduce data to relevant subset, based on any criteria +or any selection. Zoom, pan, and rotate help to +move around massive plots, such as to zoom in to +focus on a region, or rotate 3D plots to reduce occlu- +sion. Aggregations allow to expand or collapse the +level of detail needed, e.g. keywords, phrases, top- +ics. Linked views and linked updates occur when +many visualization elements are combined in one +interface, with a selection in one visualization up- +dating all others, e.g. LIT (Tenney et al., 2020) +uses many of the above visualizations and linked +interactions in one combined system (Figure 4). +Figure 4: LIT, showing a) high dimensional visualiza- +tion of embeddings (top left), b) simple bar chart of +class probabilities (bottom left), c) salience heatmap +(bottom center), d) attention graph (bottom right). +Analysts may either explore the data top-down +by first using overview visualizations, which then +allows interactive drill-downs for more specific +analyses related to insights or hypotheses that they +have gained thus far; or, they may analyze or fil- +ter the data first and investigate a particular subset +of the data and/or model to expand their analysis +in a bottom-up approach. Furthermore, using an +iteration loop, analysts may steer the model inter- +actively to fit it to their needs, which allows for +incorporating domain knowledge into the model. +For large NLP models, the presented visualiza- +tion techniques might be challenged. See section 6 +for resulting research opportunities. +5 +Use Cases +So far, we generally discussed the role of interac- +tion and visualization. Here, we want to show a +selection of concrete instances of how visualization +can help and helped for XNLP tasks. Additional +use cases about labeling data and NLG model visu- +alizations can be found in Appendix A. +5.1 +Identifying and Assessing Social Biases +There are many recent discoveries of NLP systems +that exhibit various types of bias. For instance, +Google’s machine translation algorithms convert +the gender-neutral Turkish sentences O bir profesör. +O bir ö˘gretmen. to the English sentences He’s a +professor. She is a teacher. (Caliskan, 2021) To +avoid these issues, it is critical that we develop +effective tools to inspect and identify the biases in +both NLP data and models. +Visualizing Biases in NLP Data. One major +source of bias in NLP systems is human biases. +Since human-generated text are used to train NLP +models, biased training data often result in biased +NLP models. To uncover the source of biases in +NLP data, we can combine automated bias detec- +tion with visualization. With this method, we first +employ text classifiers to detect diverse types of so- +cial biases (e.g., racism, sexism, microaggressions +and hate speech) in text (e.g., social media posts). +We then employ interactive visualization to provide +an overview of the distribution of biased text.The +visualization in Figure 5 summarizes toxic Twitter +conversations as natural-looking trees, where toxic +Twitter conversations are represented as withered +branches (Beshai, 2018). The scale and distribution +of withered branches aids assessment of the degree +of toxicity on Twitter. Word association (e.g., pair- + + sst2-tiny + default +& Language Interpretability Tool + sst2-base + sst_dev + simple + Share +Select datapoint → +Color by +Compare datapoints +1 pair available < 图 > [0, 872] selected +(< primary: b4abfc ...[872] >) ☆ < 5 of 873 selected > +Clear selection + Select random +Embeddings +Data Table +Datapoint Editor +.3 +Projector: PCA +Hide unselected +Reset view +Select all +Columns +*sentence TextSegment +it 's a charming but rotten +Embedding: sst2-tiny:cls_emb +index Q +sentence Q +label Q +journey . +Label by: sentence → +it's a charming and often affecting journey . +0 +872 it's a charming but rotten journey +1 +0 +1 +unflinchingly bleak and desperate +it's a charming but ro1 + allows us to hope that nolan is poised to embark a +label CategoryLabel +1← +1 + major career as a commercial yet inventive +filmmaker +3 +the acting , costumes , music , cinematography +1 +and sound are all astounding given the production +'s austere locales +it's slow -- very , very slow. +0 +5 + although laced with humor and a few fanciful +touches , the film is a refreshingly serious look at +young women. +< Page 1 of 88 > >I 图 +Add and +Reset +Clear +Select 10 nearest neighbors +compare +< +> +Predictions +Explanations +Metrics +Counterfactuals +TCAV +Classification Results +Salience Maps +Attention + probas + Grad L2 Norm Grad · Input Integrated Gradients LIME +layer_1/attention → Head:1 +Class +Label +Predicted +Score +token_grad_sentence +Grad L2 Norm +[CLs] it ' s a charming but rotten journey +it.sa +charming +but +rotten +journey +0 +0.561 +1 +0.439 +Grad - Input +token_grad_sentence +it.sa + charming +but +rotten +journey +[CLs] it ' + s a charming but rotten journey +Made with by the LIT team Figure 5: Visualizing the location and degree of toxic +tweet conversations as trees where the degree of toxic- +ity is represented as withering (Beshai, 2018). +Figure 6: Some adjectives in Grimms’ Fairy Tales oc- +cur more frequently in reference to gendered charac- +ters. +ing positive/negative adjectives with words about +different races) is frequently used by social psychol- +ogists to assess implicit human biases (Greenwald +et al., 1998), which can be visualized. Figure 6 +shows adjectives associated with gender-specific +words in Grimms’ Fairy Tales (Grimm et al., 1823), +revealing gender-related representation bias (e.g., +old more frequently describes woman while young +more likely describes man). +Visualizing Biases in NLP models In addition, +large pre-trained language models such as BERT +and GPTs are used in a large number of down- +stream applications. To prevent bias propagation, +it is critical that we identify, assess and mitigate +the biases in these models. As most pre-trained +models are language models that can estimate the +likelihood of words appearing in a context, we can +visualize the predicted likelihood of a word in a +specific context to reveal the biases encoded in +these models. Figure 7 visualizes the probability +of words in the blanks: Jane worked as a [ ] ver- +sus Jim worked as a [ ] (Pearce, 2021). Based on +the visualization, the top predicted occupations for +Jane are waitress, teacher, and nurse while the top +occupations for Jim are mechanic, carpenter, and +salesman. +In addition, many of the pre-trained language +models are transformer-based, which relies on +self-attention to represent and interpret word se- +Figure 7: Occupation-related gender bias in NLP mod- +els shown by visualizing the estimated probability of +words occurring in given contexts. +Figure 8: Occupation-related gender bias in NLP mod- +els shown by attention visualization: same sentence +with different pronouns attend to different occupations. +quences. Figure 8 shows the words that a BERT +model pays attention to when performing pronoun +resolution (Vig, 2019a). When the pronoun is +she, the top words that the model pays attention +to include nurse and The, while for he, the top +words are The and doctor. This visualisation re- +veals occupation-related gender bias encoded in the +BERT model (Tenney et al., 2020). +5.2 +Linguistic Information from Embeddings +While LLMs are typically trained and evaluated on +specific NLP tasks such as question answering, an- +other motivation for XNLP is to understand linguis- +tic phenomena and gain insights into language as +a system. Incremental pipeline architectures com- +bining several (neural) task-specific systems allow +for a certain degree of insight based on the outputs +of each stage: Which linguistic features were pre- +dicted in lower stages? Which stages contribute +valuable information? Should a certain stage be by- +passed as it tends to introduce errors? Is syntactic + +Adjective +Characters by frequency: 2-4 5-9 10-19 20-29 30+ +little +man tailor mother girl daughter son +Adjective bias +plo +woman man king witch cook mother +80% female +beautiful +princess daughter king queen Woman +great +king mother +even +good +man Woman mother +young + man king princess girl +80% male +dear +mother son princess woman wife +0 +2 +4 +6 +Approximate number of charactersJane worked +as a +Jim worked +as a +waitress +teacher +nurse +journalist +secretary. +'librarian +model. +lawyer +reporter. +bartender +receptionist, +photographer +sentence +nanny. +courier +maid +Chet +translator +painter +cook +therapist, +musician +chemist +ne : +carpenwaiter +prostitute. +mechanic +dancer +designer +consultant +volunteer +psychiatrist +'butcher +psychic +messenger +pianist. 'steward: +salesman +kel +sculptor + trader +healer +farmer +playwrigh +medium +barber +choreographer +nistorian +coach +printer +publisher +weaver +builder +performer +Bygsist +bookstore +policeman +guard . +‘businessman +sailor +draper +fisherman +soldier +barker +restaurant +sheriff +boxer +likelihood. Jim sentenceLayer: +54 +Layer: +5 +The +The +The +The +doctor +doctor +doctor +doctor +asked +asked +asked +asked +the +the +the +the +nurse +nurse +nurse +nurse +a +a +a +a +question +question +question +question +She +She +He +HeFigure 9: Self-similarity of token embeddings across +layers (last layer in purple) for several tokens as radial +chart (Sevastjanova et al., 2022). High self-similarity +of numbers even in higher layers indicate less contextu- +alization of numbers in BERT. +analysis helpful? With the rise of end-to-end ap- +proaches, though, these types of linguistic insights +can no longer be obtained easily. +Hence, a considerable body of work focuses +on the analysis of hidden layer representations or +specifically trained text embeddings. If models +capture linguistic knowledge, one should be able +to decode it from these representations. We can +train diagnostic classifiers on linguistic tasks based +on internal (possibly intermediate) representations +in our models and investigate if, when, and with +which representations this is possible (Belinkov +et al., 2017). Similarly, probing approaches (Ten- +ney et al., 2019b) have been developed to find out +whether and in which layers Encoder-based Trans- +former architectures capture linguistic information. +For instance, it has been shown that we can already +predict part-of-speech tags sufficiently well using +the lower layers in BERT, whereas semantic roles +seem to be captured in higher layers (Tenney et al., +2019a). Other approaches try to correlate internal +representations with the representations explicitly +trained for a task, assuming that these representa- +tions should be somewhat similar if they capture +the same linguistic properties (Saphra and Lopez, +2019). +However, we need a labeled training set of +pre-defined tasks to understand LLMs this way. +More advanced interactive visualizations can help +to explore what the model has learned without +specifically designed benchmark tasks and datasets. +LMFingerprints (Sevastjanova et al., 2022) is +one exemplary approach that aims at exploring +Transformer-based language models without ex- +plicit probing tasks. Many recent LLMs are based +on the Encoder-Decoder Transformer architec- +ture (Vaswani et al., 2017) that computes contextu- +alized token embeddings based on the other tokens +in a sequence and their embeddings in the corre- +sponding layer. LMFingerprints computes numer- +ous scores for each pair of token and corresponding +input sequence based on the contextualized token +representations in each layer (e.g., self-similarity +of token representations between layers). The ap- +proach then visualizes aggregations of these scores +in matrix-like charts and radial area charts (Fig- +ure 9) so that analysts can assess and compare the +degree of contextualization as well as the capturing +of semantic information across layers and models. +For instance, similar representations in early layers +in BERT typically correspond to lexical and seman- +tic similarities whereas middle layers correspond to +similar named entity or part-of-speech categories. +Several word-based linguistic tasks can be inves- +tigated with interactive visualizations using embed- +dings (Heimerl and Gleicher, 2018): we may find +analogies and synonyms with neighborhood views +and projections, we can explore captured concepts, +we can visualize the shift of meaning over time by +training models on specific subsets, we can com- +pare how different models have captured semantic +relatedness, and we can assess which words often +co-occur. Other approaches have a stronger focus +on the comparison of embeddings generated by dif- +ferent models. For instance, Embedding Compara- +tor (Boggust et al., 2022) utilizes multiple visual- +izations to show two-dimensional projections of the +local neighborhoods of a word for each model and +highlights similarities. Many of these approaches +employ dimensionality reduction methods to vi- +sualize internal states and computed embeddings +(e.g., Figure 4a), which is backed by the promis- +ing finding that some linguistic tasks can also be +solved based on low-dimensional subspaces of the +representations (Hernandez and Andreas, 2021). +5.3 +Using Attention to Debug for Machine +Translation +A common approach for neural machine transla- +tion is to encode an input language with a language +model and use these encoder embeddings to steer +a decoder model that generates text in the target +language. The connection between encoder and + +BERT +ADP +self-similarity +N +over +.through +not +two +three +euntil +thousand +owith +six. +ewithin +one +ewithout +million. +care +cbe +hundred. +.been +four. +ecan +five. +ecould +billion. +edid +edo +area: difference to +the previous / +following layer +tokens in the corpus +CCON +token-groups +NUMdecoder can be an attention model. While encoder +and decoder are black-box models themselves, in- +terpreting their hidden representations can give an +intuition about which of them might be failing in +case an error occurs. Similarly, the connecting at- +tention mechanism might fail. And finally, the text +generation itself might fail. +3/27/2018 +S2S Attention +http://localhost:8080/client/index.html?in=die%20l%C3%A4ngsten%20reisen%20fangen%20an%20,%20wenn%20es%20auf%20den%20stra%C3%9Fen%20dunkel%20wird%20. +1/1 +Start entering some encoder sentence (enter triggers request)... +die längsten reisen fangen an , wenn es auf den straßen dunkel wird . +Enc words: +Attention: +topK: +die +längsten +reisen +fangen +an +, +wenn +es +auf +den +straßen +dunkel +wird +. +the +longest travel +begins when +it +gets +to +the +streets +. +the +longest travel when when +it +'s +to +the +streets +. +and +oldest +trips +will +if +they +gets +dark +a +roads +in +so +tallest +journeys +begins +, +the +becomes +buried shore +road +of +well +russians +travels begin +as +there +grows +into +heaven street +, +you +icons +journey +start +in +this +comes +in +its +city +to +pivot +� change: +word +attn +�compare: +sentence +swap: +� + +the +and +so +well +longest +the +travel +longest +when +begins +begin +will +it +when +when +start +it +it +when +'s +gets +'s +it +going +to +going +'s +gets +to +the +to +going +to +streets +to +the +. +be +go +streets +buried +in +to +. +in +on +the +the +the +the +streets +streets +streets +streets +. +. +in +. +. +the +Figure 10: +An example for visually debugging a +sequence-to-sequence model called Seq2Seq-Vis (Stro- +belt et al., 2019). The highlighted attention can be ex- +cluded as likely cause for a missing context in the out- +put sentence. +Seq2Seq-Vis (Strobelt et al., 2019) is an early +example of a debugging tool that helps identify +which part of a sequence-to-sequence model is fail- +ing for a given instance. In this case, the encoder +and decoder are LSTMs that are connected by a +simple attention model. The decoder produces text +using beam search. Figure 10 shows an example of +the user interface that exposes the tokenized input +sequence (blue boxes), the output sequence (yellow +boxes), the attention between encoder and decoder +as a bipartite graph, and the beam search tree as +node-link diagram (bottom). The input sentence +in German “die laengsten reisen fangen an, wenn +es auf den strassen dunkel wird” should translate +to something like the longest journeys begin when +it gets dark in the streets, but the context of dark +streets is not reproduced in the output. The red +highlighted lines show that at the appropriate posi- +tion in the output sentence the attention is correctly +focusing on the context of dunkel (dark). So it +seems that the attention mechanism is not likely +the cause of error in this case. After excluding the +encoder and decoder embeddings as error-causing +factors (not shown here), it seems that the beam +search is not doing a good job in avoiding local +optima - it prefers the slightly more likely word to +over the word dark at the highlighted position. +After identifying a probable cause for error, the +user now can conduct what-if testing by constrain- +ing the beam tree to use the word dark instead of +the word to. The resulting sentence The longest +travel begins when it gets dark in the streets is a +very good translation. A model analyst can now +add this case to a list of well-described failure ex- +amples that can later help to improve the model. +This use case exemplifies how visual analysis of +multiple parts in a complex model can help identify +errors in a translation model. While in this case, +only one attention head had to be investigated, the +need for more advanced methods to find and investi- +gate relevant attention patterns is immanent in light +of the rise of importance of transformer models. +Visual analysis systems like BertViz (Vig, 2019b) +or RXNmapper-VIS (Schwaller et al., 2021) have +shown early successes in relating self-attention pat- +terns to features in language or properties of chem- +ical reactions encoded as SMILES strings. +6 +Research Opportunities +We have shown early evidence that interactive visu- +alization can play an important role to help explore +and explain NLP models. While the existing body +of work is already impressive, we think that there is +potential for much more collaborative work ahead. +We base our prediction on a set of research ques- +tions that we want to highlight in this section. +Before explaining research question related to +new advances in NLP, we want to highlight a set +of visualization challenges that are known to be +long-standing and important to revisit in the future: +Text summarization is a classic task in NLP +and visualization. In both domains the goal is to +generate abstractions/summaries for longer texts to +facilitate consuming the most important informa- +tion in a compressed form. One exemplary chal- +lenge is, contrary to image content, the discrete +nature of text prevents the use of simple zoom out +techniques for text visualization. +For language or model analysis, it is common +to have multiple annotations for the same token +in a text (e.g., POS tag and NER tag). Visual- +izing the overlay of multiple tags for tokens is +perceptually hard and a well-known, yet unsolved, +challenge in visualization. +When dealing with LLMs, the amount and va- +riety of data that is produced during inference +and training challenges the visual and interaction +scalability of any visual analytics system. Interac- +tive visual analytic techniques for massive number + +of data points, such as embeddings, networks, clus- +tering, and so on have to be optimized for large +data, partial data, or progressive data updates. +The following selection of research questions +has no aspiration to be complete, but we would like +to highlight some of the more recent challenges +and opportunities for interactive visualization from +data to inference: +Tokenization of the input text is a common first +step for training and inference. Using sub-word +tokens limits the growth of the vocabulary to fea- +sible sizes and allows models to handle previously +unseen words. However, it also raises questions +about the semantic nature of tokens and what they +actually represent since, e.g., a slight variation of a +word can lead to completely different tokens. +One of the main architectural ingredient in many +recent LLMs that are based on the Transformer +architecture is the heavy use of multiple dot product +attentions across most of the layers. How can we +aggregate and visualize attention processes in +models with a rapidly increasing number of layers? +During model training or fine-tuning, check- +points are being created that are evaluated for their +task performance. But how can we quickly com- +pare model checkpoints to determine qualitative +progress? How can we compare models beyond +highly abstract overall measures such as the com- +puted loss? Is the increase in accuracy from 95.1% +to 95.11% worth the training cost of our model? +A core question in XNLP is if and how we can +map model-internal representations to language fea- +tures. Can we identify what the model units have +learned about language? How do we represent +this knowledge? How can we make this knowledge +interactively actionable? The majority of work in +interactive XNLP has contributed to this topic by +building tools to formulate hypotheses. +Large language models are often trained propri- +etorially without access to internal model states. +Even with the release of weights (e.g., (Sanh et al., +2021; Zhang et al., 2022)), running the models +in inference is very costly. This requires new ap- +proaches for XNLP methods. We believe that in- +teractive what-if testing can play a major part in +formulating hypothesis about model behavior. Re- +lated research questions are: How can users mean- +ingfully interact with LLMs? What are appropri- +ate user interactions and algorithmic methods to +achieve steerability in LLMs? If inference times +are long, how can user interaction help to reduce +the amount of LLM requests while still allowing +analysts to gain insights into what the model does? +The limitations of language models range from +performance limits to learned biases. While model +cards ((Mitchell et al., 2019)) are a good start to +statically summarize how a model was trained and +which biases it might expose, we think that adding +interaction (beyond (Crisan et al., 2022)) to the +pool of model card techniques can help to discover +model limitations for real world usage. How can +we communicate these complex limitations? How +can we construct challenge datasets for models? +How can we discover systematic errors by interac- +tion? How can we find bad apples and reasons why +they might behave badly? +After having identified errors in a model, it might +not be feasible to retrain the model again but rather +apply a patch or fine-tune it very specifically. How +do we “communicate” to a model what to fix +and how? How can we generate a generalized +patch for a model? How can we observe damage +being done to models while trying to fix them? +Model architectures are typically evaluated +purely based on their performance on benchmark +datasets. However, the extent to which we can un- +derstand a model and its decision-making process +is a value in itself. Designing more comprehensi- +ble model architectures that are easier to debug +and explore with interactive visualizations yet per- +form competitively would go a long way toward +achieving more trustworthy and responsible AI. +7 +Conclusion +In this position paper, we try to motivate the use +of interactive visualization for XNLP by highlight- +ing opportunities where interactive visualization +might be helpful in NLP processing workflows. +We have also showed some existing examples of +using visualization for XNLP. 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Association for Com- +putational Linguistics. +Susan Zhang, Stephen Roller, Naman Goyal, Mikel +Artetxe, Moya Chen, Shuohui Chen, Christopher De- +wan, Mona Diab, Xian Li, Xi Victoria Lin, Todor Mi- +haylov, Myle Ott, Sam Shleifer, Kurt Shuster, Daniel +Simig, Punit Singh Koura, Anjali Sridhar, Tianlu +Wang, and Luke Zettlemoyer. 2022. Opt: Open pre- +trained transformer language models. +Yu Zhang, Peter Tino, Ales Leonardis, and Ke Tang. +2021. A survey on neural network interpretability. +IEEE Transactions on Emerging Topics in Computa- +tional Intelligence, 5(5):726–742. +A +Additional Use Cases +Besides the above use cases, we provide two ad- +ditional cases regarding the use of visualization +for data labeling and natural language generation +(NLG). +A.1 +Labeling Data +Classifying texts is a common NLP task. For exam- +ple, financial analysts have access to thousands of +news sources and hundreds of thousands of blogs +(e.g. Meltwater), from which news articles of inter- +est for narrow topics must be immediately alerted +to key users such as portfolio managers, traders, +and quantitative analysts. Too many false positives +overwhelm users, while false negatives result in +missing critical market signals. +It is therefore important for the subject matter +experts to create accurately labeled datasets to train +a well-performing classifiers. This entails getting +an understanding of the available data set (e.g., +topic distribution, level of noise) to ensure that +the training data is sufficiently clean and contains +enough examples of interest, it entails labeling +items efficiently and effectively (manually or semi- +automatically), and it also entails understanding +what the model has learned and how it decides +to classify individual stories to better assess the +quality and generalizability of the model with the +annotations made up to that point. False labels +may not only impair the performance of the trained +model, but also have wide implications for society. +For instance, tweets written by African Americans +have been wrongly labeled as hate speech dispro- +portionally (Sap et al., 2019). +Interactive visualizations may help to find these +data items to label that would improve the classifier, +for instance, by inspecting borderline stories that +are closer to the threshold between positive and neg- +ative stories (Heimerl et al., 2012). In general, two- +dimensional projections of data items (e.g., with +t-SNE) can help to find inaccurate labels or border- +line cases (Bernard et al., 2018). Additional indica- +tions explaining why the model classified a certain +document into a specific topic further help analysts +assess whether the trained classifier has learned a +plausible mapping. There are several ways to visu- +ally explain such decisions (at least parts of it), for +instance, by highlighting present or absent phrases, +by visualizing what the model attended to (DeRose +et al., 2021), by highlighting active neurons on indi- +vidual or aggregated items (Kahng et al., 2018), or +by depicting the evolution of hidden states in recur- +rent neural networks on token sequences (Strobelt +et al., 2018). For example, LIT in Figure 4 shows +a) top left, a 3D embedding of statements, color- +coded by classifier result; and, b) bottom center, a +salience heatmap of the selected statement. + +These markups can also aid users interpreta- +tion of the resulting alerts. For example, the pre- +dicted score can be used as a proxy for relevance, +and when presented visually, allows the user to +scan lists of alerts for the most relevant stories, +or, indicates model issues when irrelevant stories +score highly thereby indicating model quality is- +sues which may be due to topic drift or other +causes. +A.2 +Natural Language Generation (NLG) +Natural language text generation is used for tasks +such as news generation, descriptive business intel- +ligence (e.g. Narrative Science) or fiction. Text cre- +ated with LLM’s can result in unexpected phrases, +narrative discontinuities or factual errors (e.g. hal- +lucinations, (Rebuffel et al., 2022)), where visual- +ization can aid analysis and understanding of the +model. GLTR (Gehrmann et al., 2019) colors the +background of tokens based on their model prob- +ability to visualize model surprisal (e.g., red and +purple indicating rather unexpected tokens with +low probabilities). For instance, low overall sur- +prisal of a given text indicates that it was either +generated by the respective model or was part of its +training corpus (Figure 11). The interpretive color +overlay on the output (a) aids text skimming to +identify unexpected phrases; provides interactions, +such as (b) mouse-over on the prior word shows the +top subsequent words the model was expected to +generate, and (c) click e.g., to regenerate from this +point forward. This visualization can zoom out so +that the words are no longer visible, while colored +pixels remain. Markup with zoom and regeneration +can be used at a macro level to skim large generated +texts to facilitate editing workflows. +Furthermore, interactions with the generated text +can be used to review the training data to under- +stand the origins of unexpected text. Figure 11 +shows an example based on a document that was +generated by another GPT-2 model. Most of the +highlights are in green and yellow, indicating that +the text was indeed automatically generated by a +similar model. One sentence ends with two peaks +of rock and silver snow and the word silver has a +red background, corresponding to a low probabil- +ity. After selecting silver snow, a search against +the training data can create a clustering visualiza- +tion showing search results grouped by similarity. +While silver snow is uncommon, two large clusters +indicate its use in a Nintendo game and the name +Figure 11: Example NLG markup indicating unex- +pected words by color. +of an energy drink, thus explaining to the human +the origin and context of the phrase. + +top k count +frac(p) histogram +top 10 entropy(p) histogram + 400 +200 +90 +350 +-180 + 80 +160 +300 +- 70 +-140 + 60 +250 +120 +50 +-200 +100 +-40 +150 +80 +- 60 +-30 +100 +.40 +-20 +50 +20 +10 +L0 +LO ++ 0.492 ++ 1.65 +0 0.10.20.30.40.50.60.70.80.9 1 +0 0.20.40.60.8 1 1.21.41.61.8 2 2.22.4 +Top K +Frac P +Colors (top k): +10 +100 +1000 +In ashocking finding,scientistdiscovered aherd ofunicornslivingin a remote,previouslyunexplored valley,inthe +Andes Mountains. Even more surprising to the researchers was the fact that the unicorns spoke perfect English. +The scientist named the population, after their distinctive horn,Ovid'sUnicorn.These four-horned, silver-white unicorns +werepreviouslyunknown to science +Now, after almost two centuries, the mystery of what sparked this odd phenomenon is finally solved. +Dr. Jorge +Perez,an evolutionary biologist fromthe University of La Paz,andseveral companions,wereexploringthe +Andes Mountains when they found a small valley,with no other animals or humans.Pereznoticed that the valley hac +what + appeared to be a natural fountain, surrounded by two peaks of rock and silver +snow. +Pérez and the others then ventured further into the valley."Bythe time we reachedthe top of one peak,the watei +looked blue, with some crystals on top," said Pérez. +Pérez and his friends were astonished to see the unicornherd.Thesecreatures could be seen from the airwithout +having to movetoomuch to see them-they were so close they couldtouchtheirhorns. +While examining these bizarre creatures the scientists discovered that the creatures also spoke some fairly regular +English. Pérez stated, "We can see, for example, that they have a common 'language, something like a dialect or +dialectic: +Dr. Pérez believes that the unicorns may have originated in Argentina, where the animals were believed to be +descendants of a lost race of people who lived there before the arrival of humans in those parts of South America. +While theirorigins are still unclear,some believe that perhapsthe creatures were created whena humanand a unicorn +met eachother in a timebefore human civilization.Accordingto Perez,"InSouthAmerica,suchincidentsseemto be +quite common." +However, Pérez also pointed out that it is likely that the only way of knowing for sure if unicorns are indeed the +descendants of a lost alien race is through DNA. "But they seem to be able to communicate in English quite well, +which Ibelieve is a sign of evolution,or at least a changein social organization," said the scientist \ No newline at end of file diff --git a/ZtE3T4oBgHgl3EQfcwpV/content/tmp_files/load_file.txt b/ZtE3T4oBgHgl3EQfcwpV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb4357c5f31b80709e039bb0d83f8b649f9ffbd2 --- /dev/null +++ b/ZtE3T4oBgHgl3EQfcwpV/content/tmp_files/load_file.txt @@ -0,0 +1,814 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf,len=813 +page_content='The Role of Interactive Visualization in Explaining (Large) NLP Models: from Data to Inference Richard Brath Uncharted Software Inc rbrath@uncharted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='software Daniel Keim University of Konstanz Johannes Knittel University of Stuttgart Shimei Pan University of Maryland, BC Pia Sommerauer Vrije Universiteit Amsterdam Hendrik Strobelt IBM Research Cambridge hendrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='strobelt@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='com Abstract With a constant increase of learned parame- ters, modern neural language models become increasingly more powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Yet, explaining these complex model’s behavior remains a widely unsolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In this paper, we discuss the role interactive visualization can play in explaining NLP models (XNLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We motivate the use of visualization in relation to target users and common NLP pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We also present several use cases to provide con- crete examples on XNLP with visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Finally, we point out an extensive list of re- search opportunities in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 1 Motivation In recent years, NLP systems powered by very large neural network models, such as BERT and GPT- 3, have provided an unprecedented performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The latest models have billions of parameters and need enormous amount of data and computing re- sources for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While results in general are of high quality, there are numerous applications where explainability is of high importance, such as medical diagnosis or bias detection and mitigation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In this position paper, we examine the role of interactive visualization in explaining (large) NLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The reflections and proposals in this work are the result of intensive discussions and close collaboration of experts in NLP and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We first distinguish different user groups with varied technical and domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Each user group has different explainability needs, which may guide the design of interactive visual tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Next, we discuss the use of visualizations in explaining typical NLP pipelines, especially those employ- ing pre-trained large language models (LLM), with questions ranging from when to use visualizations and why, which visualizations to use and how to use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' It is important to note that visualizations can be used in very different ways and for very dif- ferent purposes, and probably in even more ways than they have been used in the past, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' (Belinkov and Glass, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Danilevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' However, there are also pitfalls such as a misunderstanding of what can be inferred from visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' To support our arguments, we include a few use cases ranging from identifying social bias in NLP mod- els, acquiring linguistic insight, debugging com- plex models to labeling ground truth in the main work and the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Finally, we present an out- look on research opportunities that may arise in the same context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' They cover all phases of model development starting from visualizing the data and their properties over the different stages of model development to the evaluation and interpretation of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 2 User Groups for XNLP Visualization Visualization methods for XNLP should enable users with different expertise, to solve specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' As shown in Figure 1, domain experts may have high expertise in a task and text corpus, but may not be experienced in language models and may use these models as blackboxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Model archi- tects and builders may have high degree of knowl- edge on advanced modeling techniques but might not be experts in the application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' General users, such as a casual user of Google translate, may have low knowledge of both NLP models and application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Data scientists and analysts may be in the middle, as users of general analytical and NLP toolkits, in order to perform analytical tasks on some datasets of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Explainability of NLP models through visualiza- tion can help across all user types, although each may have differing explainability needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A model builder may be more interested in locating bugs and understanding model performance which may re- quire fine-grain visualization of the model structure and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Domain experts may have a criti- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='04528v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='CL] 11 Jan 2023 INPUT/TRAINING DATA MODEL ASSESS TRAIN LARGE LANGUAGE MODEL FINE-TUNING MODEL INFERENCE OR PROMPT Raw Data Prepared Data Model Evaluate Challenge Raw Data Annotated Data Model Evaluate Challenge Input (& Steering) Model Output Prompt Observe artifacts De-bias training data Monitor training Changes over iterations Understand linguistic knowledge Inspect quality Discover boundaries Assess updates à à Audit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' inspect Bias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' active learning Find edge cases Review annotations Prompt engineering User-based manipulation to understand model sensitivity visual comparison relate back to training data EXPLORE TRAINING SPACE EXPLAIN MODEL ASSESS OUTPUT (A) MODULAR NLP PIPELINE (B) END-TO-END MODELS (C) LARGE LANGUAGE MODELS (c1) Fine-Tuned Model End-to-End NLP model Task (classify,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Q&A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' translation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' summarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='…) models for POS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' NER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' sentiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Large Language Model (c2) Few Shot Learning & Prompting LEGEND Black Box Model Other algorithms / models a b c2 c1 REVISED 20220713 MODEL KNOWLEDGE DATA KNOWLEDGE general user domain expert model architect data scientist Figure 1: User types cal need to understand how concepts are encoded in the model and require visualization of what con- cepts or linguistic units the model attends to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Gen- eral users may wish to understand if the model is biased and may desire to gain some understanding between the training data and a biased output from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Model architects and builders should understand their models and their implications in downstream applications 3 NLP Models and Interpretability Black-box neural network models are important building blocks in state-of-the-art NLP pipelines as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Classical pipeline approaches still have their place in small-data scenarios, which are common in interaction systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' when in- teractively selecting document subsets, NLP tech- niques such as NER, POS, LDA, can provide on- demand descriptive statistics, which in turn can be visualized to characterize the selected subset, in comparison to the training corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Furthermore, they can be used to combine several neural systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Another advantage is that the explicit inputs and outputs of individual models may help to interpret the combined system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We therefore include them in our considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A lot of research interest is currently directed toward large language models (LLM) and how they can be utilized to solve common NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In contrast to end-to-end models that are typically trained on specific tasks, these LLMs are trained in a self-supervised or semi-supervised fashion on very large training data and use many trainable pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In mid-2022, the largest language model reported has 540 billion parameters (Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' One of the intriguing aspects of LLMs is that they seem to capture factual knowledge to some extent (Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019), which makes them very powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Two main methods have been used recently to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='apply LLMs for specific NLP tasks: fine-tuning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='INPUT/TRAINING DATA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='MODEL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='ASSESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='TRAIN LARGE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='LANGUAGE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='MODEL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='FINE-TUNING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='MODEL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='INFERENCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='OR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='PROMPT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Raw ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='(& Steering) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Prompt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Observe artifacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='De-bias training data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Monitor ' 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+page_content='Inspect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='quality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Discover ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='boundaries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Assess ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='updates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='à ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='à ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Audit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' inspect Bias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' active learning Find edge cases Review annotations Prompt engineering User-based manipulation to understand model sensitivity visual comparison relate back to training data EXPLORE TRAINING SPACE EXPLAIN MODEL ASSESS OUTPUT (A) MODULAR NLP PIPELINE (B) END-TO-END MODELS (C) LARGE LANGUAGE MODELS (c1) Fine-Tuned Model End-to-End NLP model Task (classify,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Q&A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' translation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' summarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='…) models for POS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' NER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' sentiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Large Language Model (c2) Few Shot Learning & Prompting LEGEND Black Box Model Other algorithms / models a b c2 c1 REVISED 20220713 MODEL KNOWLEDGE DATA KNOWLEDGE general user domain expert model architect data scientist Figure 2: The role of neural network (NN) models in NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Four modes are commonly used: (a) using separate NN models to solve intermediate tasks in mod- ular NLP pipelines, (b) training end-to-end models on NLP tasks, (c1) fine-tuning with task-specific ground truth with text encoding from LLMs, and (c2) few-shot learning and prompt engineering for ad-hoc NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' task-specific models using text encodings gener- ated by LLMs (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2018) or employing LLMs as few shot learners (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2020) and adapting them to different NLP tasks using a few examples or prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' LMs have been shown to have superior perfor- mance on many complex tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' At the same time, they come with important limitations: (1) LLMs require large amounts of data and computing power and are often created in such a way that the train- ing process and the training data are not openly accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' (2) The training data often consist of large volumes of texts published online, which can reflect harmful views and biases, which are then propagated to downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' (3) The extremely large size of the models, together with the size of the training data render XNLP a very challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' From a XNLP perspective, visualization can be applied to different tasks in different NLP work- flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In addition, both the scope of data and users’ expectations can have a significant impact on the design of a visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 4 Visualization for XNLP Visualization is broadly applicable across explain- ing LLMs as indicated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Interactive visualizations can be used to explore the training data, the training processes, the resulting models, or the model output (columns in Figure 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' and can be used during model training, model adaptation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' fine-tuning), and model inference (rows in Fig- ure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The training data (first column) defines the world view of the model, a first and important step to better assess the performance of and issues in the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' domain specificity, biases, and general- izability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Simple visualizations like line charts are often used to communicate the progression of high- level measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' accuracy or loss), but more advanced approaches can allow for more thorough analyses such as whether the training is qualita- tively improving w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' From a model perspective (middle column), visualization can aid model-builders building LLMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' downstream mod- elers fine-tuning or prompt engineering LLMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' and end-users visually analyzing the output of models (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Inspecting output is a more shallow yet model-agnostic way of exploring NLP models that can lead to relevant insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, it may help model builders and domain experts to evaluate models more comprehensively instead of just relying on a small set of computed metrics on benchmark datasets (Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' INPUT/TRAINING DATA MODEL ASSESS TRAIN LARGE LANGUAGE MODEL FINE-TUNING MODEL INFERENCE OR PROMPT Raw Data Prepared Data Model Evaluate Challenge Raw Data Annotated Data Model Evaluate Challenge Input (& Steering) Model Output Prompt Observe artifacts De-bias training data Monitor training Changes over iterations Understand linguistic knowledge Inspect quality Discover boundaries Assess updates à à Audit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' inspect Bias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' active learning Find edge cases Review annotations Prompt engineering User-based manipulation to understand model sensitivity visual comparison relate back to training data EXPLORE TRAINING SPACE EXPLAIN MODEL ASSESS OUTPUT (A) MODULAR NLP PIPELINE (B) END-TO-END MODELS (C) LARGE LANGUAGE MODELS (c1) Fine-Tuned Model End-to-End NLP model Task (classify,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Q&A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' translation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' summarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='…) models for POS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' NER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' sentiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Large Language Model (c2) Few Shot Learning & Prompting LEGEND Black Box Model Other algorithms / models a b c2 c1 REVISED 20220713 MODEL KNOWLEDGE DATA KNOWLEDGE general user domain expert model architect data scientist Figure 3: Areas for visualization in LLM’s in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='1 Why and When (Not) to Use Visualization Interactive visualizations are well suited to tackle problems that are generally difficult to formulate as a mathematical problem (Keim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Exploring the inner workings of models with mil- lions or even billions of parameters is a complex, open-ended task that cannot be solved solely in an automated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Upon interacting with the visual- izations, analysts may gain a better understanding of the model or dataset at hand, and they may come up with more formal hypotheses that can be investi- gated later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In addition, visualizations often play a major role in communicating results and insights to different audiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' It is important to note that for some tasks, how- ever, there is little benefit in using interactive visu- alizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, visualizations rarely play a role in confirming formal hypotheses quantita- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For well-defined goals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', text search), automatic methods or simple visualizations such as text highlighting are better suited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='2 Visualization and Interaction While the variety of potential visualizations is large, some visualization techniques have shown to be the workhorses across many tasks: Simple charts are often used, such as line charts for loss over iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' or bar charts indicating the probability of words (at a position in a sentence), or for model performance or class prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Markup and heatmaps are frequently applied to (partly) explain model behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, we can visualize extracted rules that approximate predictions or local behavior with heatmaps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' or we can highlight salient inputs that are relevant for the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' This can be applied to narrative text, or word lists (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' one node (Dalvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' One of the advantages is that we can zoom-out, such that only the markup and structure remain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', pixel-level visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' High-dimensional visualization, such as dimen- sional reduction visualization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' PCA, t-SNE) or hierarchical clustering, may be used to visualize thousands to millions of data points by plotting sim- ilar data points in similar locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' They are often used to visualize the distribution of embeddings and hidden states, for instance, to better understand whether semantically similar input leads to similar internal representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Graph visualization can be used to show net- work relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While they are perfectly suited to show relationships between a few nodes at a local scale, it is challenging to scale them up to global structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In addition, they can be used a) to visualize alternative inputs or outputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Sen- tenTree (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2017) to view alternative outputs from one point forwards);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' b) to visualize grammat- ical structures, such as parse trees or co-referents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' (c) model internals like attention relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Fig- ure 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Text visualization techniques beyond markup are important for exploring the underlying training data set but they can also play an important role in visu- alizing the inner workings of a model, for instance, by showing visual summaries of the layers or nodes the model seems to focus on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Word clouds are popu- lar but controversial (in the traditional layout, word position, color, orientation are non-meaningful), al- though there are improvements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' (Hearst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Knittel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Other techniques exist for visualizing text with text, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' (Kucher and Ker- ren, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Brath, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Lang and Nacenta, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' On top of an effective visual encoding, some classes of interaction techniques are critical when exploring large scale datasets and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Selec- tions and tooltips help to explore subsets and access detailed data on demand, without cluttering the pri- mary representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Filters and facets allow to reduce data to relevant subset, based on any criteria or any selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Zoom, pan, and rotate help to move around massive plots, such as to zoom in to focus on a region, or rotate 3D plots to reduce occlu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Aggregations allow to expand or collapse the level of detail needed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' keywords, phrases, top- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Linked views and linked updates occur when many visualization elements are combined in one interface, with a selection in one visualization up- dating all others, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' LIT (Tenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2020) uses many of the above visualizations and linked interactions in one combined system (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 4: LIT, showing a) high dimensional visualiza- tion of embeddings (top left), b) simple bar chart of class probabilities (bottom left), c) salience heatmap (bottom center), d) attention graph (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Analysts may either explore the data top-down by first using overview visualizations, which then allows interactive drill-downs for more specific analyses related to insights or hypotheses that they have gained thus far;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' or, they may analyze or fil- ter the data first and investigate a particular subset of the data and/or model to expand their analysis in a bottom-up approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Furthermore, using an iteration loop, analysts may steer the model inter- actively to fit it to their needs, which allows for incorporating domain knowledge into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For large NLP models, the presented visualiza- tion techniques might be challenged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' See section 6 for resulting research opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 5 Use Cases So far, we generally discussed the role of interac- tion and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Here, we want to show a selection of concrete instances of how visualization can help and helped for XNLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Additional use cases about labeling data and NLG model visu- alizations can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='1 Identifying and Assessing Social Biases There are many recent discoveries of NLP systems that exhibit various types of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, Google’s machine translation algorithms convert the gender-neutral Turkish sentences O bir profesör.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' O bir ö˘gretmen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' to the English sentences He’s a professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' She is a teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' (Caliskan, 2021) To avoid these issues, it is critical that we develop effective tools to inspect and identify the biases in both NLP data and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Visualizing Biases in NLP Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' One major source of bias in NLP systems is human biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Since human-generated text are used to train NLP models, biased training data often result in biased NLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' To uncover the source of biases in NLP data, we can combine automated bias detec- tion with visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' With this method, we first employ text classifiers to detect diverse types of so- cial biases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', racism, sexism, microaggressions and hate speech) in text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', social media posts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We then employ interactive visualization to provide an overview of the distribution of biased text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='The visualization in Figure 5 summarizes toxic Twitter conversations as natural-looking trees, where toxic Twitter conversations are represented as withered branches (Beshai, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The scale and distribution of withered branches aids assessment of the degree of toxicity on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Word association (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', pair- sst2-tiny default & Language Interpretability Tool sst2-base sst_dev simple Share Select datapoint → Color by Compare datapoints 1 pair available < 图 > [0, 872] selected (< primary: b4abfc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='[872] >) ☆ < 5 of 873 selected > Clear selection Select random Embeddings Data Table Datapoint Editor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content="3 Projector: PCA Hide unselected Reset view Select all Columns sentence TextSegment it 's a charming but rotten Embedding: sst2-tiny:cls_emb index Q sentence Q label Q journey ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=" Label by: sentence → it's a charming and often affecting journey ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=" 0 872 it's a charming but rotten journey 1 0 1 unflinchingly bleak and desperate it's a charming but ro1 allows us to hope that nolan is poised to embark a label CategoryLabel 1← 1 major career as a commercial yet inventive filmmaker 3 the acting , costumes , music , cinematography 1 and sound are all astounding given the production 's austere locales it's slow -- very , very slow." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 0 5 although laced with humor and a few fanciful touches , the film is a refreshingly serious look at young women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=" < Page 1 of 88 > >I 图 Add and Reset Clear Select 10 nearest neighbors compare < > Predictions Explanations Metrics Counterfactuals TCAV Classification Results Salience Maps Attention probas Grad L2 Norm Grad · Input Integrated Gradients LIME layer_1/attention → Head:1 Class Label Predicted Score token_grad_sentence Grad L2 Norm [CLs] it ' s a charming but rotten journey it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='sa charming but rotten journey 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='561 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='439 Grad - Input token_grad_sentence it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content="sa charming but rotten journey [CLs] it ' s a charming but rotten journey Made with by the LIT team Figure 5: Visualizing the location and degree of toxic tweet conversations as trees where the degree of toxic- ity is represented as withering (Beshai, 2018)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 6: Some adjectives in Grimms’ Fairy Tales oc- cur more frequently in reference to gendered charac- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' ing positive/negative adjectives with words about different races) is frequently used by social psychol- ogists to assess implicit human biases (Greenwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 1998), which can be visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 6 shows adjectives associated with gender-specific words in Grimms’ Fairy Tales (Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 1823), revealing gender-related representation bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', old more frequently describes woman while young more likely describes man).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Visualizing Biases in NLP models In addition, large pre-trained language models such as BERT and GPTs are used in a large number of down- stream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' To prevent bias propagation, it is critical that we identify, assess and mitigate the biases in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' As most pre-trained models are language models that can estimate the likelihood of words appearing in a context, we can visualize the predicted likelihood of a word in a specific context to reveal the biases encoded in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 7 visualizes the probability of words in the blanks: Jane worked as a [ ] ver- sus Jim worked as a [ ] (Pearce, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Based on the visualization, the top predicted occupations for Jane are waitress, teacher, and nurse while the top occupations for Jim are mechanic, carpenter, and salesman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In addition, many of the pre-trained language models are transformer-based, which relies on self-attention to represent and interpret word se- Figure 7: Occupation-related gender bias in NLP mod- els shown by visualizing the estimated probability of words occurring in given contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 8: Occupation-related gender bias in NLP mod- els shown by attention visualization: same sentence with different pronouns attend to different occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 8 shows the words that a BERT model pays attention to when performing pronoun resolution (Vig, 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' When the pronoun is she, the top words that the model pays attention to include nurse and The, while for he, the top words are The and doctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' This visualisation re- veals occupation-related gender bias encoded in the BERT model (Tenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='2 Linguistic Information from Embeddings While LLMs are typically trained and evaluated on specific NLP tasks such as question answering, an- other motivation for XNLP is to understand linguis- tic phenomena and gain insights into language as a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Incremental pipeline architectures com- bining several (neural) task-specific systems allow for a certain degree of insight based on the outputs of each stage: Which linguistic features were pre- dicted in lower stages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Which stages contribute valuable information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Should a certain stage be by- passed as it tends to introduce errors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Is syntactic Adjective Characters by frequency: 2-4 5-9 10-19 20-29 30+ little man tailor mother girl daughter son Adjective bias plo woman man king witch cook mother 80% female beautiful princess daughter king queen Woman great king mother even good man Woman mother young man king princess girl 80% male dear mother son princess woman wife 0 2 4 6 Approximate number of charactersJane worked as a Jim worked as a waitress teacher nurse journalist secretary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=" 'librarian model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' lawyer reporter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' bartender receptionist, photographer sentence nanny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' courier maid Chet translator painter cook therapist, musician chemist ne : carpenwaiter prostitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=" mechanic dancer designer consultant volunteer psychiatrist 'butcher psychic messenger pianist." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=" 'steward: salesman kel sculptor trader healer farmer playwrigh medium barber choreographer nistorian coach printer publisher weaver builder performer Bygsist bookstore policeman guard ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' ‘businessman sailor draper fisherman soldier barker restaurant sheriff boxer likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Jim sentenceLayer: 54 Layer: 5 The The The The doctor doctor doctor doctor asked asked asked asked the the the the nurse nurse nurse nurse a a a a question question question question She She He HeFigure 9: Self-similarity of token embeddings across layers (last layer in purple) for several tokens as radial chart (Sevastjanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' High self-similarity of numbers even in higher layers indicate less contextu- alization of numbers in BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' analysis helpful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' With the rise of end-to-end ap- proaches, though, these types of linguistic insights can no longer be obtained easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Hence, a considerable body of work focuses on the analysis of hidden layer representations or specifically trained text embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' If models capture linguistic knowledge, one should be able to decode it from these representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We can train diagnostic classifiers on linguistic tasks based on internal (possibly intermediate) representations in our models and investigate if, when, and with which representations this is possible (Belinkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Similarly, probing approaches (Ten- ney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019b) have been developed to find out whether and in which layers Encoder-based Trans- former architectures capture linguistic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, it has been shown that we can already predict part-of-speech tags sufficiently well using the lower layers in BERT, whereas semantic roles seem to be captured in higher layers (Tenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Other approaches try to correlate internal representations with the representations explicitly trained for a task, assuming that these representa- tions should be somewhat similar if they capture the same linguistic properties (Saphra and Lopez, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' However, we need a labeled training set of pre-defined tasks to understand LLMs this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' More advanced interactive visualizations can help to explore what the model has learned without specifically designed benchmark tasks and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' LMFingerprints (Sevastjanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2022) is one exemplary approach that aims at exploring Transformer-based language models without ex- plicit probing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Many recent LLMs are based on the Encoder-Decoder Transformer architec- ture (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2017) that computes contextu- alized token embeddings based on the other tokens in a sequence and their embeddings in the corre- sponding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' LMFingerprints computes numer- ous scores for each pair of token and corresponding input sequence based on the contextualized token representations in each layer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', self-similarity of token representations between layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The ap- proach then visualizes aggregations of these scores in matrix-like charts and radial area charts (Fig- ure 9) so that analysts can assess and compare the degree of contextualization as well as the capturing of semantic information across layers and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, similar representations in early layers in BERT typically correspond to lexical and seman- tic similarities whereas middle layers correspond to similar named entity or part-of-speech categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Several word-based linguistic tasks can be inves- tigated with interactive visualizations using embed- dings (Heimerl and Gleicher, 2018): we may find analogies and synonyms with neighborhood views and projections, we can explore captured concepts, we can visualize the shift of meaning over time by training models on specific subsets, we can com- pare how different models have captured semantic relatedness, and we can assess which words often co-occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Other approaches have a stronger focus on the comparison of embeddings generated by dif- ferent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, Embedding Compara- tor (Boggust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2022) utilizes multiple visual- izations to show two-dimensional projections of the local neighborhoods of a word for each model and highlights similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Many of these approaches employ dimensionality reduction methods to vi- sualize internal states and computed embeddings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', Figure 4a), which is backed by the promis- ing finding that some linguistic tasks can also be solved based on low-dimensional subspaces of the representations (Hernandez and Andreas, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='3 Using Attention to Debug for Machine Translation A common approach for neural machine transla- tion is to encode an input language with a language model and use these encoder embeddings to steer a decoder model that generates text in the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The connection between encoder and BERT ADP self-similarity N over .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='through not two three euntil thousand owith six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' ewithin one ewithout million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' care cbe hundred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='been four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' ecan five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' ecould billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' edid edo area: difference to the previous / following layer tokens in the corpus CCON token-groups NUMdecoder can be an attention model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While encoder and decoder are black-box models themselves, in- terpreting their hidden representations can give an intuition about which of them might be failing in case an error occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Similarly, the connecting at- tention mechanism might fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' And finally, the text generation itself might fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 3/27/2018 S2S Attention http://localhost:8080/client/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='html?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='in=die%20l%C3%A4ngsten%20reisen%20fangen%20an%20,%20wenn%20es%20auf%20den%20stra%C3%9Fen%20dunkel%20wird%20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 1/1 Start entering some encoder sentence (enter triggers request).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' die längsten reisen fangen an , wenn es auf den straßen dunkel wird .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Enc words: Attention: topK: die längsten reisen fangen an , wenn es auf den straßen dunkel wird .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' the longest travel begins when it gets to the streets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' the longest travel when when it '' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='s to the streets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' and oldest trips will if they gets dark a roads in so tallest journeys begins , the becomes buried shore road of well russians travels begin as there grows into heaven street , you icons journey start in this comes in its city to pivot � change: word attn �compare: sentence swap: � the and so well longest the travel longest when begins begin will it when when start it it when '' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='s gets '' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='s it going to going '' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='s gets to the to going to streets to the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' be go streets buried in to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' in on the the the the streets streets streets streets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' the Figure 10: An example for visually debugging a sequence-to-sequence model called Seq2Seq-Vis (Stro- belt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The highlighted attention can be ex- cluded as likely cause for a missing context in the out- put sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Seq2Seq-Vis (Strobelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019) is an early example of a debugging tool that helps identify which part of a sequence-to-sequence model is fail- ing for a given instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In this case, the encoder and decoder are LSTMs that are connected by a simple attention model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The decoder produces text using beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 10 shows an example of the user interface that exposes the tokenized input sequence (blue boxes), the output sequence (yellow boxes), the attention between encoder and decoder as a bipartite graph, and the beam search tree as node-link diagram (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The input sentence in German “die laengsten reisen fangen an, wenn es auf den strassen dunkel wird” should translate to something like the longest journeys begin when it gets dark in the streets, but the context of dark streets is not reproduced in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The red highlighted lines show that at the appropriate posi- tion in the output sentence the attention is correctly focusing on the context of dunkel (dark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' So it seems that the attention mechanism is not likely the cause of error in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' After excluding the encoder and decoder embeddings as error-causing factors (not shown here), it seems that the beam search is not doing a good job in avoiding local optima - it prefers the slightly more likely word to over the word dark at the highlighted position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' After identifying a probable cause for error, the user now can conduct what-if testing by constrain- ing the beam tree to use the word dark instead of the word to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The resulting sentence The longest travel begins when it gets dark in the streets is a very good translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A model analyst can now add this case to a list of well-described failure ex- amples that can later help to improve the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' This use case exemplifies how visual analysis of multiple parts in a complex model can help identify errors in a translation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While in this case, only one attention head had to be investigated, the need for more advanced methods to find and investi- gate relevant attention patterns is immanent in light of the rise of importance of transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Visual analysis systems like BertViz (Vig, 2019b) or RXNmapper-VIS (Schwaller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2021) have shown early successes in relating self-attention pat- terns to features in language or properties of chem- ical reactions encoded as SMILES strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 6 Research Opportunities We have shown early evidence that interactive visu- alization can play an important role to help explore and explain NLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While the existing body of work is already impressive, we think that there is potential for much more collaborative work ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We base our prediction on a set of research ques- tions that we want to highlight in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Before explaining research question related to new advances in NLP, we want to highlight a set of visualization challenges that are known to be long-standing and important to revisit in the future: Text summarization is a classic task in NLP and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In both domains the goal is to generate abstractions/summaries for longer texts to facilitate consuming the most important informa- tion in a compressed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' One exemplary chal- lenge is, contrary to image content, the discrete nature of text prevents the use of simple zoom out techniques for text visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For language or model analysis, it is common to have multiple annotations for the same token in a text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', POS tag and NER tag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Visual- izing the overlay of multiple tags for tokens is perceptually hard and a well-known, yet unsolved, challenge in visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' When dealing with LLMs, the amount and va- riety of data that is produced during inference and training challenges the visual and interaction scalability of any visual analytics system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Interac- tive visual analytic techniques for massive number of data points, such as embeddings, networks, clus- tering, and so on have to be optimized for large data, partial data, or progressive data updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The following selection of research questions has no aspiration to be complete, but we would like to highlight some of the more recent challenges and opportunities for interactive visualization from data to inference: Tokenization of the input text is a common first step for training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Using sub-word tokens limits the growth of the vocabulary to fea- sible sizes and allows models to handle previously unseen words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' However, it also raises questions about the semantic nature of tokens and what they actually represent since, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', a slight variation of a word can lead to completely different tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' One of the main architectural ingredient in many recent LLMs that are based on the Transformer architecture is the heavy use of multiple dot product attentions across most of the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we aggregate and visualize attention processes in models with a rapidly increasing number of layers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' During model training or fine-tuning, check- points are being created that are evaluated for their task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' But how can we quickly com- pare model checkpoints to determine qualitative progress?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we compare models beyond highly abstract overall measures such as the com- puted loss?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Is the increase in accuracy from 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='1% to 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='11% worth the training cost of our model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A core question in XNLP is if and how we can map model-internal representations to language fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Can we identify what the model units have learned about language?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How do we represent this knowledge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we make this knowledge interactively actionable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The majority of work in interactive XNLP has contributed to this topic by building tools to formulate hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Large language models are often trained propri- etorially without access to internal model states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Even with the release of weights (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2022)), running the models in inference is very costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' This requires new ap- proaches for XNLP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We believe that in- teractive what-if testing can play a major part in formulating hypothesis about model behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Re- lated research questions are: How can users mean- ingfully interact with LLMs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' What are appropri- ate user interactions and algorithmic methods to achieve steerability in LLMs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' If inference times are long, how can user interaction help to reduce the amount of LLM requests while still allowing analysts to gain insights into what the model does?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The limitations of language models range from performance limits to learned biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While model cards ((Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019)) are a good start to statically summarize how a model was trained and which biases it might expose, we think that adding interaction (beyond (Crisan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2022)) to the pool of model card techniques can help to discover model limitations for real world usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we communicate these complex limitations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we construct challenge datasets for models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we discover systematic errors by interac- tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we find bad apples and reasons why they might behave badly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' After having identified errors in a model, it might not be feasible to retrain the model again but rather apply a patch or fine-tune it very specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How do we “communicate” to a model what to fix and how?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we generate a generalized patch for a model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' How can we observe damage being done to models while trying to fix them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Model architectures are typically evaluated purely based on their performance on benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' However, the extent to which we can un- derstand a model and its decision-making process is a value in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Designing more comprehensi- ble model architectures that are easier to debug and explore with interactive visualizations yet per- form competitively would go a long way toward achieving more trustworthy and responsible AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 7 Conclusion In this position paper, we try to motivate the use of interactive visualization for XNLP by highlight- ing opportunities where interactive visualization might be helpful in NLP processing workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We have also showed some existing examples of using visualization for XNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' So far, the research on applying interactive visualization in XNLP is still at its early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' To help reach its full po- tential, the NLP and data visualization community need to work closely together to overcome the chal- lenges posed by siloed domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' We hope that this position paper by researchers from both the NLP and visualization communities can help encourage future interdisciplinary collaborations on this important topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Acknowledgements Thanks to Dagstuhl for coordination of Seminar 22191 Visual Text Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' References Yonatan 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+page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' BERT rediscovers the classical NLP pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Asso- ciation for Computational Linguistics, pages 4593– 4601, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Ian Tenney, James Wexler, Jasmijn Bastings, Tolga Bolukbasi, Andy Coenen, Sebastian Gehrmann, Ellen Jiang, Mahima Pushkarna, Carey Radebaugh, Emily Reif, et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In Advances in Neural Information Pro- cessing Systems, volume 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Jesse Vig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A multiscale visualization of at- tention in the transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='05714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Jesse Vig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A multiscale visualization of at- tention in the transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics: System Demonstrations, pages 37–42, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Association for Com- putational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher De- wan, Mona Diab, Xian Li, Xi Victoria Lin, Todor Mi- haylov, Myle Ott, Sam Shleifer, Kurt Shuster, Daniel Simig, Punit Singh Koura, Anjali Sridhar, Tianlu Wang, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Opt: Open pre- trained transformer language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Yu Zhang, Peter Tino, Ales Leonardis, and Ke Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A survey on neural network interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' IEEE Transactions on Emerging Topics in Computa- tional Intelligence, 5(5):726–742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A Additional Use Cases Besides the above use cases, we provide two ad- ditional cases regarding the use of visualization for data labeling and natural language generation (NLG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='1 Labeling Data Classifying texts is a common NLP task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For exam- ple, financial analysts have access to thousands of news sources and hundreds of thousands of blogs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Meltwater), from which news articles of inter- est for narrow topics must be immediately alerted to key users such as portfolio managers, traders, and quantitative analysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Too many false positives overwhelm users, while false negatives result in missing critical market signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' It is therefore important for the subject matter experts to create accurately labeled datasets to train a well-performing classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' This entails getting an understanding of the available data set (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', topic distribution, level of noise) to ensure that the training data is sufficiently clean and contains enough examples of interest, it entails labeling items efficiently and effectively (manually or semi- automatically), and it also entails understanding what the model has learned and how it decides to classify individual stories to better assess the quality and generalizability of the model with the annotations made up to that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' False labels may not only impair the performance of the trained model, but also have wide implications for society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, tweets written by African Americans have been wrongly labeled as hate speech dispro- portionally (Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Interactive visualizations may help to find these data items to label that would improve the classifier, for instance, by inspecting borderline stories that are closer to the threshold between positive and neg- ative stories (Heimerl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' In general, two- dimensional projections of data items (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', with t-SNE) can help to find inaccurate labels or border- line cases (Bernard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Additional indica- tions explaining why the model classified a certain document into a specific topic further help analysts assess whether the trained classifier has learned a plausible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' There are several ways to visu- ally explain such decisions (at least parts of it), for instance, by highlighting present or absent phrases, by visualizing what the model attended to (DeRose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2021), by highlighting active neurons on indi- vidual or aggregated items (Kahng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2018), or by depicting the evolution of hidden states in recur- rent neural networks on token sequences (Strobelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For example, LIT in Figure 4 shows a) top left, a 3D embedding of statements, color- coded by classifier result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' and, b) bottom center, a salience heatmap of the selected statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' These markups can also aid users interpreta- tion of the resulting alerts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For example, the pre- dicted score can be used as a proxy for relevance, and when presented visually, allows the user to scan lists of alerts for the most relevant stories, or, indicates model issues when irrelevant stories score highly thereby indicating model quality is- sues which may be due to topic drift or other causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='2 Natural Language Generation (NLG) Natural language text generation is used for tasks such as news generation, descriptive business intel- ligence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Narrative Science) or fiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Text cre- ated with LLM’s can result in unexpected phrases, narrative discontinuities or factual errors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' hal- lucinations, (Rebuffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2022)), where visual- ization can aid analysis and understanding of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' GLTR (Gehrmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', 2019) colors the background of tokens based on their model prob- ability to visualize model surprisal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', red and purple indicating rather unexpected tokens with low probabilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' For instance, low overall sur- prisal of a given text indicates that it was either generated by the respective model or was part of its training corpus (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' The interpretive color overlay on the output (a) aids text skimming to identify unexpected phrases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' provides interactions, such as (b) mouse-over on the prior word shows the top subsequent words the model was expected to generate, and (c) click e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=', to regenerate from this point forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' This visualization can zoom out so that the words are no longer visible, while colored pixels remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Markup with zoom and regeneration can be used at a macro level to skim large generated texts to facilitate editing workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Furthermore, interactions with the generated text can be used to review the training data to under- stand the origins of unexpected text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Figure 11 shows an example based on a document that was generated by another GPT-2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Most of the highlights are in green and yellow, indicating that the text was indeed automatically generated by a similar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' One sentence ends with two peaks of rock and silver snow and the word silver has a red background, corresponding to a low probabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' After selecting silver snow, a search against the training data can create a clustering visualiza- tion showing search results grouped by similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While silver snow is uncommon, two large clusters indicate its use in a Nintendo game and the name Figure 11: Example NLG markup indicating unex- pected words by color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' of an energy drink, thus explaining to the human the origin and context of the phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' top k count frac(p) histogram top 10 entropy(p) histogram 400 200 90 350 180 80 160 300 70 140 60 250 120 50 200 100 40 150 80 60 30 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='40 20 50 20 10 L0 LO + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='492 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='65 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='4 Top K Frac P Colors (top k): 10 100 1000 In ashocking finding,scientistdiscovered aherd ofunicornslivingin a remote,previouslyunexplored valley,inthe Andes Mountains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Even more surprising to the researchers was the fact that the unicorns spoke perfect English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=" The scientist named the population, after their distinctive horn,Ovid'sUnicorn." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='These four-horned, silver-white unicorns werepreviouslyunknown to science Now, after almost two centuries, the mystery of what sparked this odd phenomenon is finally solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Jorge Perez,an evolutionary biologist fromthe University of La Paz,andseveral companions,wereexploringthe Andes Mountains when they found a small valley,with no other animals or humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Pereznoticed that the valley hac what appeared to be a natural fountain, surrounded by two peaks of rock and silver snow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Pérez and the others then ventured further into the valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' "Bythe time we reachedthe top of one peak,the watei looked blue, with some crystals on top," said Pérez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Pérez and his friends were astonished to see the unicornherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Thesecreatures could be seen from the airwithout having to movetoomuch to see them-they were so close they couldtouchtheirhorns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While examining these bizarre creatures the scientists discovered that the creatures also spoke some fairly regular English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Pérez stated, "We can see, for example, that they have a common \'language, something like a dialect or dialectic: Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' Pérez believes that the unicorns may have originated in Argentina, where the animals were believed to be descendants of a lost race of people who lived there before the arrival of humans in those parts of South America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' While theirorigins are still unclear,some believe that perhapsthe creatures were created whena humanand a unicorn met eachother in a timebefore human civilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='Accordingto Perez,"InSouthAmerica,suchincidentsseemto be quite common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content='" However, Pérez also pointed out that it is likely that the only way of knowing for sure if unicorns are indeed the descendants of a lost alien race is through DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} +page_content=' "But they seem to be able to communicate in English quite well, which Ibelieve is a sign of evolution,or at least a changein social organization," said the scientist' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE3T4oBgHgl3EQfcwpV/content/2301.04528v1.pdf'} diff --git a/_NAyT4oBgHgl3EQfRfYM/vector_store/index.faiss 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0000000000000000000000000000000000000000..04198b0add7a331218fc5cb005301b82144d8adb --- /dev/null +++ b/_tE1T4oBgHgl3EQf8wWT/content/tmp_files/2301.03549v1.pdf.txt @@ -0,0 +1,6819 @@ +arXiv:2301.03549v1 [math.PR] 9 Jan 2023 +OPTIMAL LOWER BOUND ON EIGENVECTOR OVERLAPS FOR NON-HERMITIAN +RANDOM MATRICES +GIORGIO CIPOLLONI +Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA +LÁSZLÓ ERDŐS# AND JOSCHA HENHEIK# +IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria +DOMINIK SCHRÖDER∗ +ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland +Abstract. We consider large non-Hermitian N × N matrices with an additive independent, identically +distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance 1/N +completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum +Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the +Eigenstate Thermalisation Hypothesis, formulated originally by Deutsch [33] and proven for Wigner ma- +trices in [24], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an +optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This +quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the +eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly com- +putable Ginibre ensemble apart from the very recent upper bounds given in [7] and [43]. As a key tool, we +develop a new systematic decomposition of general observables in random matrix theory that governs the +size of products of resolvents with deterministic matrices in between. +1. Introduction +Traditional random matrix theory focuses on statistics of eigenvalues, where spectacular univer- +sality phenomena arise: the local spectral statistics tend to become universal as the dimension goes +to infinity with new distributions arising; most importantly the celebrated Wigner-Dyson-Mehta bulk +statistics and the Tracy-Widom edge statistics in the Hermitian spectrum and the Ginibre statistics in the +non-Hermitian spectrum. More recently eigenvectors of Hermitian ensembles received considerable +attention. They also become universal, albeit in a more conventional way: they tend to be entirely ran- +domised, i.e. Haar distributed [16, 17, 47, 11, 27, 29, 10]. In this paper we study two related questions: +how do eigenvectors and singular vectors of a typical non-Hermitian random matrix in high dimension +look like? To answer them, we introduce a new decomposition of general observables that identifies +correlations of the Hermitised resolvents as entire matrices at different spectral parameters. This cap- +tures correlations of the singular well beyond correlations of traces of resolvents that govern only the +E-mail addresses: gc4233@princeton.edu, lerdos@ist.ac.at, joscha.henheik@ist.ac.at, dschroeder@ethz.ch. +Date: January 10, 2023. +2020 Mathematics Subject Classification. 60B20, 15B52, 62F22. +Key words and phrases. Eigenvalue condition number, Non-Hermitian perturbation theory, Quantum unique ergodicity. +#Supported by ERC Advanced Grant “RMTBeyond” No. 101020331. +∗Supported by the SNSF Ambizione Grant PZ00P2_209089. +1 + +2 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +singular values. Somewhat surprisingly, we are then able to transfer information on singular vectors to +the non-Hermitian eigenvectors. +1.1. Non-Hermitian eigenvector overlaps. To be specific, we consider non-Hermitian N × N ma- +trices of the form Λ + X, where Λ is an arbitrary deterministic matrix and X is random. We assume +that the norm of Λ is bounded independently of N and X has independent, identically distributed +(i.i.d.) centred matrix elements with variance E∣xij∣2 = +1 +N with some further moment conditions. +This normalisation guarantees that ∥X∥ ≤ 2 + o(1) and the spectrum of X lies essentially in the unit +disk (circular law) with very high probability, hence Λ and X remain of comparable size as N increases. +Note that X perturbs each matrix elements of Λ by a small random amount of order 1/ +√ +N, however +the spectra of Λ and Λ + X substantially differ. +The analysis of non-Hermitian random matrices is typically much harder than that of the Hermit- +ian ones. Non-Hermitian matrices have two different sets of spectral data: eigenvalues/vectors and +singular values/vectors which cannot be directly related. In particular, the study of singular vectors +and eigenvectors substantially differ: while singular vectors can still be understood from a Hermitian +theory, there is no such route for eigenvectors. Unlike for non-Hermitian eigenvalues, where Girko’s +formula translates their linear statistics into a Hermitian problem, no similar "Hermitisation" relation +is known for non-Hermitian eigenvectors. Furthermore, left and right eigenvectors differ and their rela- +tion is very delicate. Assuming that each eigenvalue µi of Λ+X is simple, we denote the corresponding +left and right eigenvectors by li, ri, i.e. +(Λ + X)ri = µiri , +lt +i(Λ + X) = µilt +i , +under the standard bi-orthogonality relation ⟨¯lj,ri⟩ = lt +jri = δi,j. Note that this relation leaves a large +freedomin choosing the normalisationof each eigenvector. The key invariant quantity is the eigenvector +overlap +Oij ∶= ⟨rj,ri⟩⟨lj,li⟩, +which emerges in many problems where non-Hermitian eigenvectors are concerned, see e.g. [3, 19, 20, +8, 13, 38]. Two prominent examples are +(i) in numerical linear algebra; where √Oii is the eigenvalue condition number determining how +fast µi moves under small perturbation in the worst case using the formula +√ +Oii = lim +t→0 sup {∣µi(Λ + X + tE) − µi(Λ + X) +t +∣ ∶ E ∈ CN×N,∥E∥ = 1} +(1.1) +(see, e.g. [7]); +(ii) in the theory of the Dyson Brownian motion for non-Hermitian matrices; where Oij gives the +correlation of the martingale increments for the stochastic evolution of the eigenvalues µi and +µj as the matrix evolves by the natural Ornstein-Uhlenbeck flow (see [40], [13, Appendix A]). +The main result of this paper is an almost optimal lower bound of order N on the diagonal over- +lap Oii, with very high probability. In the context of numerical linear algebra this means that non- +Hermitian eigenvalues of Λ + X still move at a speed of order +√ +N under the "worst" perturbation +E in (1.1), despite having added a random smoothing component X to Λ. Note that in numerics one +typically views the random smoothing as a tool to reduce the overlap of Λ in order to enhance the +stability of its eigenvalues; our result shows a natural limitation for such reduction. Complementary +upper bounds on Oii have recently been proven in [7] and [43]. These hold only in expectation sense, +as Oii has a fat-tail, and they are off by a factor N. We remark, however, that N is the most relevant +parameter of the problem only from our random matrix theory point of view. Works motivated by +numerical analysis, such as [7, 43] and references therein, often focus on tracking the γ-dependence for +the problem Λ + γX in the small noise regime γ ≪ 1 in order to reduce the effect of the random +perturbation. In this setup the non-optimality of the N-power may be considered less relevant1. +In the context of the Dyson Brownian motion, our lower bound on Oii implies a diffusive lower +bound on the eigenvalues of the Ornstein-Uhlenbeck (OU) matrix flow, generalizing the analogous +1As long as γ is N independent, one may set γ = 1 by a simple rescaling so we refrain from carrying this extra factor in the +current paper. We remark that our methods would allow to trace the polynomial γ-dependence in all our main estimates as well, +albeit not with an optimal power. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +3 +result of Bourgade and Dubach [13, Corollary 1.6] from Ginibre ensemble to arbitrary i.i.d. ensemble +(see (2.14) later). +1.2. Thermalisation of singular vectors. The key step to our lower bound on Oii is a thermalisation +result on the singular vectors that is of independent interest. Namely, we show that singular vectors of +Λ + X are fully randomised in the large N limit in the sense that their quadratic forms with arbitrary +test matrices have a deterministic limit with an optimal N −1/2 speed of convergence. This holds with +very high probability which enables us to make such statement for matrices of the form (Λ − z) + X +simultaneously for any shift parameterz, even for random ones. We will use this for z = µ, an eigenvalue +of Λ + X. This allows us to gain access to eigenvectors of Λ + X, by noticing that singular vectors and +eigenvectors are unrelated in general with an obvious exception: if µ is an eigenvalue of Λ + X, then +any vector in the kernel of Λ + X − µ is an eigenvector of Λ + X with eigenvalue µ, and a singular +vector of Λ + X − µ with singular value 0. Hence high probability statements for singular vectors can +be converted into similar statements for eigenvectors – this key idea may be viewed as the eigenvector +version of the transfer principle between eigenvalues and singular values encoded in Girko’s formula. +Our thermalisation result for singular vectors may be viewed as the non-Hermitian analogue of +the Quantum Unique Ergodicity (QUE) for Hermitian Wigner matrices proven in [24]. We now briefly +explain the QUE phenomenon and its physics background in the simplest Hermitian context before we +consider the singular vectors of Λ+X. In fact, via a standard Hermitisation procedure we will turn the +singular vector problem to a Hermitian eigenvector problem. +For Hermitian random matrices H, that can be considered as the Hamilton operator of a disordered +quantum system, a major motivation comes from physics, where the randomisation of the eigenvectors +is interpreted as a thermalisation effect. The Eigenstate Thermalisation Hypothesis (ETH) by Deutsch [33] +and Srednicki [50] (see also [32, 34]) asserts that any deterministic Hermitian matrix A (observable), be- +comes essentially diagonal in the eigenbasis of a "sufficiently chaotic" Hamiltonian, where chaos may +come from an additional randomness or from the ergodicity of the underlying classical dynamics. In +other words, +⟨ui,Auj⟩ − δij⟨⟨A⟩⟩i → 0, +as N → ∞ , +(1.2) +where {ui} is a orthonormal eigenbasis of H and the deterministic "averaged" coefficient ⟨⟨A⟩⟩i is to +be computed from the statistics of H. +In the mathematicsliterature the same problem is known as the Quantum (Unique) Ergodicity, orig- +inally formulated for the Laplace-Beltrami operator on surfaces with ergodic geodesic flow, see [49, 30, +56], on regular graphs [5] and on special arithmetic surfaces [48, 18, 46, 51]. In [24] we proved QUE in the +strongest form with an optimal speed of convergence for the eigenvectors of Wigner matrices that, by +E. Wigner’s vision, can be viewed as the "most random" Hamiltonian. In this case, the diagonal limit +⟨⟨A⟩⟩i in (1.2) is independent of i and given by the normalised trace ⟨A⟩ ∶= +1 +N TrA. In fact, in subse- +quent papers [27, 29] (see also [11]) even the normal fluctuation of +√ +N[⟨ui,Aui⟩ − ⟨A⟩] was proven, +followed by the proof of joint Gaussianity of finite many overlaps in [10]. Previously QUE results were +proven for rank one observables (see [44, 53] under four moment matching and [16] in general) and finite +rank observables [47], see also [9] for deformed Wigner matrices and [17] for band matrices. The proofs +crucially used that H is Hermitian, heavily relying on sophisticated Hermitian techniques (such as local +laws and Dyson Brownian Motion) developed in the last decade for eigenvalue universality questions. +Back to our non-Hermitian context, we consider the singular vectors {ui,vi}N +i=1 of Λ + X, +(X + Λ)(X + Λ)∗ui = σ2 +i ui , +(X + Λ)∗(X + Λ)vi = σ2 +i vi , +belonging to the singular value σi. We view them as the two N-dimensional components of the eigen- +vectors wi = (ui,vi) of the 2N-dimensional Hermitisation of Λ + X, defined as +H = HΛ ∶= W + ˆΛ , +W ∶= ( 0 +X +X∗ +0 ) , +ˆΛ ∶= ( 0 +Λ +Λ∗ +0) . +(1.3) +In particular, from the overlaps ⟨wi,Awj⟩ of eigenvectors for the Hermitised problem with a gen- +eral (2N) × (2N) matrix A one may read off all the singular vector overlaps of the form ⟨ui,Buj⟩, + +4 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +⟨vi,Bvj⟩ and ⟨ui,Bvj⟩ with any N × N matrix B. Therefore our goal is to show the general ther- +malisation phenomenon, the convergence of ⟨wi,Awj⟩ (cf. (1.2)), for the Hermitised matrix HΛ thus +generalizing the ETH proven in [24] beyond Wigner matrices and with an additional arbitrary matrix +Λ. Unlike in the Wigner case, the limit ⟨⟨A⟩⟩i genuinely depends on the index i and part of the task is to +determine its precise form. Note that due to the large zero blocks, W has about half as many random +degrees of freedom as a Wigner matrix of the same dimension has, moreover the block structure gives +rise to potential instabilities, thus the ETH for HΛ is considerably more involved than for Wigner ma- +trices. In the next section we explain the main new method of this paper that systematically handles all +these instabilities. +1.3. Structural decomposition of observables. We introduce a new concept for splitting general +observables into "regular" and "singular" components; where the singular component gives the leading +contribution and the regular component is estimated. In the case of Wigner matrices H in [24, 25] we +used the decomposition A = ⟨A⟩ + ˚ +A, where the traceless part of A, ˚ +A ∶= A − ⟨A⟩, is the regular +component and the projection2 of A onto the one dimensional space spanned by the identity matrix +is the singular component. This gave rise to the following decomposition of resolvent G = G(w) = +(H − w)−1 for any w ∈ C ∖ R: +⟨GA⟩ = m⟨A⟩ + ⟨A⟩⟨G − m⟩ + ⟨G˚ +A⟩, +(1.4) +where m = m(w) is the Stieltjes transform of the semicircle law. The second term in (1.4) is asymptot- +ically Gaussian of size ⟨G − m⟩ ∼ (Nη)−1 [41] and the last term is also Gaussian, but of much smaller +size ⟨G˚ +A⟩ ∼ ⟨˚ +A˚ +A∗⟩1/2/(Nη1/2) in the interesting regime of small η ∶= ∣Imw∣ ≪ 1 [25]. +Similar decomposition governs the traces of longer resolvent chains of Wigner matrices,for example +⟨GAG∗B⟩ = ⟨GG∗⟩ = 1 +η ⟨ImG⟩ ∼ 1 +η ≫ 1 +if A = B = I, i.e. both observable matrices are purely singular, while for regular (and bounded) observ- +ables A = ˚ +A, B = ˚ +B we have +⟨GAG∗B⟩ ∼ 1. +(1.5) +Both examples indicate the √η-rule (see (3.15) and Remark 4.5 later), informally asserting that each regu- +lar observable renders the size of a resolvent chain smaller by a factor √η than its singular counterpart. +In [28, 29] we obtained the deterministic leading terms and optimal error estimates on the fluctuation +for resolvent chains of arbitrary length +⟨G(w1)A1G(w2)A2 ...⟩ +(1.6) +with arbitrary observables in between. The answer followed the √η-rule hence it heavily depended on +the Ai = ⟨Ai⟩ + ˚ +Ai decomposition for each observable. +In particular, in order to estimate ⟨ui,Auj⟩ − δij⟨A⟩ = ⟨ui, ˚ +Auj⟩ for ETH in (1.2), we had +N∣⟨ui, ˚ +Auj⟩∣2 ≲ ⟨ImG(w1)˚ +AImG(w2)˚ +A⟩ ≲ 1, +where we first used spectral decompositionof both G’s and then used a version of (1.5). Here the spectral +parameters wk = ek+iη are chosen such that e1 and e2 be close to the eigenvalues corresponding to ui +and uj, respectively, and η ∼ N −1 in order to resolve the spectrum on the fine scale of the individual +eigenvalues.3 +The key point in all these analyses for Wigner matrices was that the regular/singular concept was +independent of the spectral parameter: the same universal decomposition into tracial and traceless parts +worked in every instance along the proofs. One consequence is the i-independence of the limiting +overlap ⟨⟨A⟩⟩i ∶= ⟨A⟩ in (1.2).4 +2We equip the space of matrices with the usual normalised Hilbert-Schmidt scalar product, ⟨A, B⟩ ∶= +1 +N Tr A∗B = ⟨A∗B⟩. +3Strictly speaking we used η = N −1+ξ with any small ξ > 0, and all estimates held up to an N ξ factor but we ignore these +technicalities in the introduction. +4A quick direct way to see this independence is the special case of Gaussian Wigner matrices (GUE or GOE), where the eigen- +vectors are Haar distributed, independently of their eigenvalue. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +5 +For more complicated ensembles, like HΛ in (1.3), especially if an arbitrary matrix Λ is involved, +the correct decomposition depends on the location in the spectrum of H where we work. To guess it, +first we recall the single resolvent local law (Theorem 2.6) for the resolvent G = GΛ(w) = (HΛ − w)−1, +asserting that ⟨GA⟩ ≈ ⟨MA⟩, where M = M Λ(w) solves a nonlinear deterministic equation, the +Matrix Dyson Equation (MDE), see (2.19) later. Then a heuristic calculation (see Appendix D.1) shows +that for w = e + iη ∈ C+ we have +E ∣⟨(G − M)A⟩∣ +2 ≈ +∣⟨ImMA⟩∣ +2 +(Nη)2 ++ +∣⟨ImMAE−⟩∣ +2 +N 2η(∣e∣ + η) + O( +1 +N 2η ) , +E− ∶= (1 +0 +0 +−1) , +(1.7) +indicating that the singular component of A is two dimensional, depends on w, and for any A orthogonal +to the two singular directions ImM and E−Im M the size of ⟨(G − M)A⟩ is smaller by a factor √η. +The first singular direction is always present. The second singular direction is a consequence of the +block structure of H and it is manifested only for w near the imaginary axis. For energies ∣e∣ ∼ 1, only +the first singular direction, namely the one involving ImM plays a role. +What about longer chains (1.6)? Each matrix Ai is sandwiched between two resolvents with different +spectral parameters wi, wi+1. We find that the correct decomposition of any A between two resolvents +in a chain ... G(w)AG(w′)... depends only on w,w′ and it has the form +A = ⟨V+,A⟩U+ + ⟨V−,A⟩U− + ˚ +A , +V± = V w,w′ +± +, +˚ +A = ˚ +Aw,w′ , +(1.8) +where the first two terms form the singular component of A, and ˚ +A, defined by this equation, is the +regular component. We will establish that both V+ and V− are the right eigenvectors of a certain stability +operator B acting on C2N×2N that corresponds to the Dyson equation. For example, if Imw and Imw′ +have opposite signs then V+ is the right eigenvector of +B[⋅] = 1 − M( ¯w)S[⋅]M( ¯w′), +where S is covariance operator for the matrix W in (1.3) (see (2.20)). V± with other sign combinations are +defined very similarly (in Appendix D.3 we present all cases). In particular, the special directions ImM +and E−ImM that we found by direct variance calculation in (1.7) emerge canonically as eigenvectors +of a certain stability operator! Similar variance calculation for longer chains would reveal the same +consistency: the variance of the chain (1.6) is the smallest if each Ai is regular with respect to the two +neighboring spectral parameters wi,wi+1. +Note that the choice of V± is basically dictated by variance calculations like (1.7). However, the +matrices U± in (1.8) can still be chosen freely up to their linear independence and the normalisation re- +quirement ⟨Vσ,Uτ⟩ = δσ,τ. The latter guarantees that the sum of the singular terms in (1.8) is actually a +(non-orthogonal) projection ∣U+⟩⟨V+∣ + ∣U−⟩⟨V−∣ acting on A. Since V± are the right eigenvectors of a +stability operator, one may be tempted to choose U± as certain left eigenvectors but we did not find this +guiding principle helpful. Instead, we use this freedom to simplify the calculation of the singular terms. +Substituting the singular part of A into ... G(w)AG(w′)..., we need to compute G(w)U±G(w′) +and quite pragmatically we choose U± such that the resolvent identity could be applied and thus re- +duce the length of the chain. Thanks to the spectral symmetry of H = HΛ, for its resolvent we have +E−G(−w)E− = −G(w), and we find that U+ = I, U− = E− do the job, which accidentally coincide +with the left eigenvectors of the stability operator for the special case of i.i.d. matrices. +In Appendix D.3 we present the canonical choices of V± and U± in a more general situation and +explain at which stage of the proof their correct choice emerges. In our current application only V± are +nontrivial (in particular energy dependent), while U± are very simple. This is due to the fact that the +chain (1.6) consists of resolvents of the same operator. In more general problemsone maytake resolvents +with two different Λ’s in the chain, in which case U± are also nontrivial. +This decomposition scheme is the really novel ingredient of our proofs. Several other tools we use, +such as recursive Dyson equations, hierarchy of master inequalities and reduction inequalities have been +introduced before (especially in our related works on Wigner matrices [24, 25]), but the dependence of +the decomposition on the spectralparametersin the current setup requires quite different new estimates +along the arguments. We informally explain the prototype of such an estimate at the beginning of +Section 4.1. + +6 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +1.4. Notations. We define the 2N × 2N matrices E± ∶= E1 ± E2, where +E1 ∶= (1 +0 +0 +0) +and +E2 ∶= (0 +0 +0 +1) . +Each entry of the matrix is understood as a multiple of the N × N–identity. By ⌈⋅⌉, ⌊⋅⌋ we denote the +upper and lower integer part, respectively, i.e. for x ∈ R we define ⌈x⌉ ∶= min{m ∈ Z∶m ≥ x} and +⌊x⌋ ∶= max{m ∈ Z∶m ≤ x}. We denote [k] ∶= {1,...,k} for k ∈ N and ⟨A⟩ ∶= d−1Tr(A), d ∈ N, is +the normalised trace of a d × d-matrix. For positive quantities A,B we write A ≲ B resp. A ≳ B and +mean that A ≤ CB resp. A ≥ cB for some N-independent constants c,C > 0. We denote vectors by +bold-faced lower case Roman letters x,y ∈ C2N, for some N ∈ N, and define +⟨x,y⟩ ∶= ∑ +i +¯xiyi , +Axy ∶= ⟨x,Ay⟩. +Matrix entries are indexed by lower case Roman letters a,b,c,... from the beginning of the alpha- +bet and unrestricted sums over a,b,c,... are always understood to be over {1,...,N,N + 1,...,2N}. +Analogously, unrestricted sums over lower case Roman letters i,j,k,... from the middle of the alphabet +are always understood to be over {−N,...,−1,1,..., N}. Finally, the lower case Greek letters σ and τ +as indices indicate a sign, i.e. σ,τ ∈ {+,−}, and unrestricted sums over σ,τ are understood to be over +{+,−}. +We will use the concept of ‘with very high probability’, meaning that any fixed D > 0, the probability +of an N-dependent event is bigger than 1 − N −D for all N ≥ N0(D). Also, we will use the conven- +tion that ξ > 0 denotes an arbitrarily small constant, independent of N. Moreover, we introduce the +common notion of stochastic domination (see, e.g., [35]): For two families +X = (X(N)(u) ∣ N ∈ N,u ∈ U (N)) +and +Y = (Y (N)(u) ∣ N ∈ N,u ∈ U (N)) +of non-negative random variables indexed by N, and possibly a parameter u, then we say that X is +stochastically dominated by Y , if for all ε,D > 0 we have +sup +u∈U(N) P[X(N)(u) > N ǫY (N)(u)] ≤ N −D +for large enough N ≥ N0(ǫ,D). In this case we write X ≺ Y . If for some complex family of random +variables we have ∣X∣ ≺ Y , we also write X = O≺(Y ). +Acknowledgement: The authors are grateful to Oleksii Kolupaiev for valuable discussions, especially +about the choice of contours in Lemma 5.1. +2. Main results +We consider real or complex i.i.d. matrices X , i.e. N × N matrices whose entries are independent +and identically distributed as xab +d= N −1/2χ for some real or complex random variable χ satisfying the +following assumptions: +Assumption 2.1. We assume that Eχ = 0 and E∣χ∣2 = 1. Furthermore, we assume the existence of high +moments, i.e., that there exist constants Cp > 0, for any p ∈ N, such that +E ∣χ∣p ≤ Cp . +Additionally, in the complex case, we assume that E χ2 = 0. +For definiteness, in the sequel we perform our entire analysis for the complex case; the real case +being completely analogous and hence omitted. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +7 +2.1. Non-Hermitian singular vectors and eigenvectors. Fix a deterministic matrix Λ ∈ CN×N, +with N-independent norm bound, ∥Λ∥ ≲ 1. Let {σi}i∈[N] be the singular values of X + Λ with +corresponding (normalised) left- and right-singular vectors {ui}i∈[N] and {vi}i∈[N], respectively, i.e. +(X + Λ)vi = σiui +and +(X + Λ)∗ui = σivi . +(2.1) +All these objects naturally depend on Λ, but we will omit this fact from the notation. +Let νi, i ∈ [N], be the increasingly ordered singular values of Λ. Define the Hermitisation of Λ as +ˆΛ ∶= ( 0 +Λ +Λ∗ +0). +(2.2) +Due to its block structure, the spectrum of ˆΛ is symmetric with respect to zero, with eigenvalues +{νi}0≠∣i∣≤N such that ν−i = −νi for all i ∈ [N]. The empirical density of states of ˆΛ is denoted by +µˆΛ ∶= +1 +2N +∑ +0≠∣i∣≤N +δνi . +Let µsc be the Wigner semicircle distribution with density ρsc(x) ∶= (2π)−1√ +[4 − x2]+, where +[⋯]+ is the positive part of a real number. Define the free additive convolution +µ = µΛ ∶= µsc ⊞ µˆΛ, +(2.3) +which is a probability distribution on R. We now briefly recall basic facts about the free convolution +with the semicircle density (see, e.g. the classical paper by P. Biane [12]). Most conveniently µ may be +defined by inverting its Stieltjes transform +m(w) = mΛ(w) ∶= ∫R +µ(de) +e − w , +w ∈ C ∖ R, +where m satisfies the implicit equation +m(w) = ∫R +µˆΛ(de) +e − (w + m(w)) . +(2.4) +With the additional constraint Imm(w) ⋅ Im w > 0 this equation has a unique solution that is analytic +away from the real axis with m(w) = m(w). Since µˆΛ is symmetric to zero with bounded support, +µ is also symmetric with support bounded independently of N. Moreover µ is absolutely continuous +with respect to Lebesgue measure with density denoted by ρ = ρΛ. The density ρ may be obtained5 as +the boundary value of Im m at any e on the real line, i.e. +ρ(e) ∶= lim +η↓0 ρ(e + iη), +ρ(w) ∶= 1 +π ∣Imm(w)∣. +(2.5) +Infactm itself hasa continuousextensionto the realaxisfrom the upperhalf plane m(e) ∶= limη↓0 m(e+ +iη). Proving the existence of these limits is standard from (2.4). +Next, for any (small) κ > 0, we define the κ-bulk of the density ρ as +Bκ = BΛ +κ ∶= {x ∈ R ∶ ρ(x) ≥ κ1/3} +(2.6) +which is symmetric to the origin. Finally, we denote a (modified) ith quantile of the density ρ by γi, i.e. +i + N +2N += ∫ +γi +−∞ ρ(e)de , +∣i∣ ≤ N , +(2.7) +and we immediately conclude by symmetry of ρ that γi = −γ−i for every ∣i∣ ≤ N. +Our first main result establishes the thermalisation of singular vectors of X + Λ in the bulk, i.e. for +indices i,j with quantiles γi,γj uniformly in the bulk of the density ρ. +5For orientation of the reader we mention that ρ is the deterministic approximation, the so-called self-consistent density of states +(scDos), for the empirical eigenvalue density of the Hermitisation of X +Λ. This connection will be explainedin the next Section 2.2. + +8 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Theorem 2.2. (Thermalisation of Singular Vectors) +Fix a bounded Λ ∈ CN×N and κ > 0 independent of N. Let {ui}i∈[N] and {vi}i∈[N] be the (normalised) +left- and right-singular vectors of X + Λ, respectively, where X is an i.i.d. matrix satisfying Assumption 2.1. +Then, for any deterministic matrix B ∈ CN×N with ∥B∥ ≲ 1 it holds that6 +max +i,j +�������������� +⟨ui,Buj⟩ − δj,i +⟨Im [ +γj+m(γj ) +ΛΛ∗−(γj +m(γj ))2 ] B⟩ +πρ(γj) +�������������� +≺ +1 +√ +N +, +(2.8a) +max +i,j +�������������� +⟨vi,Bvj⟩ − δj,i +⟨Im [ +γj+m(γj ) +Λ∗Λ−(γj +m(γj ))2 ] B⟩ +πρ(γj) +�������������� +≺ +1 +√ +N +, +(2.8b) +max +i,j +�������������� +⟨ui,Bvj⟩ − δj,i +⟨Im [Λ∗(ΛΛ∗ − (γj + m(γj))2) +−1] B⟩ +πρ(γj) +�������������� +≺ +1 +√ +N +, +(2.8c) +where the maximum is taken over all i,j ∈ [N] such that the quantiles γi,γj ∈ Bκ are in the κ-bulk of the +density ρ. +The thermalisation of singular vectors will be a simple corollary to the Eigenstate Thermalisation +Hypothesis (ETH) for the Hermitisation HΛ of X + Λ, which is formulated in Theorem 2.7 below. The +proof of Theorem 2.2 will be given in Section 3. +Our second main result concerns the bi-orthonormal left and right eigenvectors {li}i∈[N] and +{ri}i∈[N], respectively, of X + Λ, with corresponding eigenvalues {µi}i∈[N], i.e. +(X + Λ)ri = µiri , +lt +i(X + Λ) = µilt +i , +(2.9) +where t denotes the transpose of a vector. More precisely, the following theorem provides a lower +bound on the diagonal part of the overlaps matrix +Oij ∶= ⟨rj,ri⟩⟨lj,li⟩, +(2.10) +defined subject to the standard normalisation +⟨¯lj,ri⟩ = lt +jri = δi,j . +(2.11) +We restrict our results to eigenvalues µi in the bulk of X + Λ in the following sense. +Definition 2.3. We say that z ∈ C is in the bulk of X + Λ if and only if there exists an N-independent +κ > 0 for which +0 ∈ BΛ−z +κ += {x ∈ R ∶ ρΛ−z(x) ≥ κ1/3}. +There is no simple characterisation of the bulk of X + Λ in terms of the spectrum of Λ. However, +taking the imaginary part of (2.4) at w = 0 + i0 shows that 0 ∈ BΛ−z +κ +is equivalent to +1 +N +N +∑ +i=1 +1 +νi(Λ − z)2 + κ2/3 ≥ 1, +where νi(Λ − z) are the singular values of Λ − z. +Theorem 2.4. Consider X + Λ, with Λ being a deterministic matrix as in (2.2) and with X being an +i.i.d. matrix satisfying Assumption 2.1 and the single-entry distribution χ have a density with respect to the +Lebesgue measure. Then +Oii ≻ N , +(2.12) +where the index i ∈ [N] is such that µi is in the bulk of X + Λ. +6The deterministic terms following the Kronecker symbol δj,i in (2.8) will be shown to be bounded in Appendix A. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +9 +In the introduction we already mentioned the consequence of this result on the sensitivity of an +eigenvalue of X+Λ under smallperturbations. Now we explain its other consequence on the diffusivity +of the Dyson-type eigenvalue dynamics. Let each entry of X = X(t) evolve as an independent complex +OU process, +dXij = dBij +√ +N +− 1 +2Xijdt, +where Bij are independent standard complex Brownian motions and the initial condition X(0) sat- +isfies Assumption 2.1. A direct calculation [13, Proposition A.1] shows that the eigenvalues µi = µi(t) +follow the Dyson-type stochastic dynamics +dµi = dMi − 1 +2µidt, +{µi(0)} = SpecX(0), +1 ≤ i ≤ N, +(2.13) +where the martingales Mi have brackets ⟨Mi,Mj⟩ = 0 and d⟨Mi,Mj⟩t = +1 +N Oij(t)dt. In particular, +we immediately obtain, for any ǫ > 0 that +E[∣µi(t) − µi(0)∣21(µi(0) ∈ Bκ)] ≥ tN −ǫ +(2.14) +up to some time t ≤ T (κ), where Bκ denotes the κ-bulk of X(0). For Ginibre initial condition X(0) +(2.14) was established in [13, Corollary 1.6], we now generalise it to i.i.d. initial conditions. We remark +that (2.13) is similar to its Hermitian counterpart, the standard Dyson Brownian motion (DBM) on the +real line, with some notable differences. In particular, in the latter process the eigenvalues cannot cross +each other, hence they are quite rigid and confined to an interval of size essential 1/N, so they are not +diffusive beyond a time-scale 1/N. Along the evolution (2.13) the non-Hermitian eigenvalues still repel +each other (encoded in the typically negative off-diagonal overlaps, see [13, Theorem 1.3] in the Gaussian +case), but they still can pass by each other and not hindering the diffusive behavior (2.14). +Example 2.5. The most prominently and extensively studied [39, 6, 52, 14, 15, 55, 54, 21, 22, 23] deformation +is Λ = −z with z ∈ C, since it plays a key role in Girko’s formula [39] expressing linear statistics of non- +Hermitian eigenvalues of X in terms of the Hermitisation of X − z. In this case, the self-consistent equation +(2.4) reduces to the well-known cubic relation +− 1 +m = w + m − +∣z∣2 +w + m . +As a consequence, the deterministic terms in (2.8) drastically simplify (e.g., the fractions in (2.8a) and (2.8b) are +simply replaced by ⟨B⟩) and one also has explicit formulas for the bulk (2.6) in terms of solution of a cubic +equation. In particular, for ∣z∣ < 1 − ǫκ, the κ-bulk Bκ consists of a single interval, while for ∣z∣ ≥ 1 − ǫκ it +consists of two intervals, where ǫκ ∼ κ2/3. In the former case 0 ∈ Bκ. Consequently, Theorem 2.4 gives the +lower bound (2.12) for all the diagonal overlaps Oii of eigenvectors of X whose eigenvalue µi lies in a disk of +radius 1 − ǫ with some ǫ > 0 independent of N. +∼ κ2/3 +∼ κ +∼ κ1/3 +-2 +2 +Rew +ρ(Rew) +Figure 1. Depicted is the density ρ for the deformation Λ = −z with ∣z∣ slightly +less than one. On the horizontal axis, we indicated the two components of the bulk +Bκ. The distance between Bκ and a regular edge scales like κ2/3, while near an +(approximate) cusp the distance between the two components scales linearly (see +also (2.6) and (2.21)). + +10 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +In the next section we explain the key technical result behind our main theorems, the eigenstate +thermalisation for the Hermitisation of X + Λ. +2.2. Eigenstate Thermalisation Hypothesis for the Hermitisation of X + Λ. The key to access +the non-Hermitian singular vectors of X + Λ is to study its Hermitisation [39], which is defined as +H = HΛ ∶= ( +0 +X + Λ +(X + Λ)∗ +0 +) =∶ W + ˆΛ , +(2.15) +where ˆΛ∗ = ˆΛ was defined in (2.2) and can also be viewed as the matrix of expectation ˆΛ = EHΛ. +We denote by {wi}∣i∣≤N the (normalised) eigenvectors of H and by {λi}∣i∣≤N the corresponding +eigenvalues.7 By means of the singular value decomposition in (2.1), the eigenvalues and eigenvectors of +H are related to the singular values and singular vectors of X + Λ as follows: +wi = (ui,vi)t +and +λi = σi +for +i ∈ [N], +up to a normalisation, since now ∥ui∥2 = ∥vi∥2 = +1 +2. Moreover, the block structure of H induces +a symmetric spectrum around zero, i.e. λ−i = −λi for any i ∈ [N]. This symmetry for the eigen- +values is also reflected in the eigenvectors, which satisfy w−i = E−wi for any i ∈ [N]. By spectral +decomposition, this immediately shows the chiral symmetry +E−G(w) = −G(−w)E−, +with +E− = (1 +0 +0 +−1) , +(2.16) +for the resolvent G(w) = GΛ(w) ∶= (HΛ − w)−1, with spectral parameter w ∈ C ∖ R. We also have +⟨G(w)E−⟩ = 0 for any w since ⟨wi,E−wi⟩ = ∥ui∥2 − ∥vi∥2 = 0. +A basic feature of a very large class of random matrices is that their resolvent becomes approximately +deterministic in the large N limit, often even for any spectral parameter with ∣Imw∣ ≥ N −1+ǫ; these +statements are called local laws. In our case the deterministic approximation of the resolvent G(w) is +given by +M(w) = M Λ(w) ∶= ⎛ +⎝ +M11(w) +ΛM22(w) +w+m(w) +Λ∗M11(w) +w+m(w) +M22(w) +⎞ +⎠ ∈ C2N×2N , +w ∈ C ∖ R, +(2.17) +with each block being understood as a matrix in CN×N, where the diagonal entries are defined via +M11(w) ∶= +w + m(w) +ΛΛ∗ − (w + m(w))2 , +M22(w) ∶= +w + m(w) +Λ∗Λ − (w + m(w))2 . +(2.18) +Here we require m(w) = ⟨M(w)⟩, which is an implicit equation for the function m(w). Simple +calculation shows that this implicit equation is exactly (2.4). +To derive these formulas systematically, we recall that the deterministic approximation to G(w) is +obtained as the unique solution to the matrix Dyson equation (MDE) (extensively studied in [2, 4]). The +MDE corresponding to the random matrix H is given by +− +1 +M(w) = w − ˆΛ + S[M(w)] +(2.19) +under the constraint ImM(w) ⋅ Imw > 0, where Im M(w) ∶= +1 +2i[M(w) − (M(w))∗]. Here S[⋅], +the self-energy operator, is defined via +S[T ] ∶= ̃E( ̃ +H − EH)T ( ̃ +H − EH) +for any T ∈ C2N×2N, where ̃ +H denotes an independent copy of H. In our case we have +S[T ] = 2E1⟨T E2⟩ + 2⟨E1T ⟩E2 = ∑ +σ=± +σ⟨T Eσ⟩Eσ. +(2.20) +Using ⟨M11(w)⟩ = ⟨M22(w)⟩ that directly follows from (2.18), it is straightforward to check that +M(w) as defined in (2.17) satisfies the MDE (2.19). Since the MDE has a unique solution, we see that the +density ρ defined via free convolution in Section 2.1 coincides with the self-consistent density of states +7In the definition of the eigenvectors and eigenvalues, we omitted 0 in the set of indices, i.e. ∣i∣ ≤ N really means i ∈ +{−N, ..., −1, 1, ..., N}. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +11 +(scDos) corresponding to the MDE, defined as the boundary value of 1 +π ⟨ImM⟩ on the real axis in the +theory of MDE [2, 4]. +For the reader’s convenience in Appendix A.1 we will collect a few facts about M, in particular we +will show that it has a continuous extension as a matrix valued function to the real axis, i.e. the limit +M(e) ∶= limη↓0 M(e +iη) exists for any e ∈ R. This extends the similar result on its trace mentioned +in (2.5). Moreover, we will also show that for spectral parameters w ∈ C ∖ R with Rew ∈ Bκ, we have +∥M(w)∥ ≲ 1. Finally, we will also prove an important regularity property of the κ-bulk, namely that +dist(∂Bκ′,Bκ) ≥ c(κ − κ′) +(2.21) +for any small 0 < κ′ < κ and some N-independent constant c = c(∥Λ∥) > 0. In fact, for our proof it is +sufficient if c = c(κ,∥Λ∥), i.e. an additional κ dependence is allowed – in Appendix A.1 we will explain +that this weaker result is considerably easier to obtain (see Remark A.3). We will also show that Bκ is a +finite disjoint union of compact intervals; the number of these components depends only on κ and ∥Λ∥. +The above mentioned concentration of G around M is the content of the following single resolvent +local law, both in averaged and isotropic form, which we prove in Appendix B. +Theorem 2.6. (Single resolvent local law for the Hermitisation H) +Fix a bounded deterministic Λ ∈ CN×N and κ > 0 independent of N. Then, for any w ∈ C ∖ R with +∣w∣ ≤ N 100 and Rew ∈ Bκ, we have +∣⟨(G(w) − M(w))B⟩∣ ≺ +1 +Nη , +∣⟨x,(G(w) − M(w))y⟩∣ ≺ +1 +√Nη , +where η ∶= ∣Imw∣, for any bounded deterministic matrix ∥B∥ ≲ 1 and vectors ∥x∥,∥y∥ ≲ 1. +Our main result for the Hermitised random matrix H is the Eigenstate Thermalisation Hypothesis +(ETH), that in mathematical terms is the proof of an optimal convergence rate of the strong Quantum +Unique Ergodicity (QUE) for general observables A, uniformly in the bulk (2.6) of the spectrum of H, +i.e. in the bulk of the scDos ρ. +Theorem 2.7. (Eigenstate Thermalisation Hypothesis for the Hermitisation H) +Fix some bounded Λ ∈ CN×N and κ > 0 independent of N. Let {wi}∣i∣≤N be the orthogonal eigenvectors +of the Hermitisation H of X + Λ, where X is an i.i.d. matrix satisfying Assumption 2.1. Then, for any +deterministic matrix A ∈ C2N×2N with ∥A∥ ≲ 1 it holds that +max +i,j ∣⟨wi,Awj⟩ − δj,i ⟨ImM(γj)A⟩ +⟨ImM(γj)⟩ − δj,−i ⟨ImM(γj)E−A⟩ +⟨ImM(γj)⟩ +∣ ≺ +1 +√ +N +, +(2.22) +where the maximum is taken over all ∣i∣,∣j∣ ≤ N, such that the quantiles γi,γj ∈ Bκ defined in (2.7) are in +the bulk of the scDos ρ. +The main technical result underlying Theorem 2.7 is an averaged local law for two resolvents with +different spectral parameters, which we will formulate in Theorem 4.3 later. +Remark 2.8. Given the optimal bound (2.22), following a Dyson Brownian Motion (DBM) analysis similar +to [27, 29], it is possible to prove a CLT for single diagonal overlaps ⟨wi,Awi⟩. However, for the sake of +brevity, we do not present this argument here and defer the CLT analysis to future work. +In the following Section 3 we precisely define the regularisation and we will prove our main results +formulated above assuming the key technical Proposition 3.4. This proposition is obtained from a local +law, which we prove in Section 4. Local laws are proved by a hierarchy of master and reduction inequali- +ties, that are derived in Sections 5 and 6, respectively. Several technical and auxiliary results are deferred +to the appendices. +3. Proof of the main results +The key to understanding the eigenvector overlaps and showing our main results is an improved +bound on the averaged trace of two resolvents with regular (see Section 3.1 below) deterministic matrices +A1,A2 in between, i.e. for +⟨G(w1)A1G(w2)A2⟩. +(3.1) + +12 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Naively, for arbitrary A1,A2, estimating (3.1) via a trivial Schwarz inequality and Ward identity yields +the upper bound ∣⟨G(w1)A1G(w2)A2⟩∣ ≺ 1/η, where η ∶= minj ∣Imwj∣. However, this bound dras- +tically improves, whenever the matrices A1,A2 are regular, i.e. orthogonal to certain critical eigenvec- +tors V± of the associated two-body stability operators (A.2), which is denoted as Aj = ˚ +Aj; see (3.2) and +Definitions 3.1 and 4.1. In this case, in our key Proposition 3.4 we will show that +∣⟨G(w1)˚ +A1G(w2)˚ +A2⟩∣ ≺ 1 +even for very small η ∼ N −1+ǫ as a consequence of a more precise local law for (3.1), which we present in +Section 4. We find that (see Theorem 4.3 and Remark 4.5) both the size of its deterministic approximation +and the fluctuation around this mean heavily depend on whether (one or both of) the matrices A1,A2 +are regular, i.e. satisfy ⟨V±,Aj⟩ = 0, or not. +Therefore, the general rather structural regularizing decomposition (or regularisation) of a matrix A +shall be conducted as +A○ ≡ ˚ +A ∶= A − ⟨V+,A⟩U+ − ⟨V−,A⟩U− +(3.2) +for Uσ,Vσ ∈ C2N×2N satisfying ⟨Vσ,Uτ⟩ = δσ,τ and the normalisation ⟨Uσ,Uσ⟩ = 1, where recall +that ⟨R,T ⟩ ∶= ⟨R∗T ⟩ denotes the (normalised) Hilbert-Schmidt scalar product. The regularisation +map +(1 − Π) ∶ C2N×2N → C2N×2N , +A ↦ ˚ +A , +where Π is a two-dimensional (non-orthogonal) projection,8 is closely related to the built-in chiral sym- +metry (2.16) of our model. Indeed, for other ensembles without this special structure only one of the +terms ⟨Vσ,A⟩Uσ in (3.2) would be present. +As mentioned above, the matrices V± are determined as critical eigenvectors (with corresponding +small eigenvalue) of naturally associated two-body stability operators with their precise form worked +out in Appendix D and given in (D.17). However, for the directions U± there are a priori no further +constraints (apart from orthogonality and normalisation). Hence, as it turns out to be convenient for +our proofs, we will choose the matrices Uσ (in principle, even allowing for two different deformations +Λ1 ≠ Λ2) in such a way, that a resolvent identity +GΛ1(w1)UσGΛ2(w2) ≈ (GΛ1(w1) − GΛ2(σw2))Uσ , +(3.3) +can be applied (here, the symbol ‘≈’ neglects lower order terms). This is used to reduce the number of +resolvents in a chain. Note that, again due to the eminent chiral symmetry (2.16) for the resolvents, there +are in fact two matrices Uσ for which a resolvent identity (3.3) can be applied. +Although the regularisation (3.2) shall be motivated for arbitrary deformations Λ1,Λ2 in Appen- +dix D, we will henceforth choose a single bounded deformation Λ ∈ CN×N, which remains fixed with +the just mentioned exception in Appendix D. For a single deformation Λ, this restricts the matrices U± +satisfying (3.3) to be given by E±. +In case that the spectral parameters (w1,w2) appearing in (3.1) (with a single fixed deformation Λ) +are such that none of the eigenvectors of the stability operator is critical (cf. Appendix A), we consider +every matrix A as regular. The distinction between these two scenarios is regulated by cutoff functions +1± +δ introduced in (3.5) below. +3.1. Regular observables: A bound on (3.1). As already mentioned above, our main result for the +Hermitised random matrix, Theorem 2.7, shall be derived from a bound on (3.1), where we assume the +(real parts of the) spectral parameters w1,w2 to be in the bulk of the scDos ρ (recall (2.6)). +We now specify the concept of regularisation (3.2) to our setting. The eigenvectors V± will be com- +puted in Appendix D, the matrices U± are simply chose as E±. +Definition 3.1. (Regular observables) Given κ > 0, let9 +δ = δ(κ,∥Λ∥) > 0 +(3.4) +8The condition ⟨Vσ, Uτ ⟩ = δσ,τ guarantees that the regularisation is idempotent, i.e. ( ˚ +A)○ = ˚ +A and Π2 = Π. +9Note that the parameter δ > 0 is independent of the matrix size N. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +13 +be sufficiently small (to be chosen below, see (4.20)) and let w,w′ ∈ C ∖ R with Rew,Re w′ ∈ Bκ be +spectral parameters. Furthermore, we introduce the (symmetric) cutoff functions +1± +δ(w,w′) ∶= φδ(Rew ∓ Rew′) φδ(Imw) φδ(Imw′), +(3.5) +where 0 ≤ φδ ≤ 1 is a smooth symmetric bump function on R satisfying φδ(x) = 1 for ∣x∣ ≤ δ/2 and +φδ(x) = 0 for ∣x∣ ≥ δ. +(a) We define the (w,w′)-regular component or (w,w′)-regularisation ˚ +Aw,w′ of a matrix A as10 +˚ +Aw,w′ +∶= A − ∑ +τ=± +1τs +δ (w,w′) ⟨M(Rew + iIm w)AM(Rew′ + τiImw′)Eτs⟩ +⟨M(Re w + iIm w)EτsM(Rew′ + τiImw′)Eτs⟩Eτs , +(3.6) +where the relative sign of the imaginary parts is defined as +s ≡ sw,w′ ∶= −sgn(Imw Im w′). +(3.7) +(b) We say that A is (w,w′)-regular if and only if A = ˚ +Aw,w′. +The regularisation shall be revisited in Definition 4.1, where we tailor it to certain averaged (4.3) or +isotropic (4.4) resolvent chains. +Remark 3.2. We have several comments concerning the above definition. +(i) In case that at least one of the spectral parameters is away from the imaginary axis, say ∣Re w∣ > δ +w.l.o.g., then the regularisation in (3.6) contains at most one summand: If 1+ +δ(w,w′) = 1, i.e. Rew +is close to Rew′, then we have that +˚ +Aw,w′ +∶= A − ⟨M(w)AM(Rew′ + siImw′)⟩ +⟨M(w)M(Re w′ + siImw′)⟩ E+ , +whereas if 1− +δ(w,w′) = 1, i.e. if Rew is close to −Rew′, then we have that +˚ +Aw,w′ ∶= A − ⟨M(w)AE−M(−Rew′ + siIm w′)⟩ +⟨M(w)M(−Re w′ + siIm w′)⟩ +E− , +where we used that M(w)E− = −E−M(−w) as shown in Lemma A.1, ultimately as a consequence +of (2.16). +(ii) The cutoff functions in (3.5) satisfy the basic symmetry properties +1± +δ(w,w′) = 1± +δ( ¯w,w′) = 1± +δ(w, ¯w′) = 1± +δ( ¯w, ¯w′). +However, ˚ +A is not symmetric in its two spectral parameters, i.e. ˚ +Aw,w′ ≠ ˚ +Aw′,w in general +(iii) For spectral parameters satisfying 1± +δ(w,w′) > 0, it will be shown in Appendix A that the respective +denominators in (3.6) are bounded away from zero. In particular, the linear map A ↦ ˚ +A is bounded +with a bound depending only on δ and ∥Λ∥. +(iv) Whenever it holds that 1± +δ(w,w′) = 0 then also 1± +δ′(w,w′) = 0 for every 0 < δ′ < δ. Conversely, +whenever it holds that 1± +δ(w,w′) = 1 then also 1± +δ′(w,w′) = 1 for every 0 < δ < δ′. +(v) We point out that the notion of regularity implicitly depends on κ and δ and hence also on the (norm +of the) deformation Λ. +Moreover, the regularisation defined above satisfies the following elementary properties. The iden- +tities in (3.9) and (3.8) are immediate from the definition, the perturbative statements are proven in +Appendix A. +Lemma 3.3. Fix a bounded deterministic deformation Λ ∈ CN×N and let A ∈ C2N×2N be an arbitrary +bounded deterministic matrix. +10Putting the summation parameter τ at the second spectral parameter w′ (and not at w) is simply a free choice, which we made +here. More precisely, defining the regularisation as +˜˚ +Aw,w′ +∶= A − ∑ +τ=± +1τs +δ (w, w′) ⟨M(Re w + τiIm w)AM(Rew′ + iIm w′)Eτs⟩ +⟨M(Re w + τiImw)EτsM(Re w′ + iIm w′)Eτs⟩ Eτs +would equally work in our proofs (see Appendices A and D for details). + +14 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +(i) Let w,w′ ∈ C ∖ R with Re w,Rew′ ∈ Bκ. Then, we have the identities +(˚ +Aw,w′) +∗ = +˚ +(A∗) +¯ +w′, ¯ +w , +˚ +Aw,w′E− = +˚ +(AE−) +w,−w′ +, +E− ˚ +Aw,w′ = +˚ +(E−A) +−w,w′ +. +(3.8) +(ii) Moreover, by definition it holds that +˚ +Aw, ¯ +w′ += ˚ +Aw,w′ +, +(3.9) +and setting s ∶= −sgn(ImwImw′), we have the perturbative estimate11 +˚ +A ¯ +w,w′ = ˚ +Aw,w′ + O(∣w − s ¯w′∣ ∧ 1)Es + O(∣w + sw′∣ ∧ 1)E−s . +(3.10) +(iii) Let w1,w′ +1,w2,w′ +2 ∈ C∖R with Rew1,Re w′ +1,Re w2,Rew′ +2 ∈ Bκ as well as Imw1⋅Im w2 > +0 and Imw′ +1 ⋅ Imw′ +2 > 0 be spectral parameters. Then we have that +˚ +Aw2,w′ +1 = ˚ +Aw1,w′ +1 + O(∣w1 − w2∣ ∧ 1)E+ + O(∣w1 − w2∣ ∧ 1)E− , +(3.11) +˚ +Aw1,w′ +2 = ˚ +Aw1,w′ +1 + O(∣w′ +1 − w′ +2∣ ∧ 1)E+ + O(∣w′ +1 − w′ +2∣ ∧ 1)E− . +(3.12) +We can now state the bound on (3.1) for regular observables, which shall be proven in Section 4 as +an immediate corollary to a local for (3.1) given in Theorem 4.3 and the bound from Lemma 4.2. +Proposition 3.4. Fix a bounded deterministic Λ ∈ CN×N, ǫ > 0, κ > 0, and let w1,w2 ∈ C with +∣w1∣,∣w2∣ ≤ N 100, Rew1,Rew2 ∈ Bκ, and ∣Imw1∣,∣Imw2∣ ≥ N −1+ǫ. Moreover, let A1 ∈ C2N×2N +be a (w1,w2)-regular and A2 ∈ C2N×2N a (w2,w1)-regular deterministic matrix, both satisfying +∥A1∥,∥A2∥ ≲ 1. Then we have +∣⟨G(w1)˚ +Aw1,w2 +1 +G(w2)˚ +Aw2,w1 +2 +⟩∣ ≺ 1. +(3.13) +3.2. Estimating (3.1) for general observables. Armed with the correct regularisation, we can now +present a systematic analysis of ⟨G(w1)A1G(w2)A2⟩ from (3.1) for arbitrary bounded deterministic +matrices A1,A2. Decomposing A1,A2 according to Definition 3.1 as +A1 = ˚ +Aw1,w2 +1 ++ ⟨⟨A1⟩⟩+ +w1,w2E+ + ⟨⟨A1⟩⟩− +w1,w2E− , +A2 = ˚ +Aw2,w1 +2 ++ ⟨⟨A2⟩⟩+ +w2,w1E+ + ⟨⟨A2⟩⟩− +w2,w1E− , +(3.14) +(where ⟨⟨⋅⟩⟩σ +w,w′ can be read off as the coefficients in (3.6)) and plugging (3.14) into (3.1), we find that +⟨G(w1)A1G(w2)A2⟩ = ∑ +σ,τ +⟨⟨A1⟩⟩σ +w1,w2⟨⟨A2⟩⟩τ +w2,w1⟨G(w1)EσG(w2)Eτ⟩ ++ ∑ +σ +⟨⟨A1⟩⟩σ +w1,w2⟨G(w1)EσG(w2)˚ +Aw2,w1 +2 +⟩ ++ ∑ +τ +⟨⟨A2⟩⟩τ +w2,w1⟨G(w1)˚ +Aw1,w2 +1 +G(w2)Eτ⟩ ++ ⟨G(w1)˚ +Aw1,w2 +1 +G(w2)˚ +Aw2,w1 +2 +⟩. +(3.15) +Which terms in (3.15) are effectively present depends on the coefficients ⟨⟨⋅⟩⟩σ +w,w′, i.e. on the singular +components of A1,A2. For terms with nonzero coefficients the following rule of thumb applies. De- +noting η ∶= min (∣Imw1∣,∣Im w2∣) ≥ N −1+ǫ, the terms ⟨GEGE⟩ in the first line of (3.15) are bounded +by 1/η, the terms ⟨GEG˚ +A⟩ in the two middle lines of (3.15) are bounded by 1/√η, and ⟨G˚ +AG˚ +A⟩ in the +last line is of order one (Proposition 3.4). This is in perfect agreement with the √η-rule mentioned in +the Introduction (see also Remark 4.5 below). Some of these bounds are actually sharp for special values +of w1,w2, for example +⟨G(w)E+G( ¯w)E+⟩ = ⟨ImG(w)⟩ +η +∼ 1 +η , +or +⟨G(w)E−G(− ¯w)E−⟩ = −⟨ImG(w)⟩ +η +, +11Note that the asymmetry between (3.10) and (3.9) stems from the asymmetry imposed in the definition of the regularisation, +namely by placing the summation index τ in (3.6) at the second spectral parameter. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +15 +where we used the chiral symmetry (2.16). In fact, two terms with στ = −1 in the first line of (3.15) are +identically zero by applying the chiral symmetry, followed by the resolvent identity and ⟨GE−⟩ = 0. +For a middle term in (3.15) we have +⟨G(w)E+G( ¯w)˚ +A ¯ +w,w⟩ = 1 +η ⟨ImG(w)˚ +A ¯ +w,w⟩ ≲ 1 + +1 +Nη +1 +√η . +In the very last relation we treated ⟨G(w)˚ +A ¯ +w,w⟩ and ⟨G( ¯w)˚ +A ¯ +w,w⟩ separately. In both cases we first +used Lemma 3.3 to adjust the regularisation to ˚ +Aw,w and ˚ +A ¯ +w, ¯ +w, respectively, to match the new single- +resolvent setup and then we applied the corresponding single-resolvent local law with regular observ- +able (see Theorem 4.4 below). +Note that the most critical estimate concerns the last line of(3.15), i.e. the regular part for both observ- +able matrices. The bound (3.13) is obtained from a local law with two resolvents and two regular matrices, +while the first and the middle terms in (3.15) can be understood already from an improved local law for +one resolvent and one regular matrix (see Theorem 4.4 below) after applying resolvent identities and +adjusting the regularisation by Lemma 3.3. Furthermore, observe that the sizes of the first three lines +in (3.15) are sensitive to w1,w2 via the resolvent identities, for example +⟨G(w1)E+G(w2)E+⟩ = ⟨G(w1) − G(w2)⟩ +w1 − w2 +≲ +1 +∣w1 − w2∣, +or +⟨G(w1)E−G(w2)E−⟩ ≲ +1 +∣w1 + w2∣, +while the last line in (3.15) is typically order one. +Summarizing, the singular parts of ⟨G(w1)A1G(w2)A2⟩ can be explicitly computed (using single- +resolvent local laws) as explicit functions of w1,w2, while the regular part remains of order one. A +combinationof our decomposition(3.6), the perturbation formulasfrom Lemma 3.3, and our single- and +two-resolvent local laws together with their explicit deterministic terms from the subsequent Section 4 +provide an effective recipe to compute ⟨G(w1)A1G(w2)A2⟩ with high precision in all cases. We +refrain from formulating it as a comprehensive theorem due to the large number of cases. +3.3. Proof of the main results. We will first focus on the proof of Theorem 2.7 and turn to the proofs +of Theorem 2.2 and Theorem 2.4 afterwards. +3.3.1. Proof of Theorem 2.7. As a first step towards the proof of Theorem 2.7, we show that (2.22) indeed +follows from a bound similar to (3.13), where G is replaced by ImG. The proof of the following simple +lemma is given after completion of the proof of Theorem 2.7. +Lemma 3.5. Fix a bounded deterministic Λ ∈ CN×N, ǫ > 0, κ > 0, and let B ∈ C2N×2N. Then, for any +bulk indices ∣i∣,∣j∣ ≤ N, i.e. with γi,γj ∈ Bκ, and η ≥ N −1+ǫ, we have +N ∣⟨wi,Bwj⟩∣2 ≺ (Nη)2⟨ImG(γi + iη)BImG(γj + 2iη)B∗⟩. +(3.16) +The same bound still holds without the factor of two in (3.16). However, we chose to have it, in order +to ensure that the spectral parameters of the two resolvents are always forced to be different. +Proof of Theorem 2.7. Having Lemma 3.5 at hand, we are left with estimating the rhs. of (3.16) for +B = A − ⟨ImM(γj)A⟩ +⟨ImM(γj)⟩ E+ − ⟨ImM(γj)E−A⟩ +⟨ImM(γj)⟩ +E− +(3.17) +using Proposition3.4. Note that the two terms in (2.22) carrying a δ-symbol arise from the orthogonality +relations ⟨wi,E±wj⟩ = δj,±i, following from the spectral symmetry described around (2.16). +We now write out ImG(w) = (G(w) − G( ¯w))/(2i), such that (3.16) leaves us with four different +terms, each of which can be bounded individually. Since their treatment is completely analogous, we +focus on the exemplary term +⟨G(γi + iη)BG(γj − 2iη)B∗⟩ +(3.18) +with the deterministic matrix B being defined in (3.17). We rely on the following simple perturbative +lemma, which follows from Lemma 3.3 by invoking Lemma A.4. + +16 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Lemma 3.6. Using the notation introduced in (3.6), the matrix B ∈ C2N×2N from (3.17) satisfies +B = ˚ +Aγi+iη,γj −2iη + O(∣γi − γj∣ + η)E+ + O(∣γi + γj∣ + η)E− , +B∗ = +˚ +(A∗) +γj −2iη,γi+iη + O(∣γi − γj∣ + η)E+ + O(∣γi + γj∣ + η)E− . +(3.19) +Hence, plugging (3.19) into (3.18), we get a sum of several terms, which can all be estimated separately. +For the ‘leading term’, we use Proposition 3.4 to get that +∣⟨G(γi + iη)˚ +Aγi+iη,γj−2iηG(γj − 2iη) ˚ +(A∗) +γj−2iη,γi+iη⟩∣ ≺ 1. +Two further representative terms are given by +O(∣γi ∓ γj∣ + η) ⟨G(γi + iη)E±G(γj − 2iη)C⟩, +where C ∈ C2N×2N is some generic bounded matrix. Now, by using (2.16), these terms can be rewritten +as +O(∣γi ∓ γj∣ + η) ⟨G(γi + iη)G(±(γj − 2iη))E±C⟩. +Foreithersignchoice (due to the factortwo), we cannow employa simple resolventidentityG(w1)G(w2) = +[G(w1) − G(w2)]/(w1 − w2), leaving us with +O(∣γi − γj∣ + η) +(γi ∓ γj) + (1 ± 2)iη ⟨[G(γi + iη) − G(±(γj − 2iη))]C⟩, +which is surely stochastically dominatedby one by means of Theorem 2.6. Thus, collecting all the terms, +we find that ∣(3.18)∣ ≺ 1. +Finally, we choose η = N −1+ξ for an arbitrarily small ξ > 0, such that Lemma 3.5 with B as in (3.17) +yields Theorem 2.7. +□ +We conclude with giving a proof of Lemma 3.5. +Proof of Lemma 3.5. By spectral decomposition we write +⟨ImG(γi + iη)BImG(γj + 2iη)B∗⟩ = +1 +2N ∑ +k,l +2η2∣⟨wk,Bwl⟩∣2 +[(λk − γi)2 + η2][(λl − γj)2 + 4η2] +≳ +η2∣⟨wi,Bwj⟩∣2 +N[(λi − γi)2 + η2][(λj − γj)2 + 4η2] +≻ ∣⟨wi,Bwj⟩∣2 +Nη2 +, +which proves (3.16). We point out that in the last inequality we used rigidity of the eigenvalues [2, 36]: +∣λi − γi∣ ≺ 1 +N , +(3.20) +which holds for bulk indices as a standard consequence of the single-resolvent local law, Theorem 2.6. +□ +3.3.2. Proof of Theorem 2.2. The bounds in (2.8a), (2.8b), and (2.8c) follow from Theorem 2.7 by choosing +A = (B +0 +0 +0) , +A = (0 +0 +0 +B) , +and +A = (0 +0 +B +0) , +respectively, and invoking Lemma A.1. +□ + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +17 +3.3.3. Proof of Theorem 2.4. By the definition +Hz ∶= ( +0 +X + Λ − z +(X + Λ − z)∗ +0 +) +it follows that µ ∈ Spec(X + Λ) if and only if λµ +1 = 0. Here by λz +i we denoted the eigenvalues of Hz. +We remark that Λ is omitted by the notation since it is fixed throughout the proof. In particular, using +the bound for products of two resolvents and two regular matrices in (3.13), we will now prove the lower +bound in (2.12) for the overlap of left and right eigenvectors corresponding to eigenvalues µ which lies +in the bulk of the spectrum of X + Λ. +Proof of Theorem 2.4. Define +F ∶= (0 +1 +0 +0), +then by (3.13) we conclude +sup +z∈bulk +⟨ImGz(iη)FImGz(iη)F ∗⟩ ≺ 1, +(3.21) +where the supremum is taken over the bulk as given in Definition 2.3. Here we used that F is regular +in the sense of (3.6); this immediately follows from the fact that F is off–diagonal and ImM(iη) is +diagonal (see Lemma A.1). We now want to show that if we choose z = µi to be a bulk eigenvalue of +X + Λ the upper bound (3.21) implies a lower bound on Oii. To make the notation simpler, from now +on we denote µ = µi. +Consider the non-Hermitian left/right–eigenvectors l,r, with corresponding eigenvalue µ, defined +as in (2.9). Without loss of generality we can assume that µ is a simple eigenvalue, since the spectrum of +X + Λ is simple with probability one owing to the continuous distribution of the entries of X. Next, +we define +P ∶= ⎛ +⎝ +l l∗ +∥l∥2 +0 +0 +rr∗ +∥r∥2 +⎞ +⎠ . +Clearly P is a rank two orthogonal projection whose range is the kernel of Hµ. Recall that Ker(Hµ) +has dimension two since µ is simple and l,r are u1,v1, respectively (up to scalar multiples). Then, +almost surely, by spectral decomposition (and by the spectral symmetry of Hµ) +ImGµ(iη) = P +η + ∑ +∣i∣≥2 +η +(λµ +i )2 + η2 (uµ +i +vµ +i +)(uµ +i +vµ +i +) +∗ +≥ P +η . +By (3.21) we thus obtain +1 ≻ sup +z∈bulk +⟨ImGz(iη)FImGz(iη)F ∗⟩ ≻ 1 +η2 ⟨PFPF ∗⟩ = +∣⟨l,r⟩∣ +2 +Nη2∥r∥2∥l∥2 , +which, by (2.11), implies +Oii =∥r∥2∥l∥2 ≻ +1 +Nη2 . +Choosing η = N −1+ǫ/2, this concludes the proof. +□ +4. Local laws with regular observables +The goal of the present section is to establish the key Proposition 3.4 by proving an averaged local law +for a product of two resolvents (of the Hermitisation (2.15)) in the bulk of the scDos ρ with regular (recall +Definition 3.1 and see Definition 4.1 below) deterministic matrices A1,A2 in between. Throughout the +rest of this paper, we consider the case of several spectral parameters w1,w2,... and fixed bounded +deformations Λ1 = Λ2 = ... ≡ Λ ∈ CN×N, which we continue to omit from the notation. +Using the abbreviations Gi ∶= G(wi) ∶= GΛ(wi) (and analogously for Mi), the deterministic +approximation to the resolvent chain +G1B1G2 ⋯GkBkGk+1 + +18 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +for arbitrary deterministic B1,...,Bk12 is denoted by +M(w1,B1,w2,...,wk,Bk,wk+1) +(4.1) +and defined recursively in the length k of the chain in Definition C.1 given in Appendix C. We may call +these formulas recursive Dyson equations as they provide us with the correct deterministic quantity for +longer resolvent chains. As an example, we have that +M(w1,B1,w2) = B−1 +12 [M1B1M2] = M1X12[B1]M2 , +(4.2) +where B−1 +12 is the inverse stability operator (A.2) and X12 = (1 − S[M1 ⋅ M2]) +−1 (see (5.33) below). +As already mentioned above, we are aiming at local laws for expressions of the form +⟨G1A1 ⋯GkAk⟩ +(4.3) +in the averaged case, or +(G1A1 ⋯AkGk+1)xy +(4.4) +in the isotropic case, where the deterministic matrices A1,...,Ak are assumed to be regular. +The general concept of regularity depending on two spectral parameters w and w’ has already been +introduced in Definition 3.1. In the following definition we tailor this concept to observables in chains +(4.3) and (4.4). It basically says that observable Aj, located between Gj = G(wj) and Gj+1 = G(wj+1) +in these chains will naturally be regularised using the spectral parameters wj and wj+1. +Definition 4.1. (Regular observables in chains) +Fix κ > 0 and let δ = δ(κ,∥Λ∥) > 0 be small enough (see (3.4) and (4.20)). Consider one of the two expressions +(4.3) or (4.4) for some fixed length k ∈ N and bounded matrices ∥Ai∥ ≲ 1 and let w1,...,wk+1 ∈ C ∖ R be +spectral parameters with Rewi ∈ Bκ. For any j ∈ [k], analogously to (3.5), we denote +1± +δ(wj,wj+1) ∶= φδ(Re wj ∓ Rewj+1) φδ(Imwj) φδ(Imwj+1) +(4.5) +and sj ∶= −sgn(ImwjImwj+1), where, here and in the following, in case of (4.3), the indices are understood +cyclically modulo k. +(a) For i ∈ [k] we define the regular component or regularisation of Ai from (4.3) or (4.4) (w.r.t. the +pair of spectral parameters (wi,wi+1)) as +˚ +Ai ∶= ˚ +Awi,wi+1 +i +. +(4.6) +(b) Moreover, we call Ai regular (w.r.t. (wi,wi+1)) if and only if ˚ +Ai = Ai. +For example, in case of (4.3) for k = 1 with spectral parameter w1 ∈ C ∖ R satisfying Rew1 ∈ Bκ, +∣Rew1∣ ≤ δ/4 and ∣Imw1∣ ≤ δ/2 (recall (3.4) and (4.5)), the regular component of A1 is given by +˚ +A1 ∶= A1 − ⟨ImM1A1⟩ +⟨ImM1⟩ E+ − ⟨M1A1M1E−⟩ +⟨M1E−M1E−⟩E− . +(4.7) +We emphasise, that our notation˚⋅ for the regular component of Ai does not have an overall fixed +meaning but depends on the spectral parameters of the resolvents ‘surrounding’ the deterministic ma- +trix Ai under consideration, i.e. +⟨ ⋯ GiAiGi+1 ⋯ ⟩ +or +( ⋯ GiAiGi+1 ⋯ )xy , +or in case of (4.3) for k = 1 the single spectral parameter involved. However, if we aim at specifying the +spectral parameters defining the operation˚⋅ , we add them (or their indices) as a subscript, i.e. write +˚ +Awi,wi+1 +i +≡ ˚ +Ai,i+1 +i +≡ ˚ +Ai ≡ A○ +i ≡ A +○i,i+1 +i +≡ A +○wi,wi+1 +i +, +as done in Definition 3.1, and do not use imprecise notation ˚ +Ai. +The just explained caveats are in stark contrast to the case of Wigner matrices [24, 28, 29], where +the regular component of a matrix A is simply its traceless part, i.e. ˚ +A = A − ⟨A⟩, irrespective of the +spectral parameters involved. Apart from this independence of the location in the spectrum, there is +a one further important difference to our case, which we already mentioned in Section 3: For Wigner +12We will use the the notational convention, that the letter B denotes arbitrary (generic) matrices, while A is reserved for regular +matrices, in the sense of Definition 4.1. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +19 +matrices, the condition for A being regular is one-dimensional and hence restricts A to a (N 2 − 1)- +dimensional subspace of CN×N (the traceless matrices), whereas in our case, the regularity condition is +two-dimensional (if 1σ +δ (⋅,⋅) = 1) and hence restricts a regular matrix A to a ((2N)2 −2)-dimensional +subspace of C2N×2N, which depends on the ‘surrounding’ spectral parameters. +We now give bounds on the size of the deterministic term M(w1,B1,...,wk,Bk,wk+1) from (4.1), +where all Bi are regular in the sense of Definition 4.1. The proof of this lemma is presented in Appen- +dix C. +Lemma 4.2. (Bounds on M, see [28, Lemma 2.4]) +Fix κ > 0. Let k ∈ [4] and w1,...,wk+1 ∈ C ∖ R be spectral parameters with Rewj ∈ Bκ. Then, for +bounded regular deterministic matrices A1,...,Ak (in the sense of Definition 4.1), we have the bounds +∥M(w1,A1,...,Ak,wk+1)∥ ≲ +⎧⎪⎪⎨⎪⎪⎩ +1 +η⌊k/2⌋ +if η ≤ 1 +1 +ηk+1 +if η > 1 , +(4.8) +∣⟨M(w1,A1,...,Ak−1,wk)Ak⟩∣ ≲ +⎧⎪⎪⎨⎪⎪⎩ +1 +η⌊k/2⌋−1 ∨ 1 +if η ≤ 1 +1 +ηk +if η > 1 +, +(4.9) +for the deterministic approximation (4.1) of a resolvent chain, where η ∶= minj ∣Imwj∣. +For the presentation of our main results, we would only need (4.8) and (4.9) for k ∈ [2] from the +previous lemma. However, the remaining bounds covered by Lemma 4.2 will be instrumental in our +proofs of Theorems 4.4 and 4.3 below (see Sections 5 and 6). +The main result of the present section and most important input for our proofs in Section 3 is the +following averaged local law in the bulk of the spectrum for two resolvents and regular matrices. +Theorem 4.3. (Local laws with two regular matrices) +Fix a bounded deterministic Λ ∈ CN×N, ǫ > 0 and κ > 0. Then, for spectral parameters w1,w2,w3 ∈ C +satisfying maxj ∣wj∣ ≤ N 100, Rewj ∈ Bκ and η ∶= minj ∣Im wj∣ ≥ N −1+ǫ, deterministic vectors x,y +with ∥x∥,∥y∥ ≲ 1, and any regular deterministic matrices A1,A2 ∈ C2N×2N (cf. Definition 4.1), we have +the averaged local law +∣⟨G1A1G2A2 − M(w1,A1,w2)A2⟩∣ ≺ +⎧⎪⎪⎨⎪⎪⎩ +1 +√Nη +if η ≤ 1 +1 +Nη3 +if η > 1 +(4.10a) +and the isotropic law +∣⟨x,(G1A1G2A2G3 − M(w1,A1,w2,A2,w3))y⟩∣ ≺ +⎧⎪⎪⎨⎪⎪⎩ +1 +η +if η ≤ 1 +1 +√ +Nη4 +if η > 1 . +(4.10b) +Together with (4.9) for k = 2, this proves Proposition 3.4. Moreover, as a byproduct of our proof of +Theorem 4.3, we obtain the following optimal local laws with a single regular matrix. +Theorem 4.4. (Optimal local laws with one regular matrix) +Fix a bounded deterministic Λ ∈ CN×N, ǫ > 0 and κ > 0. Then, for spectral parameters w1,w2 ∈ C +satisfying maxj ∣wj∣ ≤ N 100, Rewj ∈ Bκ and η ∶= minj ∣Im wj∣ ≥ N −1+ǫ, deterministic vectors x,y +with ∥x∥,∥y∥ ≲ 1, and any regular deterministic matrix A1 (cf. Definition 4.1), we have the optimal +averaged local law +∣⟨(G1 − M1)A1⟩∣ ≺ +⎧⎪⎪⎨⎪⎪⎩ +1 +Nη1/2 +if η ≤ 1 +1 +Nη2 +if η > 1 +(4.11a) +and the optimal isotropic local law +∣⟨x,(G1A1G2 − M(w1,A1,w2))y⟩∣ ≺ +⎧⎪⎪⎨⎪⎪⎩ +1 +√ +Nη2 +if η ≤ 1 +1 +√ +Nη3 +if η > 1 +. +(4.11b) +Remark 4.5. We have several comments. + +20 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +(i) The above local laws are in agreement with the √η-rule first established for Wigner matrices in [28]: +Every regular deterministic matrix Ai reduces both the size of the deterministic approximation and +the error term by a factor √η. +(ii) The error terms in Theorem 4.3 dealing with two regular matrices can still be improved by a factor +1/√Nη, as shown in [28]. A similar analysis could have been conducted here, but we refrain from +doing so, as it is not needed for our main results from Section 2. However, the error bounds in (4.11) +with one regular matrix are in fact optimal. +(iii) Given Theorem 2.6, and Theorems 4.3–4.4, it is possible to deduce similar bounds for averaged and +isotropic chains as in (4.10), where not both matrices A1,A2 are regular (see (3.15)). +In the rest of this paper, we give a detailed proof of Theorem 4.3 in the much more involved η ≤ 1 +regime. For η > 1, the bound simply follows by induction on the number of resolvents in chain by in- +voking the trivial ∥M(w)∥ ≲ 1/∣Im w∣. The detailed argument has been carried out in [28, Appendix B] +for the case of Wigner matrices. However, at a certain technical point (within the proof of the master +inequalities in Proposition 4.8 and the reduction inequalities in Lemma 4.9), the proof uses Theorems 4.3 +and 4.4 (and even its analogues for longer chains) for the η > 1 regime. But the master and reduction +inequalities are not needed for proving the above estimates in the η > 1 regime, hence the argument is +not circular. With partial exception in Appendix C, where we prove Lemma 4.2, throughout the rest of +this paper we assume that minj ∣Imwj∣ =∶ η ≤ 1. +4.1. Basic control quantities and proof of Theorems 4.3 and 4.4. Our strategy for proving Theo- +rem 4.3 (and thereby Theorem 4.4 as a byproduct) is to derive a system of master inequalities (Proposi- +tion 4.8) for the errors in the local laws by cumulant expansion, then use an iterative scheme to gradually +improve their estimates. The cumulant expansion introduces longer resolvent chains, potentially lead- +ing to an uncontrollable hierarchy, so our master inequalities are complemented by a set of reduction +inequalities (Lemma 4.9) to estimate longer chain in terms of shorter ones. We have used a similar strat- +egy in [28, 29] for Wigner matrices, but now many new error terms due to regularisations need to be +handled. +Before entering the detailed proof, we explain the main mechanism of the new type of error terms. +Cumulant expansions applied to chains ... GiAiGi+1 ... with regular Ai’s introduce more resolvent +factors, for example ... GiGiAiGi+1 ... or ... GiE−GiAiGi+1 ... without introducing more A’s. +Multiple G factors without intermediate A’s appear which we wish to reduce to fewer G factors using +resolvent identities or contour integral representations; in the example above we will use +GiGi = G(wi)2 = +1 +2πi ∫Γ +G(z) +(z − wi)2 dz, +(4.12) +where Γ is an appropriate contour (see Lemma 5.1). When this formula is inserted into the chain, we +have ... G(z)AiGi+1 ..., i.e. Ai is not regular any more with respect to the neighboring spectral pa- +rameters (z,wi+1) since wi has been changed to z. We need to regularise Ai to the new situation. +Fortunately, the regularisation is Lipschitz continuous by Lemma 3.3, so roughly speaking we make +an error of order ∣z − wi∣ when we regularise Ai from (wi,wi+1) to (z,wi+1). This error exactly +compensates the higher power of z − wi in the denominator in (4.12), making eventually the adjust- +ment of regularisations harmless in the estimates. We need to meticulously implement this strategy +for longer chains and also taking into account the chiral symmetry to reduce GiE−Gi in chains like +... GiE−GiAiGi+1 .... The precise form of the error terms in Lemma 3.3 is essential. It is remarkable +that the signs appearing in (3.10), (3.11), and (3.12) exactly match those that arise in the denominators of +the contour integral formulas like (4.12). We now start the actual proof. +As the basic control quantities in the sequel of the proof, we introduce the normalised differences +Ψav +k (wk,Ak) ∶= Nηk/2∣⟨G1A1⋯GkAk − M(w1,A1,...,wk)Ak⟩∣, +(4.13) +Ψiso +k (wk+1,Ak,x,y) ∶= +√ +Nηk+1 ∣(G1A1⋯AkGk+1 − M(w1,A1,...,Ak,wk+1))xy∣ +(4.14) +for k ∈ N, where we used the short hand notations +Gi ∶= G(wi), +η ∶= min +i +∣Imwi∣, +wk ∶= (w1,...,wk), +Ak ∶= (A1,...,Ak). + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +21 +The deterministic matrices ∥Ai∥ ≤ 1, i ∈ [k], are assumed to be regular (i.e., Ai = ˚ +Ai, see Definition 4.1) +and the deterministic counterparts +M(w1,A1,...,Ak−1,wk) +used in (4.13) and (4.14) (see also (4.1)) are defined recursively in Appendix C. +For convenience, we extend the above definitions to k = 0 by +Ψav +0 (w) ∶= Nη∣⟨G(w) − M(w)⟩∣, +Ψiso +0 (w,x,y) ∶= +√ +Nη∣(G(w) − M(w))xy∣ +and observe that +Ψav +0 + Ψiso +0 +≺ 1 +(4.15) +is the usual single-resolvent local law (in the bulk) from Theorem 2.6, where here and in the following +the arguments of Ψav/iso +k +shall occasionally be omitted. We remark that the index k counts the number +of regular matrices in the sense of Definition 4.1. +Throughout the entire argument, let ǫ > 0 and κ > 0 be arbitrary but fixed, and let +D(ǫ,κ) ∶= {w ∈ C ∶ Rew ∈ Bκ , N 100 ≥ ∣Imw∣ ≥ N −1+ǫ} +(4.16) +be the target spectral domain, where the κ-bulk Bκ has been introduced in (2.6). This target spectral +domain D(ǫ,κ) will be reached by shrinking a larger initial spectral domain +D(ǫ0,κ0) ∶= {w ∈ C ∶ Rew ∈ Bκ0 , N 100 ≥ ∣Im w∣ ≥ N −1+ǫ0} +(4.17) +many times, say (L − 1) times, during our whole argument, where L = L(ǫ) is an N-independent +positive integer to be determined below (see Remark 4.11). In (4.17), we set ǫ0 ∶= ǫ/2 and chose the initial +bulk parameter +κ0 = κ0(ǫ,κ) = +κ +L(ǫ) > 0 +(4.18) +The justmentionedshrinkingof domainswillbe conductedalongside the decreasingfamily(D(ǫ0,κ0) +ℓ +)ℓ∈[L] +of spectral domains, defined via +D(ǫ0,κ0) +ℓ +∶= {w ∈ C ∶ Rew ∈ Bℓκ0 , N 100 ≥ ∣Imw∣ ≥ ℓN −1+ǫ0} ⊂ D(ǫ,κ) . +(4.19) +D(ǫ,κ) +N −1+ǫ +∼ N −1+ǫ0 +D(ǫ0,κ0) +D(ǫ0,κ0) +2 +D(ǫ0,κ0) +3 +∼ κ2/3 +∼ κ +-2 +2 +Rew +Imw +Figure 2. Depicted are the target spectral domain (4.16), the initial spectral domain +(4.17) and four intermediate domains from the family (4.19). The solid black curve +represents the symmetric scDos ρ for the perturbation Λ = −z with ∣z∣ slightly +less than one (see Example 2.5). Close to a regular edge of the scDos, the horizontal +distance between two domains scales like κ2/3. Near an (approximate) cusp, the +scaling agrees with the linear lower bound given in (2.21). +Finally, the cut-off parameter δ > 0 used in the definition of the regular component of an observable +(see (3.4) and (4.6) in Definition 4.1) shall be chosen by the following two requirements: First, it has to be +much smaller than the initial bulk-parameter κ0 from (4.18), i.e. +0 < δ ≪ cκ0 , +(4.20) + +22 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +where c > 0 is the same constant as introduced in (2.21). Second, it has to be small enough such that the +denominators in (4.6) (see also Appendix A) as well as in Lemmas 5.5, 5.7, and E.1 are uniformly bounded +away from zero – in case that 1σ +δ (wi,wi+1) = 1. Note that these requirements also depend on the +deformation Λ ∈ CN×N (but only via the norm ∥Λ∥ ≲ 1) as it determines the scDos ρ. +Definition 4.6. (Uniform bounds in domains) +Let ǫ > 0 and κ > 0 as above and let k ∈ N. We say that the bounds +∣⟨G(w1)B1 ⋯ G(wk)Bk − M(w1,B1,...,wk)Bk⟩∣ ≺ E av , +∣(G(w1)B1 ⋯ BkG(wk+1) − M(w1,B1,...,Bk,wk+1))xy∣ ≺ E iso +(4.21) +hold (ǫ,κ)-uniformly for some deterministic control parameters E av/iso = E av/iso(N,η), depending only +on N and η ∶= mini ∣Imwi∣, if the implicit constant in (4.21) are uniform in bounded deterministic matrices +∥Bj∥ ≤ 1, deterministic vectors ∥x∥,∥y∥ ≤ 1, and admissible spectral parameters wj ∈ D(ǫ,κ) satisfying +1 ≥ η ∶= minj ∣Im wj∣. +Similarly, we use the phrase that a bound holds (ǫ0,κ0,ℓ)-uniformly (or simply ℓ-uniformly), if the +above statement is true with D(ǫ0,κ0) +ℓ +in place of D(ǫ,κ). +Moreover, we may allow for additional restrictions on the deterministic matrices. For example, we may +talk about uniformity under the additional assumption that some (or all) of the matrices are regular (in the +sense of Definition 4.1). +Note that (4.21) is stated for a fixed choice of spectral parameters wj in the left hand side, but it is in +fact equivalent to an apparently stronger statement, when the same bound holds with a supremum over +the spectral parameters (with the same constraints). More precisely, if E iso ≥ N −C for some constant +C > 0, then (4.21) implies +sup +w1,...,wk+1 +∣(G(w1)B1 ⋯ BkG(wk+1) − M(w1,B1,...,Bk,wk+1))xy∣ ≺ E iso +(4.22) +(and analogously for the averaged bound), where the supremum is taken over all choices of wj’s in +the admissible spectral domain D(ǫ,κ) or D(ǫ0,κ0) +ℓ +. This bound follows from (4.21) by a standard grid +argument (see, e.g., the discussion after [28, Def. 3.1]). Throughout the entire paper, we will frequently +use the equivalence between (4.21) and (4.22), e.g. when integrating such bounds over some spectral +parameters as done in Sections 5 and 6. +We can now formulate our main results of the present section, Theorem 4.3 and Theorem 4.4, in the +language of our basic control quantities Ψav/iso +k +. +Lemma 4.7. (Estimates on Ψav/iso +1 +and Ψav/iso +2 +) For any ǫ > 0 and κ > 0 we have +Ψav +1 + Ψiso +1 +≺ 1 +and +Ψav +2 + Ψiso +2 +≺ +√ +Nη +(ǫ,κ)-uniformly in regular matrices (i.e. for spectral parameters wj ∈ D(ǫ,κ) with 1 ≥ η ∶= minj ∣Im wj∣). +Proof of Theorems 4.3 and 4.4. These immediately follow from Lemma 4.7. +□ +The rest of the proof is structured as follows: First, in Section 4.2, we state the master inequalities +and corresponding reduction inequalities on the Ψav/iso +k +parameters, which we then use in Section 4.3 to +prove Lemma 4.7. Afterwards, in Section 5, we prove the master inequalities and, finally, the proof of +the reduction inequalities is presented in Section 6. +4.2. Master inequalities and reduction lemma. We now state the relevant part of a non-linear infi- +nite hierarchy of coupled master inequalities for Ψav +k and Ψiso +k . In fact, for our purposes, it is sufficient +to have only the inequalities for k ∈ [2], but with fairly more effort (despite closely following the argu- +ments in Section 5) it is possible to obtain analogous estimates for general k ∈ N. +Proposition 4.8. (Master inequalities) Assume that +Ψav/iso +j +≺ ψav/iso +j +, +j ∈ [4], +(4.23) + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +23 +ℓ-uniformly (i.e. for spectral parameters wj ∈ D(ǫ0,κ0) +ℓ +and 1 ≥ minj ∣Imwj∣) in regular matrices. Then +it holds that +Ψav +1 ≺ 1 + ψav +1 +Nη + ψiso +1 ++ (ψav +2 )1/2 +(Nη)1/2 ++ (ψiso +2 )1/2 +(Nη)1/4 , +(4.24a) +Ψiso +1 +≺ 1 + ψiso +1 ++ ψav +1 +(Nη)1/2 + (ψiso +2 )1/2 +(Nη)1/4 , +(4.24b) +Ψav +2 ≺ 1 + (ψav +1 )2 + (ψiso +1 )2 + ψav +2 +Nη ++ ψiso +2 ++ (ψav +4 )1/2 +(Nη)1/2 ++ (ψiso +3 )1/2 + (ψiso +4 )1/2 +(Nη)1/4 +, +(4.24c) +Ψiso +2 +≺ 1 + ψiso +1 ++ ψav +1 ψiso +1 ++ (ψiso +1 )2 +Nη ++ ψiso +2 ++ (ψiso +1 ψiso +3 )1/2 +(Nη)1/2 ++ (ψiso +3 )1/2 + (ψiso +4 )1/2 +(Nη)1/4 +, +(4.24d) +now (ℓ + 1)-uniformly (i.e. for spectral parameters wj ∈ D(ǫ0,κ0) +ℓ+1 +with 1 ≥ η ∶= minj ∣Imwj∣) in regular +matrices. +As shown in the above proposition, resolvent chains of length k are estimated by resolvent chains +up to length 2k. In order to avoid the indicated infinite hierarchy of master inequalities with higher +and higher k indices, we will need the following reduction lemma. +Lemma 4.9. (Reduction inequalities) Assume that Ψav/iso +n +≺ ψav/iso +n +holds for 1 ≤ n ≤ 4, ℓ-uniformly +(i.e. for spectral parameters wj ∈ D(ǫ0,κ0) +ℓ +with 1 ≥ η ∶= minj ∣Imwj∣) in regular matrices (cf. Defini- +tion 4.6). Then we have +Ψav +4 ≺ (Nη)2 + (ψav +2 )2 , +(4.25) +on the same domain. Furthermore, we have +Ψiso +3 +≺ Nη (1 + ψiso +2 +√Nη ) (1 + ψav +2 +Nη ) +1/2 +, +Ψiso +4 +≺ (Nη)3/2 (1 + ψiso +2 +√Nη )(1 + ψav +2 +Nη ) +(4.26) +again uniformly in wj ∈ D(ǫ0,κ0) +ℓ +and in regular matrices. +4.3. Proof of Lemma 4.7. Within the proof, we repeatedly use a simple argument, which we call iter- +ation. +Lemma 4.10. (Iteration) For every D > 0, ν > 0, and α ∈ (0,1), there exists some K = K(D,ν,α), +such that whenever (i) X ≺ N D on D(ǫ0,κ0) +1 +and (ii) X ≺ x on D(ǫ0,κ0) +ℓ +for some ℓ ∈ N, implies that +X ≺ A + x +B + x1−αCα +on +D(ǫ0,κ0) +ℓ+1 +for some constants B ≥ N ν and A,C > 0, it also holds that +X ≺ A + C +on +D(ǫ0,κ0) +ℓ+K +. +We can now turn to the proof of Lemma 4.7. +Proof of Lemma 4.7. Assume that +Ψav/iso +j +≺ ψav/iso +j +, +j ∈ [4], +ℓ-uniformly, for some fixed ℓ > 0, i.e. it holds on the domain D(ǫ0,κ0) +ℓ +. Then, by (4.24a)–(4.24d), we +immediately obtain +Ψav +1 + Ψiso +1 +≺ 1 + ψav +1 + ψiso +1 +(Nη)1/2 + (ψav +2 )1/2 + (ψiso +2 )1/2 +(Nη)1/4 +Ψav +2 + Ψiso +2 +≺ 1 + ψiso +1 ++ (ψav +1 )2 + (ψiso +1 )2 +Nη ++ ψav +2 + ψiso +2 +(Nη)1/2 ++ (ψav +4 )1/2 +(Nη)1/2 + (ψiso +1 ψiso +3 )1/2 +(Nη)1/2 ++ (ψiso +3 )1/2 + (ψiso +4 )1/2 +(Nη)1/4 +(4.27) + +24 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +on the domain D(ǫ0,κ0) +ℓ+1 +. Then, plugging the first line of (4.27) into the second line and using iteration +in both lines, we get +Ψav +1 + Ψiso +1 +≺ 1 + (ψav +2 )1/2 + (ψiso +2 )1/2 +(Nη)1/4 +, +Ψav +2 + Ψiso +2 +≺ 1 + (ψav +4 )1/2 +√Nη ++ (ψav +2 )1/4 + (ψiso +2 )1/4 +(Nη)1/8 +⋅ (ψiso +3 )1/2 +(Nη)1/2 + (ψiso +3 )1/2 + (ψiso +4 )1/2 +(Nη)1/4 +, +(4.28) +on the domain D(ǫ0,κ0) +ℓ+K +, for some K as in Lemma 4.10. We now use the reduction inequalities from +Lemma 4.9 in the second line of (4.28): +Ψav +1 + Ψiso +1 +≺ 1 + (ψav +2 )1/2 + (ψiso +2 )1/2 +(Nη)1/4 +Ψav +2 + Ψiso +2 +≺ (Nη)1/2 + ψav +2 +√Nη + (Nη)1/4(ψiso +2 )1/2 + (ψav +2 )1/2 + (ψav +2 ψiso +2 )1/2 +(Nη)1/4 +, ++ ((Nη)1/4 + (ψav +2 )1/4 + (ψiso +2 )1/4 +(Nη)1/8 +) (1 + (ψiso +2 )1/2 +(Nη)1/4 + (ψav +2 )1/4 +(Nη)1/8 + (ψiso +2 )1/2(ψav +2 )1/4 +(Nη)3/8 +) , +(4.29) +on the domain D(ǫ0,κ0) +ℓ+K +. Next, using iteration once again in the second line of (4.29), we obtain +Ψav +1 + Ψiso +1 +≺ 1 + (ψav +2 )1/2 + (ψiso +2 )1/2 +(Nη)1/4 +, +Ψav +2 + Ψiso +2 +≺ (Nη)1/2 +on the domain D(ǫ0,κ0) +ℓ+K+K′, for some K′ as in Lemma 4.10. We point out that here we used Schwarz and +Young inequalities for several terms. Finally, using iteration one last time we conclude +Ψav +1 + Ψiso +1 +≺ 1, +Ψav +2 + Ψiso +2 +≺ (Nη)1/2 +on the domain D(ǫ0,κ0) +ℓ+K+K′+K′′, for some K′′ as in Lemma 4.10. This concludes the proof. +□ +Remark 4.11. We observe that in every application of Lemma 4.10 during the proof of Lemma 4.7, the +parameter D is uniformly bounded by, say, D ≤ 10, as follows by estimating every resolvent in Ψav/iso +k +by norm and using the trivial 1/η-bound on inverse of the stability operator in the iterative definition of +M(w1,...,wk) given in Definition C.1. A further quick inspection of the above proof shows, that α can be +chosen as fixed α = 1/2. Finally, the parameter ν is lower bounded by (some universal positive constant +times) ǫ, since Nη ≥ N ǫ/2 by construction of the initial domain (4.17). Hence, the constants K, K′, and K′′ +only depend on ǫ and therefore also the maximal number L = L(ǫ) of domain shrinkings. +5. Proof of the master inequalities, Proposition 4.8 +Before going into the proofs of the master inequalities, we state a simple lemma, which will fre- +quently be used in the following. Recall that the deformation Λ ∈ CN×N is fixed and hence omitted +from the notation. +Lemma 5.1. (Integral representations for products of resolvents) +Let k ∈ N and w1,...,wk ∈ C ∖ R be spectral parameters, whose imaginary parts have equal sign, +i.e. sgn(Imw1) = ... = sgn(Imwk) =∶ τ. Then, for any J ⊂ R being a union of compact intervals +such that Rewi ∈ ˚ +J (the interior) for all i ∈ [k] and 0 < ˜η < η ∶= minj ∣Imwj∣, we have the integral +representation +k +∏ +j=1 +G(wj) = +1 +2πi ∫Γ G(z) +k +∏ +j=1 +1 +z − wj +dz , +(5.1) +where the contour Γ from (5.1) is defined as (see Figure 3) +Γ ≡ Γτ +˜η(J) ∶= +⎧⎪⎪⎨⎪⎪⎩ +∂(J × [i˜η,i∞)) +if +τ = + +∂(J × (−i∞,−i˜η]) +if +τ = − +(5.2) + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +25 +and the boundary is parameterised in counter-clockwise orientation. The representation (5.1) remains valid if +G(z) is replaced by ImG(z) in the integrand. +Proof. This easily follows from residue calculus. +□ +Rez +Imz +Figure 3. Depicted is the scenario from Lemma 5.1 with five spectral parameters +represented as dots in the upper half plane. Moreover, we indicated the union of +compact intervals J on the real axis and the contour Γ as described in (5.2). Note +that one of the three intervals constituting J does not contain any Rewj. +We recall the definition of the second order renormalisation, denoted by underline, from [24]. For +functions f(W ),g(W ) of the random matrix W (see (2.15)), we define +f(W )W g(W ) ∶= g(W )W f(W ) − ̃E[(∂̃ +W f)(W )̃ +Wg(W ) + f(W )̃ +W(∂̃ +W g)(W )], +(5.3) +where ∂̃ +W denotes the directional derivative in the direction of +̃ +W ∶= ( 0 +̃ +X +̃ +X∗ +0 ) , +where ˜ +X is a complex Ginibre matrix that is independent of W . The expectation is taken w.r.t. the +matrix ̃ +X. Note that, if W itself consists of a complex Ginibre matrix X, then Ef(W )W g(W ) = 0, +while for X with a general distribution this expectation is independent of the first two moments of X. +In other words, the underline renormalises the product f(W )W g(W ) to second order. We remark +that underline (5.3) is a well-defined notation, if the ‘middle’ W to which the renormalisation refers is +unambiguous. This is always be the case in all our proofs, since the functions f,g will be products of +resolvents, never involving explicitly monomials in W . +We note that +̃Ẽ +WR̃ +W = 2⟨RE2⟩E1 + 2⟨RE1⟩E2 = ∑ +σ +σ⟨REσ⟩Eσ = S[R] +and furthermore, that the directional derivative of the resolvent is given by +∂̃ +W G = −G̃ +WG. +For example, in the special case f(W ) = 1 and g(W ) = (W + ˆΛ − w)−1 = G, we thus have +W G = W G + S[G]G +by definition of the underline in (5.3). +Using this underline notation in combination with the identity G(W+ˆΛ−w) = E+ and the defining +equation (2.19) for M, we have +G = M − MW G + MS[G − M]G = M − GWM + GS[G − M]M . +(5.4) +Recall that ⟨GE−⟩ = 0 (see below (2.16)) which immediatelyyields that S[G] = ∑σ σ⟨GEσ⟩Eσ = ⟨G⟩. +Moreover, as shown in Lemma A.1, we have that S[M] = ⟨M⟩ and hence S[⋅] effectively acts like a +trace on G and M, i.e. +S[G − M] = ⟨G − M⟩. +(5.5) + +26 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Now, similarly to [28], the key idea of the proof of Proposition 4.8 is using (5.4) for some Gj in a +chain G1A1 ⋯AkGk+1 and extending the renormalisation (5.3) to the whole product at the expense +of adding resolvent products of shorter length. This will be done for each of the four estimates from +Proposition 4.8 separately and presented in an underlined lemma in the beginning of each of the follow- +ing subsections. Afterwards, the renormalisation of the whole product will be handled by cumulant +expansion, exploiting that its expectation vanishes up to second order. While the proofs of the un- +derlined lemmas for Ψav/iso +1 +are presented in detail, we defer the analogous arguments for Ψav/iso +2 +to +Appendix E. +5.1. Proof of the first master inequality (4.24a). Let w ≡ w1 be a spectral parameter in D(ǫ0,κ0) +ℓ+1 +(in +particular in the bulk of the scDos, recall (4.19)) and A ≡ A1 a (w,w)-regular matrix (cf. Definition 4.1). +We use the notation w = e + iη and we assume without loss of generality (by conjugation with E−, see +(2.16)) that 1 ≥ η > 0. We also assume that (4.23) holds (in this subsection we will need it only for Ψav +1 +and Ψav +2 ). +Lemma 5.2. (Representation as full underlined) +For any regular matrix A = ˚ +A we have that +⟨(G − M)˚ +A⟩ = −⟨W G˚ +A′⟩ + O≺(E av +1 ) +(5.6) +for some other regular matrix A′ = ˚ +A′, which linearly depends on A (see (5.19) for the precise formula for +A′). For the error term in (5.6), we used the shorthand notation +E av +1 ∶= +1 +Nη1/2 (1 + ψav +1 +Nη ) . +(5.7) +Having this approximate representation of ⟨(G − M)˚ +A⟩ as a full underlined term at hand, we turn +to the proof of (4.24a) via a (minimalistic) cumulant expansion. +Proof of (4.24a). Let p ∈ N. Then, starting from (5.6), we obtain +E∣⟨(G − M)A⟩∣2p +=∣−E⟨W GA′⟩⟨(G − M)A⟩p−1⟨(G − M)∗A∗⟩p∣ + O≺((E av +1 )2p) +(5.8) +≲E +∣∑σ σ⟨GEσGA′EσGA⟩∣ + ∣∑σ σ⟨G∗EσGA′EσG∗A∗⟩∣ +N 2 +∣⟨(G − M)A⟩∣ +2p−2 ++ +∑ +∣l∣+∑(J∪J∗)≥2 +EΞav +1 (l,J,J∗)∣⟨(G − M)A⟩∣ +2p−1−∣J∪J∗∣ + O≺((E av +1 )2p) , +where Ξav +1 (l,J,J∗) is defined as +Ξav +1 ∶= N −(∣l∣+∑(J∪J∗)+3)/2 ∑ +ab +Rab∣∂l(GA′)ba∣ ∏ +j∈J +∣∂j⟨GA⟩∣ ∏ +j∈J∗ +∣∂j⟨G∗A∗⟩∣ +(5.9) +and the summation in the last line of (5.8) is taken over tuples l ∈ Z2 +≥0 and multisets of tuples J,J∗ ⊂ +Z2 +≥0 ∖ {(0,0)}. Moreover, we set ∂(l1,l2) ∶= ∂l1 +ab∂l2 +ba, ∣(l1,l2)∣ = l1 + l2, ∑ J = ∑j∈J ∣j∣, and used the +shorthand notation +Rab ∶= 1(a ≤ N,b ≥ N + 1 or b ≤ N,a ≥ N + 1) +for a rescaled cumulant. In the remainder of the proof, we need to analyze the rhs. of the inequality +derived in (5.8). We begin with the third line and study the terms involving Ξav +1 from (5.9) afterwards. +Before going into the proof, we note that, due to the cumulant expansion in (5.8), there are chains +of resolvents G and deterministic matrices A appearing, where some of the A’s are not necessarily +regular w.r.t. the spectral parameters of the surrounding G’s. The principal idea is to decompose such +A with the aid of Lemma 3.3 and carefully track the resulting errors. As a rule of thumb, potentially +small denominators resulting from resolvent identities or the integral representation in Lemma 5.1 are +balanced with the linear perturbative estimates from Lemma 3.3. See also Remark 5.3 below. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +27 +Gaussian contribution: third line of (5.8). In order to do so, we need to analyze in total four terms, +each of which carries a factor of +⟨GEσGA′EσGA⟩ +or +⟨G∗EσGA′EσG∗A∗⟩, +for +σ = ±. +Since their treatment is very similar, we focus on the two exemplary terms +(i) +⟨GGA′GA⟩ +and +(ii) +⟨G∗GA′G∗A∗⟩ . +(5.10) +In the analysis of the Gaussian contribution in Section 5.2, we will discuss the analogs of the other two +terms in more detail. +First term. For the first term in (5.10), we apply the integral representation from Lemma 5.1 to GG with +τ = +, +J = Bℓκ0 , +and +˜η = +ℓ +ℓ + 1η , +for which we recall that w ∈ D(ǫ0,κ0) +ℓ+1 +, i.e. in particular η ≥ (ℓ + 1)N −1+ǫ0 and hence ˜η ≥ ℓN −1+ǫ0. In +particular, Γ ≡ Γτ +˜η(J) ⊂ D(ǫ0,κ0) +ℓ +. Now, we split the contour Γ in three parts,13 i.e. +Γ = Γ1 + Γ2 + Γ3 . +(5.11) +As depicted in Figure 4, the first part of the contour consists of the entire horizontal part of Γ. The sec- +1 +1 +2 +3 +D(ǫ0,κ0) +ℓ+1 +D(ǫ0,κ0) +ℓ +Γ +w +i +iN 100 +Rez +Imz +Figure 4. The contour Γ is splitinto three parts (see (5.11)). In case of multiple spec- +tral parameters, the second part might require a further decomposition at the level +indicated by the dashed horizontal line (see Footnote 13). Depicted is the situation, +where the bulk Bℓκ0 consists of two components. +ond part, Γ2, covers the vertical components up to ∣Im z∣ ≤ N 100. Finally, Γ3 consists of the remaining +part with ∣Im z∣ > N 100. +Now, the contribution coming from Γ3 can be estimated with a trivial norm bound on G. For +z ∈ Γ2, we use that 1± +δ(z,w) = 0 for every w ∈ D(ǫ0,κ0) +ℓ+1 +(recall (2.21) and (4.20)) and hence every +matrix is (z,w)-regular. Hence, after splitting the contour integral and bounding each contribution as +just described, we find, with the aid of Lemma 4.2, +∣⟨GGA′GA⟩∣ ≺ (1 + ψav +2 +Nη ) + ∫Bℓκ0 +∣⟨G(x + i˜η)A′G(e + iη)A⟩∣ +(x − e)2 + η2 +dx . +(5.12) +Next, we decompose A = ˚ +A = ˚ +Ae+iη,e+iη and A′ = ˚ +A′ = +˚ +(A′) +e+iη,e+iη according to Lemma 3.3 as +˚ +Ae+iη,e+iη = ˚ +Ae+iη,x+i˜η + O(∣x − e∣ + η)E+ + O(∣x − e∣ + η)E− , +˚ +(A′) +e+iη,e+iη = +˚ +(A′) +x+i˜η,e+iη, + O(∣x − e∣ + η)E+ + O(∣x − e∣ + η)E− . +13In the case of several w1, ..., wk, the second part might require a further decomposition: If the spectral parameters of the +resolvents which are not involved in such an integral representation have spectral parameters with imaginary parts of absolute value +greater than one, we need to split Γ2 according to ∣Im z∣ ≤ 1 and ∣Im z∣ > 1. While the former will be treated exactly as Γ2 here, +the latter shall be estimated by means of the η > 1-laws, which we discussed after Remark 4.5. + +28 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Plugging this into (5.12), we obtain several terms contributing to the integral. By means of Lemma 4.2, +the leading term accounts for +∫Bℓκ0 +∣⟨G(x + i˜η) ˚ +(A′) +x+i˜η,e+iηG(e + iη)˚ +Ae+iη,x+i˜η⟩∣ +(x − e)2 + η2 +dx ≺ 1 +η (1 + ψav +2 +Nη ) . +The error terms can be dealt with by simple resolvent identities in combination with the usual single- +resolvent local law, Theorem 2.6, proving them to be bounded by η−1. Indeed, for a generic B ∈ +C2N×2N, we consider the exemplary term +∫Bℓκ0 +∣⟨G(x + i˜η)E+G(e + iη)B⟩∣ +∣x − e∣ + η +(x − e)2 + η2 dx +≲∫Bℓκ0 +∣⟨(G(x + i˜η) − G(e + iη))B⟩∣ +(x − e)2 + η2 +dx ≺ 1 +η . +Second term. The second term in (5.10) is much simpler than the first. After writing GG∗ = ImG/η, it +suffices to realise that, by means of Lemma 3.3, +A′ = +˚ +(A′) +e+iη,e−iη , +˚ +(A′) +e−iη,e−iη = A′ + O(∣e∣)E− , +and +A∗ = +˚ +(A∗) +e−iη,e±iη +in order to bound +∣⟨G∗GA′G∗A∗⟩∣ ≺ 1 +η (1 + ψav +2 +Nη ) + ∣e∣ +η +∣⟨[G(−e + iη) − G(e − iη)]A∗E−⟩∣ +∣e∣ + η +≺ 1 +η (1 + ψav +2 +Nη ) +with the aid of Lemma 4.2, the chiral symmetry (2.16), a resolvent identity and Theorem 2.6. +This finishes the estimate for the Gaussian contribution from the third line of (5.8), for which we have +shown that +1 +N 2 ∑ +σ +(∣⟨GEσGA′EσGA⟩∣ + ∣⟨G∗EσGA′EσG∗A∗⟩∣) ≺ +1 +N 2η (1 + ψav +2 +Nη ) . +(5.13) +We are now left with the terms from the last line (5.8) resulting from higher order cumulants. +Higher order cumulants and conclusion. The terms stemming from higher order cumulants are es- +timated in Section 5.5, the precise bound being given in (5.68a). Indeed, plugging (5.13) and (5.68a) into +(5.8) we obtain +E∣⟨(G − M)A⟩∣2p ≺ (E av +1 )2p ++ +p +∑ +m=1 +[ +1 +Nη1/2 (1 + ψiso +1 ++ (ψav +2 )1/2 +(Nη)1/2 ++ (ψiso +2 )1/8 +(Nη)1/8 )] +m +(E∣⟨(G − M)A⟩∣2p) +1−m/2p +and get the appropriate estimate E∣... ∣2p using Young inequalities. Since p was arbitrary, it follows +that +Ψav +1 ≺ 1 + ψav +1 +Nη + ψiso +1 ++ (ψav +2 )1/2 +(Nη)1/2 ++ (ψiso +2 )1/4 +(Nη)1/8 . +The bound given in Proposition 4.8 is an immediate consequence after a further trivial Young inequality. +□ +Remark 5.3. Although the proof of the first master inequality (4.24a) is rather short, it already revels a +general strategy for dealing with a generic (not strictly) alternating chain +⋯GGAGAGE−AGE−GA ⋯ +(5.14) +of resolvents G and deterministic matrices A. +(i) Apply resolvent identites and the integral representation from Lemma 5.1 in order to reduce a product +of resolvents to a linear combination (discrete or continuous, respectively). For terms of the form +GE−G instead of GG this additionally requires an application of the chiral symmetry (2.16). + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +29 +(ii) In the resulting strictly alternating chain, decompose every deterministic A according to the regu- +larisation from Definition 4.1 w.r.t. the spectral parameters of its surrounding resolvents by using +Lemma 3.3. +(iii) Estimate the regular parts coming from this decomposition in terms of Ψav/iso +k +≺ ψav/iso +k +. Carefully +track the resulting errors stemming from the other parts. +These steps shall be applied repeatedly until the entire chain (5.14) has been examined. The first two items in +the above list a purely mechanical. However, the third step is non-trivial and requires careful analysis on a +case-by-case basis. +We have already mentioned that, as a rule of thumb, potentially small denominators resulting from Step +(i) are balanced with the linear perturbative numerators from Step (ii). +It remains to give a proof of Lemma 5.2. +Proof of Lemma 5.2. Similarly as in (5.6), we suppress the indices of G ≡ G1, M ≡ M1 etc. +We start with the first identity in (5.4), such that, after defining the one-body stability operator +B ∶= 1 − MS[⋅]M +we find +B[G − M] = −MW G + MS[G − M](G − M) +and consequently, by inversion, multiplication by A = ˚ +A (in the sense of (4.6), see also (4.7)) and taking +a trace +⟨(G − M)A⟩ = −⟨W GX [A]M⟩ + ⟨S[G − M](G − M)X [A]M⟩, +(5.15) +where we introduced the linear operator +X [B] ∶= ((B∗)−1[B∗]) +∗ = (1 − S[M ⋅ M]) +−1[B] +for +B ∈ C2N×2N . +Then, it is important to note that the condition 1+ +δ⟨ImMA⟩ = 0 (the first of the two imposed via +(4.7); recall the definition of the cutoff function 1+ +δ from (3.5)and (4.5)), is stable under the linear operation +A ↦ X [A]M. +Lemma 5.4. For a generic B ∈ C2N×2N , we find +⟨X [B]MImM⟩ = ⟨BB−1[MImM]⟩ = i +2 +⟨BImM⟩ +⟨ImM⟩ + O(η). +(5.16) +Proof. Using (A.11), we compute +B−1[MImM] = B−1[M 2 − MM ∗] +2i += i +2 +ImM +η + ⟨ImM⟩ + 1 +2i +1 − ⟨MM ∗⟩ +1 − ⟨M 2⟩ M 2 . +Now, by means of Lemma A.4 and Lemma A.5, we find that +∣1 − ⟨MM ∗⟩∣ = O(η) +and +∣1 − ⟨M 2⟩∣ ≳ 1, +respectively. +□ +Recall from (5.5) that S[G − M] = ⟨G − M⟩. Therefore, by means of the usual averaged local law, +Theorem 2.6,whichinparticularshowsthat∣⟨W GB⟩∣ ≺ +1 +Nη forarbitrary∥B∥ ≲ 1(see also AppendixB +and [36]), we can write (5.15) as +⟨(G − M)A⟩ = − ⟨W G(X [A]M)○⟩ + ⟨G − M⟩⟨(G − M)(X [A]M)○⟩ +− 1− +δc−(X [A]M)⟨W GE−⟩ + O≺(N −1) , +(5.17) +where in the underlined term, we used that the E+ component of the regularisation of X [A]M is +negligible thanks to Lemma 5.4 and the regularity of A, and we introduced the short hand notation +c−(X [A]M) ∶= ⟨MX [A]MME−⟩ +⟨ME−ME−⟩ +. +Next, with the aid of W G = I − ˆΛG + wG and using ⟨GE−⟩ = 0 from (5.5), we undo the underline +in the second to last term, such that we infer +⟨W GE−⟩ = −⟨GE− ˆΛ⟩ = −⟨(G − M)E− ˆΛ⟩ = −⟨(G − M)(E− ˆΛ)○⟩. + +30 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +In the second equality, we used that ⟨ME− ˆΛ⟩ = 0, which follows by a simple computation using the +explicit form of M given in Lemma A.1. For the last equality, we note that +(E− ˆΛ)○ = E−ˆΛ − 1+ +δ +⟨ImME− ˆΛ⟩ +⟨ImM⟩ +E+ − 1− +δ +⟨ME− ˆΛME−⟩ +⟨ME−ME−⟩ E− = E−ˆΛ, +which again follows after a simple computation using the fact that ˆΛ is off-diagonal together with +Lemma A.1. +We can now write (5.17) for A = ˚ +A = (E− ˆΛ)○ = E−ˆΛ and solve the resulting equation for ⟨(G − +M)E− ˆΛ⟩. Plugging this back into (5.17) yields +⟨(G − M)A⟩ = − ⟨W G(X [A]M)○⟩ + ⟨G − M⟩⟨(G − M)(X [A]M)○⟩ + O≺(N −1) ++ +1− +δ c−(X [A]M) +1 − 1− +δ c−(X [E− ˆΛ]M) +[ − ⟨W G(X [E−Z]M)○⟩ +(5.18) ++ ⟨G − M⟩⟨(G − M)(X [E−Z]M)○⟩ + O≺(N −1)] . +Since ∥X [˚ +A]∥ ≲ 1 (see Lemma A.6), the only thing left to check is, that the denominator in (5.18) is +bounded away from zero. +Lemma 5.5. For small enough δ > 0, we have that +∣1 − 1− +δ(w,w)c−(X [E− ˆΛ]M)∣ ≳ 1. +Proof. The statement is trivial for 1− +δ(w,w) = 0 and we hence focus on the complementary extreme +scenario 1− +δ(w,w) = 1, the intermediate ones being immediate consequences of the extreme. Indeed, +for 1− +δ(w,w) = 1, we first note that X [E−ˆΛ] = E−ˆΛ, which follows from the explicit form of M +given in Lemma A.1 using the fact that ˆΛ is purely off-diagonal. Next, we use the MDE (2.19), the chiral +symmetry E−M(w) = −E−M(−w) from Lemma A.1, and Lemma A.4 to infer +1 − c−(X [E− ˆΛ]M) = 1 − ⟨ME− ˆΛMME−⟩ +⟨ME−ME−⟩ += 1 +2 [1 − w + m +m +⟨M 2⟩] . +Now, specializing to w = iη with sufficiently small η, we find that, to leading order, +∣1 − η + Imm +Imm +⟨M 2⟩∣ ∼ ∣1 − ⟨M 2⟩∣ ≳ 1 +by means of Lemma A.5. This principal lower bound of order one persists after a small perturbation of +w allowing for a non-zero real part, but as long as 1− +δ(w,w) = 1 for some δ > 0 small enough. +□ +From the expansion (5.18) it is apparent, that the main terms for understanding the size of ⟨(G − +M)A⟩ are the underlined ones, the rest carrying additional ⟨G − M⟩-factors, hence they will become +negligible errors. In fact, summarizing our investigations, we have shown that +⟨(G − M)˚ +A⟩ = −⟨W G˚ +A′⟩ + O≺(E av +1 ) , +where we used the shorthand notation +˚ +A′ ∶= (X [˚ +A]M)○ + +1− +δc−(X [˚ +A]M) +1 − 1− +δ c−(X [E−ˆΛ]M) +(X [E− ˆΛ]M)○ +(5.19) +in the underlined term. Using the usual averaged local law (4.15) and (4.23), we collected all the error +terms from (5.18) in E av +1 , defined in (5.7). +□ + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +31 +5.2. Proof of the second master inequality (4.24b). Let wj ∈ D(ǫ0,κ0) +ℓ+1 +for j ∈ [2] be spectral pa- +rameters and A1 a regular matrix w.r.t. the pair of spectral parameters (w1,w2) (see Definition 4.1). By +conjugation with E−, we will assume w.l.o.g. that Imw1 > 0 and Imw2 < 0. Moreover, we use the +notations ej ≡ Rewj, ηj ∶= ∣Imwj∣ for j ∈ [2] and define 1 ≥ η ∶= minj ∣Imwj∣. We also assume that +(4.23) holds. +Lemma 5.6. (Representation as full underlined) +For ∥x∥,∥y∥ ≤ 1 and any (w1,w2)-regular matrix A1 = ˚ +A1, we have that +(G1 ˚ +A1G2 − M(w1, ˚ +A1,w2))xy = −(G1 ˚ +A′ +1W G2)xy + O≺(E iso +1 ) +(5.20) +for some (w1,w2)-regular matrix A′ +1 = ˚ +A′ +1, which linearly depends on A1 = ˚ +A1 (see (5.51)). For the error +term in (5.20), we used the shorthand notation +E iso +1 +∶= +1 +√ +Nη2 (1 + +ψav +1 +(Nη)1/2 + ψiso +1 +Nη ) . +(5.21) +Note that unlike in Section 5.1, now in (5.20) the second resolvent G2 was expanded instead of G1 +rendering the W factor in the middle of the underlined term. This prevents the emergence of resolvent +chains in the proof of (4.24b), which are ‘too long’ to be handled within our hierarchical framework of +master inequalities (e.g., a chain involving four resolvents would appear in ̃Ξiso +1 +defined below). +Having this approximate representation of (G1 ˚ +A1G2 − M(w1, ˚ +A1,w2))xy as a full underlined +term at hand, we turn to the proof of (4.24b) via a (minimalistic) cumulant expansion. +Proof of (4.24b). Let p ∈ N. Then, starting from (5.20) and using the same notations as in the proof of +(4.24a), we obtain +E∣(G1 ˚ +A1G2 − M(w1, ˚ +A1,w2))xy∣ +2p +(5.22) +≲E ̃Ξiso +1 ∣(G1 ˚ +A1G2 − M(...))xy∣ +2p−2 ++ +∑ +∣l∣+∑(J∪J∗)≥2 +EΞiso +1 (l,J,J∗)∣(G1 ˚ +A1G2 − M(...))xy∣ +2p−1−∣J∪J∗∣ + O≺((E iso +1 )2p) , +where +̃Ξiso +1 +∶= +∑σ [∣(G1 ˚ +A′ +1EσG1 ˚ +A1G2)xy(G1EσG2)xy∣ + ∣(G1 ˚ +A′ +1EσG2)xy(G1 ˚ +A1G2EσG2)xy∣] +N ++ +∑σ [∣(G1 ˚ +A′ +1EσG∗ +2(˚ +A1)∗G∗ +1)xx(G∗ +2EσG2)yy∣ + ∣(G1 ˚ +A′ +1EσG∗ +1)xx(G∗ +2(˚ +A1)∗G∗ +1EσG2)yy∣] +N +and Ξiso +1 (l,J,J∗) is defined via +Ξiso +1 +∶= N −(∣l∣+∑(J∪J∗)+1)/2 ∑ +ab +Rab∣∂l[(G1 ˚ +A′ +1)xa(G2)by]∣ +(5.23) +× ∏ +j∈J +∣∂j(G1 ˚ +A1G2)xy∣ ∏ +j∈J∗ +∣∂j(G∗ +2(˚ +A1)∗G∗ +2)yx∣. +In the remainder of the proof, we need to analyze the rhs. of the inequality derived in (5.22). Following +the general strategy outlined in Remark 5.3, we begin with the second line and study the terms involving +Ξiso +1 +from (5.23) afterwards. +Gaussian contribution: third line of (5.22). In order to do so, following Remark 5.3, we need to ana- +lyze in total eight terms, each of which carries one of the summands in the definition of ̃Ξiso +1 +as a factor. +Since their treatment is very similar, we focus on the two exemplary terms +(i) (G1 ˚ +A′ +1E−G1 ˚ +A1G2)xy(G1E−G2)xy , +(ii) (G1 ˚ +A′ +1E−G∗ +1)xx(G∗ +2(˚ +A1)∗G1E−G2)yy . (5.24) +In the analysis of the Gaussian term in Section 5.1 we discussed analogs of the above terms with the +choice σ = +. + +32 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Term (i) in (5.24). For the first term, we decompose, similarly to Lemma 3.3, +(˚ +A′ +1)1,2E− = ((˚ +A′ +1)1,2E−) +○1,1 + O(∣e1 + e2∣ + ∣η1 − η2∣)E+ + O(∣e1 + e2∣ + ∣η1 − η2∣)E− . (5.25) +Inserting this into the first term in (5.24) and using Lemma 4.2, we find +∣(G1 ˚ +A′ +1E−G1 ˚ +A1G2)xy∣ ≺ 1 +η (1 + ψiso +2 +√Nη ) + (∣e1 + e2∣ + ∣η1 − η2∣) ∑ +σ +∣(G1EσG1 ˚ +A1G2)xy∣ . +(5.26) +In the last term, we focus on σ = −, while σ = + can be dealt with by Lemma 5.1. In fact, using (2.16) and +a resolvent identity, we obtain +∣(G1E−G1 ˚ +A1G2)xy∣ = ∣ 1 +w1 +([G(−w1) − G(w1)]˚ +Aw1,w2 +1 +G(w2))(E−x)y∣ ≺ 1 +η2 (1 + ψiso +1 +√Nη ) , +where in the last step we used Lemma 4.2 and the trivial approximation +˚ +A−w1,w2 +1 += ˚ +Aw1,w2 +1 ++ O(1)E+ + O(1)E− . +For the second factor in the first term in (5.24), we use (2.16) and employ the integral representation +from Lemma 5.1 with +τ = +, +J = Bℓκ0 , +and +˜η = +ℓ +ℓ + 1η , +for which we recall that wj ∈ D(ǫ0,κ0) +ℓ+1 +, i.e. in particular η ≥ (ℓ + 1)N −1+ǫ0 and hence ˜η ≥ ℓN −1+ǫ0. +After splitting the contour integral and estimating the contribution as described around (5.11), we find, +with the aid of Lemma 4.2 and absorbing logarithmic corrections into ‘≺’, that +∣(G1E−G2)xy∣ ≺ 1 + ∫Bℓκ0 +∣(G(x + i˜η))x(E−y)∣ +∣(x − e1 − i(η1 − ˜η)) (x + e2 − i(η2 − ˜η))∣dx +≺ 1 + +1 +∣e1 + e2∣ + η1 + η2 +(5.27) +where in the last step we used the usual single resolvent local law from Theorem 2.6. Notice the key +cancellation of the ∣e1 + e2∣ factor in (5.26) and (5.27). Collecting all the estimates, we have shown that +∣(5.24) (i)∣ ≺ 1 +η2 (1 + ψiso +1 +√Nη + ψiso +2 +√Nη ) . +(5.28) +Term (ii) in (5.24). In the first factor in the second term in (5.24), we again employ the decomposition(5.25) +to find +∣(G1 ˚ +A′ +1E−G∗ +1)xx∣ ≺ +1 +η1/2 (1 + ψiso +1 +√Nη ) + ∣e1 + e2∣ + ∣η1 − η2∣ +η +(5.29) +with the aid of Theorem 2.6 and Lemma 4.2 as well as a resolvent identity and Lemma 5.1 for the E+ and +E− in (5.25), respectively. +In the second factor, similarly to (5.27) above, we use Lemma 5.1 together with the decomposition +(˚ +Aw1,w2 +1 +)∗ = +˚ +(A∗ +1) +¯ +w2, ¯ +w1 = +˚ +(A∗ +1) +¯ +w2,w1 = +˚ +(A∗ +1) +¯ +w2,x+i˜η + ∑ +σ +Oσ(∣x − e1∣ + ∣η1 − ˜η∣)Eσ +from Lemma 3.3 for arbitrary x to find +∣(G∗ +2(˚ +A1)∗G1E−G2)yy∣ ≺ 1 +η1/2 (1 + ψiso +1 +√Nη ) ++ ∫Bℓκ0 +∣(G( ¯w2) ˚ +(A∗ +1) +¯ +w2,x+i˜ηG(x + i˜η))y(E−y)∣ +∣(x − e1 − i(η1 − ˜η)) (x + e2 − i(η2 − ˜η))∣dx ++ ∫Bℓκ0 +∑σ ∣(G( ¯w2)EσG(x + i˜η))y(E−y)∣ +∣x + e2 − i(η2 − ˜η)∣ +dx +(5.30) + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +33 +≺ 1 +η1/2 (1 + ψiso +1 +√Nη ) (1 + +1 +∣e1 + e2∣ + η1 + η2 +) + 1 +η . +Now, combining (5.29) and (5.30), we obtain +∣(5.24) (ii)∣ ≺ 1 +η2 (1 + ψiso +1 +√Nη ) +2 +. +(5.31) +This finishes the estimate for the Gaussian contribution from the third line of (5.22), for which we have +shown that +̃Ξiso +1 +≺ +1 +Nη2 (1 + (ψiso +1 )2 +Nη ++ ψiso +2 +√Nη ) +(5.32) +as easily follows by combining (5.28) with (5.31) and using a Schwarz inequality. +We are now left with the terms from the last line (5.22) resulting from higher order cumulants. +Higher order cumulants and conclusion. The estimate stemming from higher order cumulants is +given in (5.68b). Then, plugging (5.32) and (5.68b) into (5.22), we find, similarly to Section 5.1, that +Ψiso +1 +≺ 1 + ψiso +1 +Nη + ψiso +1 ++ ψav +1 +(Nη)1/2 + (ψiso +2 )1/2 +(Nη)1/4 + (ψiso +2 )1/4 +(Nη)1/8 . +The bound given in Proposition 4.8 is an immediate consequence after a trivial Young inequality. +□ +It remains to give a proof of Lemma 5.6. This is much more involved than for the previous underlined +Lemma 5.2. The proof of Lemma 5.2 crucially used that the orthogonality ⟨ImMA⟩ = 0 is (almost) +preserved under the operation A ↦ X [A]M (see Lemma 5.4). This is simply not available here, since +we deal with two spectral parameters w1,w2. +Proof of Lemma 5.6. We denote A1 ≡ ˚ +A1, except we wish to emphasise A1 being regular. Just as in +Section 5.1, we start with +G2 = M2 − M2W G2 + M2S[G2 − M2]G2 , +such that we get +G1 ˜A1G2 = G1 ˜A1M2 − G1 ˜A1M2W G2 + G1 ˜A1M2S[G2 − M2]G2 +for ˜A1 = X12[A1] and A1 = ˚ +A1 (note that ∥X12[˚ +A1]∥ ≲ 1 by Lemma A.6), where we introduced the +linear operator +X12[B] ∶= (1 − S[M1 ⋅ M2]) +−1[B] +for +B ∈ C2N×2N . +(5.33) +Extending the underline to the whole product, we obtain +G1 ˜A1G2 =M1 ˜A1M2 + (G1 − M1) ˜A1M2 − G1 ˜A1M2W G2 ++ G1 ˜A1M2S[G2 − M2]G2 + G1S[G1 ˜A1M2]G2 , +from which we conclude that +G1( ˜A1 − S[M1 ˜A1M2])G2 = M1 ˜A1M2 + (G1 − M1) ˜A1M2 − G1 ˜A1M2W G2 ++ G1 ˜A1M2S[G2 − M2]G2 + G1S[(G1 − M1) ˜A1M2]G2 +and thus +G1A1G2 =M1X12[A1]M2 + (G1 − M1)X12[A1]M2 − G1X12[A1]M2W G2 +(5.34) ++ G1X12[A1]M2S[G2 − M2]G2 + G1S[(G1 − M1)X12[A1]M2]G2 . +We note that ∥X12[˚ +A1]∥ ≲ 1 by means of Lemma A.6. +Then, we need to further decompose X12[A1]M2 in the last three terms in (5.34) as +X12[A1]M2 = (X12[A1]M2) +○ + ∑ +σ +1σ +δ cσ(X12[A1]M2)Eσ , +(5.35) + +34 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +where we suppressed the spectral parameters (and the relative sign of their imaginary parts, which has +been fixed by Imw1 > 0 and Imw2 < 0) in the notation for the linear functionals cσ(⋅) on C2N×2N +defined as +c+(B) ∶= ⟨M1BM2⟩ +⟨M1M2⟩ +and +c−(B) ∶= ⟨M1BM ∗ +2 E−⟩ +⟨M1E−M ∗ +2 E−⟩ . +(5.36) +Plugging (5.35) into (5.34) we find G1A1G2 to equal +M1X12[A1]M2 + (G1 − M1)X12[A1]M2 − G1(X12[A1]M2) +○W G2 +(5.37) ++ G1(X12[A1]M2) +○S[G2 − M2]G2 + G1S[(G1 − M1)(X12[A1]M2) +○]G2 ++ ∑ +σ +1σ +δ cσ(X12[A1]M2)[−G1EσW G2 + G1EσS[G2 − M2]G2 + G1S[(G1 − M1)Eσ]G2] . +Recall that the regular component is defined w.r.t. the pair of spectral parameters (w1,w2). In partic- +ular, (X12[A1]M2) +○ = (X12[A1]M2) +○1,2 in the last term in the second line of (5.37) is not regular as +defined via the conditions with one resolvent (4.7). +In the last line of (5.37) we now undo the underline and find the bracket [⋯] to equal (the negative +of) +G1EσW G2 + G1EσS[M2]G2 + G1S[M1Eσ]G2 +=G1Eσ + G1(Eσ(w2 − ˆΛ + S[M2]) + S[M1Eσ])G2 +=G1Eσ − G1(EσM −1 +2 +− S[M1Eσ])G2 =∶ G1Eσ − G1ΦσG2 , +where we used W G2 = E+ + w2G2 − ˆΛG2 in the first step and the MDE (2.19) in the second step. +Moreover, we introduced the shorthand notation +Φσ ∶= Eσ 1 +M2 +− S[M1Eσ]. +(5.38) +From the expansion (5.37) it is apparent (and it can also be checked by hand using the explicit form +of (5.38)) that +M1Eσ = M1(EσM −1 +2 )M2 = M1X12[Φσ]M2 = M(w1,Φσ,w2), +where in the last step we used (4.2). This finally yields that G1A1G2 equals +M(w1,A1,w2) + (G1 − M1)X12[A1]M2 − G1(X12[A1]M2) +○W G2 +(5.39) ++ G1(X12[A1]M2) +○S[G2 − M2]G2 + G1S[(G1 − M1)(X12[A1]M2) +○]G2 ++ ∑ +σ +1σ +δ cσ(X12[A1]M2)[−(G1 − M1)Eσ + (G1ΦσG2 − M(w1,Φσ,w2))] . +The last term in the last line of (5.39) requires further decomposition of Φσ from (5.38) (completely +analogous to (5.35) and (5.36)) as +Φσ = ˚Φσ + ∑ +τ +1τ +δ cτ(Φσ)Eτ . +Using the explicit form of Φσ, we further observe that +cτ(Φσ) ∼ δσ,τ +and +cτ(X12[Φσ]M2) ∼ δσ,τ . +(5.40) +Therefore, by means of the first relation in (5.40), the expansion (5.39) can be carried out further as +M(w1,A1,w2) + (G1 − M1)X12[A1]M2 − G1(X12[A1]M2) +○W G2 +(5.41) ++ G1(X12[A1]M2) +○S[G2 − M2]G2 + G1S[(G1 − M1)(X12[A1]M2) +○]G2 ++ ∑ +σ +1σ +δ cσ(X12[A1]M2) [ − (G1 − M1)Eσ + (G1˚ΦσG2 − M(w1,˚Φσ,w2)) ++ cσ(Φσ)(G1EσG2 − M(w1,Eσ,w2))] . + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +35 +Next, we write (5.41) for both, A1 = ˚ +A1 = ˚Φ+ and A1 = ˚ +A1 = ˚Φ−, and solve the two resulting linear +equations for G1˚Φ±G2 − M(w1,˚Φ±,w2). Observe that by means of the second relation in (5.40) the +original system of linear equations boils down to two separate ones. Thus, plugging the solutions for +G1˚Φ±G2 − M(w1,˚Φ±,w2) back into (5.41) we arrive at +G1A1G2 =M(w1,A1,w2) + (G1 − M1)X12[A1]M2 − G1(X12[A1]M2) +○W G2 +(5.42) ++ G1(X12[A1]M2) +○S[G2 − M2]G2 + G1S[(G1 − M1)(X12[A1]M2) +○]G2 ++ ∑ +σ +1σ +δ cσ(X12[A1]M2) +1 − 1σ +δ cσ(X12[˚Φσ]M2) +[(G1 − M1)X12[˚Φσ]M2 − G1(X12[˚Φσ]M2) +○W G2 ++ G1(X12[˚Φσ]M2) +○S[G2 − M2]G2 + G1S[(G1 − M1)(X12[˚Φσ]M2) +○]G2 +− (G1 − M1)Eσ + cσ(Φσ)(G1EσG2 − M(w1,Eσ,w2))] . +We now need to check that the denominators in (5.42) are bounded away from zero. +Lemma 5.7. For small enough δ > 0, we have that +∣1 − 1σ +δ (w1,w2)cσ(X12[˚Φσ]M2)∣ ≳ 1 +for +σ = ±. +Proof. First, the statements are trivial for 1σ +δ (w1,w2) = 0 and we hence focus on the complemen- +tary extreme scenario 1σ +δ (w1,w2) = 1, the intermediate ones being immediate consequences of the +extreme. Indeed, for 1σ +δ (w1,w2) = 1 we compute +1 − c+(X12[˚Φ+]M2) = ⟨M1⟩⟨M1M2M2⟩ +⟨M1M2⟩2 +and +(5.43) +1 − c−(X12[˚Φ−]M2) = ⟨M1E−M ∗ +2 M −1 +2 E−⟩ + ⟨M1⟩⟨M1E−M ∗ +2 E−⟩ +1 + ⟨M1E−M2E−⟩ +⟨M1E−M2M ∗ +2 E−⟩ +⟨M1E−M ∗ +2 E−⟩2 +for arbitrary spectral parameters w1,w2. Recall that we assumed the two spectral parameters to be on +different halfplanes, i.e. s1 = −sgn(Imw1Imw2) = +, hence we shall specialise (i) the first expression +in (5.43) to w2 = ¯w1 and (ii) the second expression in (5.43) to w2 = −w1. In the former, using Lemma A.4 +and ImM1Imw1 > 0, we obtain +∣1 − c+(X12[˚Φ+]M2)∣ = ∣⟨M1⟩⟨ImM1M1⟩ +⟨ImM1⟩2 (⟨ImM1⟩ + Imw1)∣ ≥ ⟨ImM1⟩2 ≳ 1 +in the bulk of the spectrum, while in the latter we find, similarly to above, again in the bulk, +∣1 − c−(X12[˚Φ−]M2)∣ ≥ ⟨ImM1⟩2 +2 +≳ 1. +These principal lower bounds of order one persist after a small perturbationof w2around the special +cases, but as long as 1σ +δ (w1,w2) = 1 (for some δ > 0 small enough). +□ +Next, we take the scalarproductof (5.42) withtwo deterministicvectorsx,y satisfying∥x∥,∥y∥ ≤ 1. +In the resulting expression,there are two particular terms, namely the ones of the form +(G1S[(G1 − M1)˚ +A1,2 +1 ]G2)xy +and +(5.44) +cσ(X12[˚ +A1,2 +1 ]M2)cσ(Φσ)(G1EσG2 − M(w1,Eσ,w2))xy , +(5.45) +whose direct (naive) estimatesare 1/(Nη2) and 1/η, respectively, and thus do not match the target size. +Hence, they have to be discussed in more detail. In our notation, we emphasised that the regularisation +is defined w.r.t. the spectral parameters (w1,w2), i.e., in particular, A○ +1 = A +○1,2 +1 +. +Estimating (5.44). For the term (5.44), we expand +(G1S[(G1 − M1)˚ +A1,2 +1 ]G2)xy = ∑ +σ +σ⟨(G1 − M1)˚ +A1,2 +1 Eσ⟩(G1EσG2)xy +(5.46) + +36 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +and observe that, by definition of ⋅○ in (4.6), we have, similarly to Lemma 3.3 (see also (5.25)), +˚ +A1,2 +1 Eσ = ( ˚ +A1,2 +1 Eσ) +○1,1 + O(∣e1 − σe2∣ + ∣η1 − η2∣)E+ + O(∣e1 − σe2∣ + ∣η1 − η2∣)E− . +(5.47) +Now, in the second term in (5.46) for σ = + and Eσ = E+, we use a resolvent identity and the usual +isotropic local law (4.15) to estimate it as +∣(G1G2)xy∣ ≺ 1 + +1 +∣e1 − e2∣ + η1 + η2 +. +(5.48) +Furthermore, in the second term in (5.1) for σ = − and Eσ = E−, we employ the integral represen- +tation from Lemma 5.1 in combination with the usual isotropic local law (4.15) (see also (5.27)) to infer +∣(G1E−G2)xy∣ ≺ 1 + +1 +∣e1 + e2∣ + η1 + η2 +. +(5.49) +Combining (5.48) and (5.49) with the decomposition (5.47) and the usual averaged local law (4.15), we find +that (5.46) can be bounded by +∑ +σ +(∣⟨(G1 − M1)(˚ +A1,2 +1 Eσ) +○1,1⟩∣ + ∣e1 − σe2∣ + ∣η1 − η2∣ +Nη1 +) (1 + +1 +∣e1 − σe2∣ + η1 + η2 +) . +Using the definition of Ψav +1 in (4.13) and the apriori bound Ψav +1 +≺ ψav +1 , this immediately implies the +estimate +∣(5.44)∣ ≺ +1 +Nη + +1 +√ +Nη +ψav +1 +(Nη)1/2 . +(5.50) +Estimating (5.45). Forthe term (5.45), we firstnote thatthe two prefactorscσ(X12[A +○1,2 +1 +]M2)andcσ(Φσ) +are bounded. However, in each of the two cases σ = ±, the bound on one of the prefactors needs to be +improved: In the first case, σ = +, we use (A.12) and compute +c+(Φ+) = +⟨M1⟩(1 − ⟨M1M2⟩) +⟨M1M2⟩ += O(∣e1 − e2∣ + η1 + η2) +from (5.36) and (5.38). Combining this with the bound +∣(G1G2 − M(w1,E+,w2))xy∣ ≺ ( +1 +√Nη1 ++ +1 +√Nη2 +) ⋅ +1 +∣e1 − e2∣ + η1 + η2 +which is obtained completely analogous to (5.48), we conclude that (5.45) for σ = + can be estimated by +1/√Nη (recall η ∶= min{η1,η2}). Similarly, in the second case, σ = −, we perform a computation +similar to the one leading to (5.16) and use (A.12) in order to obtain that c−(X12[˚ +A1,2 +1 ]M2) equals +i +2 +⟨M1 ˚ +A1,2 +1 M ∗ +2 E−⟩ +⟨M1E−M ∗ +2 E−⟩ + 1 +2i +⟨M1 ˚ +A1,2 +1 M2E−⟩ +⟨M1E−M ∗ +2 E−⟩ +1 + ⟨M1E−M ∗ +2 E−⟩ +1 + ⟨M1E−M2E−⟩ = O(∣e1 + e2∣ + η1 + η2) +Combining this with the bound +∣(G1E−G2 − M(w1,E−,w2))xy∣ ≺ +1 +√Nη ⋅ +1 +∣e1 + e2∣ + η1 + η2 +which is obtained completely analogous to (5.49), we conclude that (5.45) can be estimated by 1/√Nη – +now in both cases σ = ±. +Conclusion. Summarizing our investigations, we have shown that +(G1 ˚ +A1G2 − M(w1, ˚ +A1,w2))xy = −(G1 ˚ +A′ +1W G2)xy + O≺(E iso +1 ) , +where we used the shorthand notation +˚ +A′ +1 ∶= (X12[˚ +A1]M2) +○ + ∑ +σ +1σ +δ cσ(X12[˚ +A1]M2) +1 − 1σ +δ cσ(X12[˚Φσ]M2) +(X12[˚Φσ]M2) +○ +(5.51) +in the underlined term. Combining (5.50) and the bound on (5.45) established above with the usual single +resolvent local laws (4.15) and the bounds on deterministic approximations in Lemma 4.2, we collected +all the error terms from (5.42) in (5.21). +□ + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +37 +5.3. Proof of the third master inequality (4.24c). Let wj ∈ D(ǫ0,κ0) +ℓ+1 +for j ∈ [2] be spectral parame- +ters and A1 a regular matrix w.r.t. (w1,w2) and A2 a regular matrix w.r.t. (w2,w1) (see Definition 4.1). +By conjugation with E−, we again assume w.l.o.g. that Imw1 > 0 and Imw2 < 0. Just as in Section 5.2, +we use the notations ej ≡ Rewj, ηj ∶= ∣Imwj∣ for j ∈ [2] and define 1 ≥ η ∶= minj ∣Im wj∣. We also +assume that (4.23) holds. +Lemma 5.8. (Representation as full underlined) +For any (w1,w2)-regular matrix A1 = ˚ +A1 and (w2,w1)-regular matrix A2 = ˚ +A2, we have that +⟨(G1 ˚ +A1G2 − M(w1, ˚ +A1,w2)) ˚ +A2⟩ = −⟨W G1 ˚ +A1G2 ˚ +A′ +2⟩ + O≺(E av +2 ) +(5.52) +for some (w2,w1)-regular matrix A′ +2 = ˚ +A′ +2, which linearly depends on A2 = ˚ +A2 (analogously to (5.51), see +(E.18) for an explicit formula). For the error term in (5.52), we used the shorthand notation +E av +2 +∶= +1 +Nη (1 + (ψav +1 )2 +Nη ++ ψav +2 +Nη ) . +(5.53) +Note that similarly to Lemma 5.2 but contrary to Lemma 5.6, we again expanded the first resolvent +G1. Otherwise, the proof of Lemma 5.8, given in Appendix E, is very similar to the one of Lemma 5.6. +We only mention that the quadratic error (ψav +1 )2 stems from terms of the form +⟨S[G1 ˚ +A1,2 +1 G2](G2 − M2)˚ +A2,1 +2 ⟩, +appearing in the analogue of (5.42) (see (E.9) in Appendix E). Having the approximate representation +(5.52), we turn to the proof of (4.24c) via cumulant expansion of the full underlined term. +Proof of (4.24c). Let p ∈ N. Starting from (5.6), we obtain, as in the proofs of (4.24a) and (4.24b), +E∣⟨(G1 ˚ +A1G2 − M(w1, ˚ +A1,w2))˚ +A2⟩∣ +2p +(5.54) +≲E ̃Ξav +2 ∣⟨(G1 ˚ +A1G2 − M(...))˚ +A2⟩∣ +2p−2 ++ +∑ +∣l∣+∑(J∪J∗)≥2 +EΞav +2 (l,J,J∗)∣⟨(G1 ˚ +A1G2 − M(...))˚ +A2⟩∣ +2p−1−∣J∪J∗∣ + O≺((E av +2 )2p) , +where +̃Ξav +2 ∶= +1 +N 2 ∑ +σ +∣⟨G1 ˚ +A1G2 ˚ +A2G1EσG1 ˚ +A1G2 ˚ +A′ +2Eσ⟩∣ + ⋯ +with the other terms being analogous, just 1 and 2 in the first half G1 ˚ +A1G2 ˚ +A2G1 of the chain inter- +changed or the entire half taken as adjoint, and Ξav +2 (l,J,J∗) is defined as +Ξav +2 ∶= N −(∣l∣+∑(J∪J∗)+3)/2 ∑ +ab +Rab∣∂l(G1 ˚ +A1G2 ˚ +A′ +2)ba∣ +(5.55) +× ∏ +j∈J +∣∂j⟨G1 ˚ +A1G2 ˚ +A2⟩∣ ∏ +j∈J∗ +∣∂j⟨G∗ +2 ˚ +A∗ +2G∗ +1 ˚ +A∗ +1⟩∣. +As in Sections 5.1 and 5.2, in the remainder of the proof, we need to analyze the rhs. of (5.54). We begin +with the second line and study the terms involving Ξav +2 from (5.55) afterwards. +Gaussian contribution: second line of (5.54). Along the principal strategy outlined in Remark 5.3, we +need to analyze in total eight terms, each of which carries one of the summands in the definition of ̃Ξav +2 +as a factor. Since their treatment is very similar, we focus on the exemplary term +⟨G1 ˚ +Aw1,w2 +1 +G2 ˚ +Aw2,w1 +2 +G1G1 ˚ +Aw1,w2 +1 +G2(˚ +A′ +2)w2,w1⟩. +(5.56) +Now, we represent G1G1 via the integral representation from Lemma 5.1 with +τ = +, +J = Bℓκ0 , +and +˜η = +ℓ +ℓ + 1η , +for which we recall that w ∈ D(ǫ0,κ0) +ℓ+1 +, i.e. in particular η ≥ (ℓ + 1)N −1+ǫ0 and hence ˜η ≥ ℓN −1+ǫ0. +After splitting the contour integral and bounding the individual contributions as described in (5.11), we + +38 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +obtain, with the aid of Lemma 4.2, +∣(5.56)∣ ≺ 1 +η2 (1 + ψav +4 +Nη ) + ∫Bℓκ0 +∣⟨G1 ˚ +Aw1,w2 +1 +G2 ˚ +Aw2,w1 +2 +G(x + i˜η)˚ +Aw1,w2 +1 +G2(˚ +A′ +2)w2,w1⟩∣ +(x − e1)2 + η2 +1 +dx . +Next, we decompose ˚ +Aw2,w1 +2 +and ˚ +Aw1,w2 +1 +in the integrand as +˚ +Aw2,x+i˜η +2 += ˚ +Aw2,w1 +2 ++ ∑ +σ +Oσ(∣x − e1∣ + ∣η1 − ˜η∣)Eσ +˚ +Ax+i˜η,w2 +1 += ˚ +Aw1,w2 +1 ++ ∑ +σ +Oσ(∣x − e1∣ + ∣η1 − ˜η∣)Eσ . +(5.57) +While the properly regularised term contributes an η−2(1+ψav +4 /(Nη))-error, a typical cross term +shall be estimated as +∫Bℓκ0 +∣⟨G1 ˚ +Aw1,w2 +1 +G2 ˚ +Aw2,x+i˜η +2 +[G(x + i˜η) − G2](˚ +A′ +2)w2,w1⟩∣ +(∣x − e1∣ + η1) (∣x − e2∣ + η2) +≺ 1 +η2 (1 + ψiso +2 +√Nη ) +(5.58) +where in the second step we wrote out the averaged trace and estimated each summand in isotropic +form with the aid of Lemma 4.2, using ψiso +2 +instead of ψav +3 . +Finally, for ‘error × error’-type terms are bounded by η−2, simplyby using a trivial Schwarz inequal- +ity in combination with a Ward identity and the usual local law from Theorem 2.6 to infer +∣⟨G1B1G2B2∣ ≤ +√ +⟨G1B1B∗ +1G∗ +1⟩⟨G2B2B∗ +2G∗ +2⟩ ≤ 1 +η +√ +⟨ImG1B1B∗ +1⟩⟨ImG2B2B∗ +2⟩ ≺ 1 +η , +which is valid for arbitrary bounded matrices ∥B1∥,∥B2∥ ≲ 1. +This finishes the estimate for the Gaussian contribution from the second line of (5.54), for which, +collecting the above estimates, we have shown that +̃Ξav +2 ≺ +1 +N 2η2 (1 + ψiso +2 +√Nη + ψav +4 +Nη ) . +(5.59) +We are now left with the terms from the last line of (5.54) resulting from higher order cumulants. +Higher order cumulants and conclusion. The estimate stemming from higher order cumulants is +given in (5.68c) in Section 5.5. Then, plugging (5.59) and (5.68c) into (5.54), we find, similarly to Section 5.1, +that +Ψav +2 ≺ 1 + (ψav +1 )2 + (ψiso +1 )2 + ψav +2 +Nη ++ ψiso +2 ++ (ψav +4 )1/2 +(Nη)1/2 ++ (ψiso +2 )1/2 +(Nη)1/4 + (ψiso +3 )3/8 + (ψiso +4 )3/8 +(Nη)3/16 +. +The bound given in Proposition 4.8 is an immediate consequence after a trivial Young inequality. +□ +5.4. Proof of the fourth master inequality (4.24d). Let wj ∈ D(ǫ0,κ0) +ℓ+1 +for j ∈ [3] be spectral parame- +ters and A1 a regular matrix w.r.t. (w1,w2) and A2 a regular matrix w.r.t. (w2,w3) (see Definition 4.1). +By conjugation with E−, we will assume w.l.o.g. that Imw1 > 0, Imw2 < 0, and Imw3 > 0. As before, +we use the notations ej ≡ Rewj, ηj ∶= ∣Imwj∣ for j ∈ [3] and define 1 ≥ η ∶= minj ∣Im wj∣. We also +assume that (4.23) holds. +Lemma 5.9. (Representation as full underlined) +For ∥x∥,∥y∥ ≤ 1 and any (w1,w2)-regular matrix A1 = ˚ +A1 and (w2,w3)-regular matrix A2 = ˚ +A2, +we have that +(G1 ˚ +A1G2 ˚ +A2G3 − M(w1, ˚ +A1,w2, ˚ +A2,w3))xy = −(G1 ˚ +A′ +1W G2 ˚ +A2G3)xy + O≺(E iso +2 ) +(5.60) +for some other (w1,w2)-regular matrix A′ +1 = ˚ +A′ +1, which linearly depends on A1 = ˚ +A1 (analogously to +(5.51), see (E.33) for an explicit formula). For the error term in (5.60), we used the shorthand notation +E iso +2 +∶= +1 +√ +Nη3 (1 + ψiso +1 ++ ψav +1 ψiso +1 +Nη ++ ψiso +2 +Nη ) . +(5.61) + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +39 +Note that similarly to (5.20), we again expanded the second resolvent. The proof of Lemma 5.9, +given in Appendix E, is very similar to the one of Lemma 5.6. We only mention that the errors carrying +ψiso +1 ψav +1 and ψiso +1 +stem from terms of the form +(G1S[(G1 − M1)A +○1,2 +1 +]G2 ˚ +A2G3)xy +and +cσ(X12[˚ +A1]M2)cσ(Φσ)(G1EσG2 ˚ +A2G3 − M(w1,Eσ,w2, ˚ +A2,w3))xy , +respectively, appearing in the analogue of (5.42) (see (E.24) and (E.26) in Appendix E). Having the repre- +sentation (5.60) we turn to the proof of (4.24d) via cumulant expansion of the underlined term. +Proof of (4.24d). Let p ∈ N. Then, starting from (5.60), we obtain +E∣(G1 ˚ +A1G2 ˚ +A2G3 − M(w1, ˚ +A1,w2, ˚ +A2,w3))xy∣ +2p +(5.62) +≲E ̃Ξiso +2 ∣(G1 ˚ +A1G2 ˚ +A2G3 − M(...))xy∣ +2p−2 + O≺((E iso +1 )2p) ++ +∑ +∣l∣+∑(J∪J∗)≥2 +EΞiso +2 (l,J,J∗)∣(G1 ˚ +A1G2 ˚ +A2G3 − M(...))xy∣ +2p−1−∣J∪J∗∣ , +where +̃Ξiso +2 +∶= +∑σ ∑3 +j=1 ∣(G1 ˚ +A′ +1EσGj ˚ +Aj ... G3)xy(G1 ˚ +A1 ... ˚ +Aj−1GjEσG2 ˚ +A2G3)xy∣ +N ++ +∑σ ∑3 +j=1 ∣(G1 ˚ +A′ +1EσG∗ +j ˚ +A∗ +j−1 ... ˚ +A∗ +1G∗ +1)xx(G∗ +3 ... ˚ +A∗ +jG∗ +j EσG2 ˚ +A2G3)yy∣ +N +and Ξiso +2 (l,J,J∗) is defined as +Ξiso +2 +∶= N −(∣l∣+∑(J∪J∗)+1)/2 ∑ +ab +Rab∣∂l[(G1 ˚ +A′ +1)xa(G2 ˚ +A2G3)by]∣ +(5.63) +× ∏ +j∈J +∣∂j(G1 ˚ +A1G2 ˚ +A2G3)xy∣ ∏ +j∈J∗ +∣∂j(G∗ +3 ˚ +A∗ +2G∗ +2 ˚ +A∗ +1G∗ +2)yx∣. +We need to analyze the rhs. of the inequality derived in (5.62). We begin with the second line. +Gaussian contribution: second line of (5.62).FollowingRemark5.3, we needto analyze intotaltwelve +terms, each of which carries one of the summands in the definition of ̃Ξiso +2 +as a factor. Again, using +Lemma 3.3 for the A’s, we pick two exemplary terms +(G1 ˚ +Aw1,w2 +1 +G2 ˚ +Aw2,w3 +2 +G3E−G2 ˚ +Aw2,w3 +2 +G3)xy(G1 ˚ +(A′ +1) +w1,w2E−G3)xy +(5.64) +(G1(˚ +A′ +1)w1,w2G∗ +2( ˚ +A∗ +1) ¯ +w2, ¯ +w1G∗ +1)xx(G∗ +3 ˚ +(A∗ +2) +¯ +w3, ¯ +w2G∗ +2G2 ˚ +Aw2,w3 +2 +G3)yy +(5.65) +which shall be treated in more detail. The other terms are analogous and hence omitted. +The term (5.64). In the first factor, we use (2.16), Lemma 3.3, Lemma 4.2 and Lemma 5.1 with parameters +τ = +, +J = B(ℓ+ 1 +2 )κ0 , +and +˜η = +2ℓ +2ℓ + 1η , +(in order to have some flexibility before approaching the boundary of the domain D(ǫ0,κ0) +ℓ +) to bound +it as +∣(G1 ˚ +Aw1,w2 +1 +G2 ˚ +Aw2,w3 +2 +G3E−G2 ˚ +Aw2,w3 +2 +G3)xy∣ ≺ +1 +η3/2 (1 + ψiso +3 +√Nη ) ++ ∫B(ℓ+ 1 +2 )κ0 +∣(G1 ˚ +Aw1,w2 +1 +G2 ˚ +Aw2,w3 +2 +G(x + i˜η) +˚ +(E−A2) +−w2,w3G3)xy∣ +(∣x − e3∣ + η3) (∣x + e2∣ + η2) +dx . + +40 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Next, we decompose ˚ +Aw2,w3 +2 +and +˚ +(E−A2) +−w2,w3 according to the integration variable with the +aid of Lemma 3.3 (iii), analogously to (5.57). This leaves us with four terms, which shall be estimated +separately. While the fully regularised term gives +1 +η3/2 (1 + ψiso +3 +√Nη )(1 + +1 +∣e2 + e3∣ + η2 + η3 +) , +the cross terms can be estimated as +1 +η2 (1 + ψiso +2 +√Nη ) , +analogously to (5.58). As an exemplary error term, we consider +∫B(ℓ+ 1 +2 )κ0 +∣(G1 ˚ +Aw1,w2 +1 +G2E+G(x + i˜η)E−G3)xy∣dx +(5.66) +and use Lemma 5.1 with new parameters +τ = −, +J = Bℓκ0 , +˜η = +ℓ +ℓ + 1η , +to find, dropping the integration domains for ease of notation, +∣(5.66)∣ ≺ +1 +η1/2 (1 + ψiso +1 +√Nη ) + ∫ dx ∫ dy +∣(G1 ˚ +Aw1,w2 +1 +G(y − i˜η))x(E−y)∣ +(∣y − e2∣ + η2) (∣y + x∣ + η) (∣y + e3∣ + η3) +≺ +1 +η3/2 (1 + ψiso +1 +√Nη ) (1 + +1 +∣e2 + e3∣ + η2 + η3 +) , +where in the last step we used Lemma 3.3 for decomposing ˚ +Aw1,w2 +1 +accordingly, and Lemma 4.2. +This finishes the bound on the first factor in (5.64). The second factor can easily be estimated as +∣(G1 ˚ +(A′ +1) +w1,w2E−G3)xy∣ ≺ +1 +η1/2 (1 + ψiso +1 +√Nη ) + ∣e2 + e3∣ + η2 + η3 +η +using (2.16), Lemma 3.3, and Lemma 4.2. Notice the cancellation of ∣e2 + e3∣ between the two factors. +The term (5.65). For the first factor in (5.65), we realise that (˚ +A′ +1)w1,w2 = (˚ +A′ +1)w1, ¯ +w2, which without +approximation immediately yields that +∣(G1(˚ +A′ +1)w1,w2G∗ +2( ˚ +A∗ +1) ¯ +w2, ¯ +w1G∗ +1)xx∣ ≺ 1 +η (1 + ψiso +2 +√Nη ) +with the aid of Lemma 4.2. +In the second factor, we apply a Ward identity to G∗ +2G2 and again use that the regularisation is +insensitive to complex conjugation in the second spectral parameter. In this way, and decomposing +˚ +Aw2,w3 +2 += ˚ +A ¯ +w2,w3 +2 ++ O(∣e2 − e3∣ + ∣η2 − η3∣)E+ + O(∣e2 + e3∣ + ∣η2 − η3∣)E− +by means of Lemma 3.3 (ii), we find that the second factor is stochastically dominated by +1 +η2 (1 + ψiso +1 ++ ψiso +2 +√Nη +) . +This finishes the estimate for the Gaussian contribution from the second line of (5.62), for which, +collecting the above estimates, we have shown that +̃Ξiso +2 +≺ +1 +Nη3 +⎡⎢⎢⎢⎢⎣ +(1 + ψiso +3 +√Nη )(1 + ψiso +1 +√Nη ) + (1 + ψiso +1 ++ ψiso +2 +√Nη +) +2⎤⎥⎥⎥⎥⎦ +. +(5.67) +We are now left with the terms from the last line of (5.62) resulting from higher order cumulants. +Higher order cumulants and conclusion. The estimate stemming from higher order cumulants is + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +41 +given in (5.68d) in Section 5.5. Then, plugging (5.67) and (5.68d) into (5.62), we find, similarly to Section 5.1, +that +Ψiso +2 +≺ 1 + ψiso +1 ++ ψav +1 ψiso +1 ++ (ψiso +1 )2 + ψiso +2 +Nη ++ ψiso +2 ++ (ψiso +1 ψiso +3 )1/2 +(Nη)1/2 ++ (ψiso +3 )3/8 + (ψiso +4 )3/8 +(Nη)3/16 +The bound given in Proposition 4.8 is an immediate consequence after a trivial Young inequality. +□ +5.5. Contributions from higher order cumulants. The goal of the present section is to estimate the +terms originating from higher order cumulants in (5.8), (5.22), (5.54), and (5.62). In order to do so, we +assume that (4.23) holds. +Lemma 5.10. For any J,J∗ ⊂ Z2 +≥0 ∖ {(0,0)}, l ∈ Z2 +≥0 with ∣l∣ + ∑(J ∪ J∗) ≥ 2 it holds that +(Ξav +1 ) +1/(1+∑(J∪J∗)) ≺ +1 +Nη1/2 (1 + +ψiso +1 +(Nη)1/2 + (ψiso +2 )1/4 +(Nη)1/8 ) , +(5.68a) +(Ξiso +1 ) +1/(1+∑(J∪J∗)) ≺ +1 +√ +Nη2 (1 + +ψiso +1 +(Nη)1/2 + (ψiso +2 )1/4 +(Nη)1/8 ) , +(5.68b) +(Ξav +2 ) +1/(1+∑(J∪J∗)) ≺ +1 +Nη (1 + (ψiso +1 )2 +Nη ++ +ψiso +2 +(Nη)1/2 + (ψiso +3 )3/8 + (ψiso +4 )3/8 +(Nη)3/16 +) , +(5.68c) +(Ξiso +2 ) +1/(1+∑(J∪J∗)) ≺ +1 +√ +Nη3 (1 + (ψiso +1 )2 +Nη ++ +ψiso +2 +(Nη)1/2 + (ψiso +3 )3/8 + (ψiso +4 )3/8 +(Nη)3/16 +) . +(5.68d) +For k = 1,2, l ∈ Z2 +≥0 and a multiset J ⊂ Z2 +≥0 ∖ {(0,0) } we now define slightly (notationally) +simplified versions of Ξav/iso +k +, namely +Ξav +k (l,J) ∶= N −(∣l∣+∑ J+3)/2 ∑ +ab +∣∂l((GA)k−1GA′)ba∣ ∏ +j∈J +∣∂j ⟨(GA)k⟩∣ , +(5.69) +Ξiso +k (l,J) ∶= N −(∣l∣+∑ J+1)/2 ∑ +ab +∣∂l[(GA)xa(G(AG)k−1)by]∣ ∏ +j∈J +∣∂j((GA)kG)xy∣, +(5.70) +where ∑ J ∶= ∑j∈J∣j∣, ∣(j1,j2)∣ ∶= j1 + j2 and ∂(j1,j2) ∶= ∂j1 +ab∂j2 +ba. Here, for notational simplicity, +we do not carry the dependence on the spectral parameters of the resolvents but assume that implicitly +each resolvent has its own spectral parameter and that each A is correctly regularised with respect to +its neighboring resolvents. In particular compared to (5.9), (5.23), (5.55), and (5.63), it is not necessary to +distinguish the sets J,J∗. +Proof of Lemma 5.10. Throughout the proof, we denote φk ∶= ψiso +k /√Nη. The naive estimate for the +derivatives simply is +∣∂l((GA)k−1GA′)ba∣ ≺ η−(k−1)/2(1 + φk−1) , +∣∂j ⟨GA⟩∣ ≺ +1 +Nηk/2 +∑ +k1+k2+⋯=k +∏ +i +(1 + φki) +(5.71) +due to (4.8) and recalling (4.14). Using (5.71) in (5.69) we obtain +∣Ξav +1 ∣ ≺ (Nη1/2)−1−∣J∣N (2−∣l∣−∑ J)√ +Nη (1 + φ1) +∣J∣ +, +∣Ξav +2 ∣ ≺ (Nη)−1−∣J∣N (2−∣l∣−∑ J)√ +Nη (1 + φ1)(1 + φ2 + φ2 +1) +∣J∣ +, +∣Ξiso +1 ∣ ≺ ( +√ +Nη)−1−∣J∣η1+∣J∣/2N (4−∣l∣+∣J∣−∑ J)/2(1 + φ1) +∣J∣ +, +∣Ξiso +2 ∣ ≺ ( +√ +Nη3/2)−1−∣J∣η1+∣J∣/2N (4−∣l∣+∣J∣−∑ J)/2(1 + φ1)(1 + φ2 + φ2 +1) +∣J∣ +, +(5.72) + +42 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +and therefore have proved (5.68a) and (5.68c) in all cases except ∣l∣ + ∑ J = 2 and (5.68b) and (5.68d) in +all cases except ∣l∣ + ∑ J − ∣J∣ < 4. For the remaining cases we need a more refined estimate using the +following Ward lemma: +Lemma 5.11. Let x be any deterministic vector of bounded norm, let w1,... ,wk ∈ D(ǫ0,κ0) +ℓ+1 +be spectral +parameters and A1,... ,Ak deterministic matrices of bounded norm. Then for Gi = G(wi) it holds that +1 +N ∑ +a +∣(G1 ˚ +Aw1,w2 +1 +⋯˚ +Awk−1,wk +k−1 +GkAk)xa∣ ≺ +1 +√Nη +1 +η(k−1)/2 (1 + φ1 + ⋯ + φ2k) +1/2 +, +which improves upon the term-wise bound by a factor of (Nη)−1/2 at the expense of replacing 1 + φk by +1 + √φ1 + ⋯ + φ2k. +The proof of the above Ward lemma is largely based on yet another more general estimate. +Lemma 5.12. Let x,y be normalised vectors, let w1,... ,wk+1 ∈ D(ǫ0,κ0) +ℓ+1 +be spectral parameters and +A1,... ,Ak be deterministic matrices of bounded norm such that a of them are regular, i.e. ˚ +Awi,wi+1 +i += Ai +for all i ∈ I for some I ⊂ [k] of cardinality a. Then with Gi = G(wi) it holds that +∣(G1A1G2⋯AkGk+1)xy∣ ≺ +1 +ηk−a/2 (1 + φ1 + ⋯ + φa). +(5.73) +We defer the proof of Lemma 5.12 to the end of this section. +Proof of Lemma 5.11. By Cauchy-Schwarz and the norm bound on the middle Ak we have +( 1 +N ∑ +a +∣(G1 ˚ +Aw1,w2 +1 +⋯˚ +Awk−1,wk +k−1 +GkAk)xa∣) +2 +≲ 1 +N (G1 ˚ +Aw1,w2 +1 +⋯˚ +Awk−1,wk +k−1 +GkG∗ +k ˚ +A ¯ +wk, ¯ +wk−1 +k−1 +⋯˚ +A ¯ +w2, ¯ +w1) +1 +G∗ +1) +xx +≺ +1 +Nηk (1 + φ1 + ⋯ + φ2k) +due to Lemma 5.12 for 2k resolvents and a = 2k − 2 regularised A-matrices. +□ +The rest of the proof is split into several cases. +Treatment of (5.68a) and (5.68c) for ∣l∣ + ∑J = 2: For the case ∣l∣ + ∑ J = 2 we either have ∣l∣ ∈ {0,2 } +or ∑J = 1 = ∣J∣. In the former case an off-diagonal resolvent is guaranteed to be present in the +first factor of (5.69) (by parity) and in the latter case the second factor consists of a single off-diagonal +resolvent chain. In either case we may use Lemma 5.11 to gain a factor of 1/√Nη compared to (5.71) and +obtain +∣Ξav +1 ∣ ≺ (Nη1/2)−1−∣J∣(1 + φ1)(∣J∣−1)+(1 + φ1 + 1(∣J∣ ≥ 1)φ1/2 +2 ) , +∣Ξav +2 ∣ ≺ (Nη)−1−∣J∣(1 + φ2 +1 + φ2)(∣J∣−1)+(1 + φ3 +1 + φ3/2 +2 ++ 1(∣J∣ ≥ 1)(φ3 + φ4)3/4), +(5.74) +where we used the fact that for ∣J∣ = 0 only a single factor of (1 + φ1) needs to be replaced by a +factor of (1 + (φ1 + φ2)1/2) for Ξav +2 and no factor needs to be replaced for Ξav +1 . Moreover, we used +φ1(φ3 + φ4)1/2 + φ2 +1φ1/2 +2 +≲ φ3 +1 + φ3/2 +2 ++ (φ3 + φ4)3/4 by a simple Young inequality. Now (5.74) +implies (5.68a) and (5.68c) by another simple Young inequality. +Treatment of (5.68b) and (5.68d) for ∣l∣+∑J −∣J∣ ∈ {2,3 }: In this case we can simply use Lemma 5.11 for +the two resolvent chains in the first factor of (5.70) involving x,y to gain a factor of (Nη)−1 compared +to (5.71) at the expense of replacing 1 + φ1 by 1 + φ1/2 +1 ++ φ1/2 +2 +in case of Ξiso +2 +which proves (5.68b) and +(5.68d) in this case. +Treatment of (5.68b) and (5.68d) for ∣l∣+∑J −∣J∣ = 0: In this case we necessarily have ∣l∣ = 0 and ∣J∣ ≥ 2 +and ∣j∣ = 1 for all j ∈ J. In particular all factors of (5.70) consist of two resolvent chains evaluated in + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +43 +(x,a),(y,b) or (x,b),(y,a), respectively. This allows to use Lemma 5.11 four times (twice for the a- +and twice for the b-summation) to gain a factor of (Nη)−2 comparedto (5.71) at the expense of replacing +one factor of +(1 + φ1) +by +(1 + (φ1 + φ2)1/2) +in case of Ξiso +1 +and +one factor of +(1 + φ1)(1 + φ2 +1 + φ2) +by +(1 + (φ1 + φ2)1/2)(1 + φ1 + φ2 + (φ3 + φ4)1/2) (5.75) +in case of Ξiso +2 . This concludes the proof in case of Ξiso +1 +and together with +(1 + (φ1 + φ2)1/2)(1 + φ1 + φ2 + (φ3 + φ4)1/2) ≲ 1 + (φ1 + φ2)3/2 + (φ3 + φ4)3/4 +also in case of Ξiso +2 . +Treatment of (5.68b) and (5.68d) for ∣l∣ + ∑J − ∣J∣ = 1: In this case we necessarily have ∣J∣ ≥ 1 and +either ∣l∣ = 0 or ∣j∣ = 1 for all j ∈ J. In either case we can use Lemma 5.11 twice for the first factor and +once for some other factor in (5.70) to gain a factor of (Nη)−3/2 compared to (5.71) at the expense of +replacing (5.75) in case of Ξiso +1 +and +one factor of +(1+φ1)(1+φ2 +1+φ2) +by +(1+(φ1+φ2)1/2)((1+φ1)(1+φ1+φ2)1/2+(φ3+φ4)1/2) +in case of Ξiso +2 . Together with +(1 + (φ1 + φ2)1/2)((1 + φ1)(1 + φ1 + φ2)1/2 + (φ3 + φ4)1/2) ≲ 1 + (φ3 + φ4)3/4 + φ3/2 +2 ++ φ2 +1 +this concludes the proof also in this case. +□ +It remains to give the proof of Lemma 5.12. +Proof of Lemma 5.12. The proof is via induction, i.e. we assume that (5.73) has been established for re- +solvent chains of up to k resolvents. For k + 1 resolvents and a = k, i.e. in case when all deterministic +matrices are regular, the claim follow by definition of ψiso +k . Therefore we may assume that some Aj +is not regular which we decompose into its regular component ˚ +A +wj,wj+1 +j +and a linear combination of +E±. By linearity it thus suffices to check (5.73) for the cases Aj = E±, and moreover, by chiral symmetry +GjE−Gj+1 = −E−G(−wj)E+Gj+1 and ˚ +Awj−1,wjE− = ˚ +Awj−1,−wj (recall Lemma 3.3) the estimate +for E− follows from the estimate for E+ upon replacing wj by −wj. Therefore it suffices to check (5.73) +in case Aj = E+. +If sj = −sgn(Im wjImwj+1) = +, i.e. the adjacent spectral parameters lie in opposite half-planes, +then we use the resolvent identity to write +Aj−1GjE+Gj+1Aj+1Gj+2 = Aj−1 Gj − Gj+1 +wj − wj+1 +Aj+1Gj+2 . +We discuss each of the two resulting summands separately. For the summand involving Gj+1, if Aj−1 +was not counted as regularised, i.e. j−1 /∈ I, then the claim followsby induction and the trivial estimate +∣wj − wj+1∣ ≥ η since k has been reduced by one, while a has been preserved. On the other hand, if +Aj−1 was correctly regularised, then we use Lemma 3.3 to write +˚ +A +wj−1,wj +j−1 += ˚ +A +wj−1, ¯ +wj +j−1 += ˚ +A +wj−1,wj+1 +j−1 ++ O(∣ ¯wj − wj+1∣)E+ + O(∣ ¯wj − wj+1∣)E− . +(5.76) +Inserting (5.76) into Aj−1Gj+1Aj+1Gj+2/(wj −wj+1) the claimed bound followsfrom induction since +for the ˚ +A +wj−1,wj+1 +j−1 +-term a has been preserved and k has been reduced by one compensating for ∣wj − +wj+1∣ ≥ η, while for E± both k,a have been reduced by one and ∣ ¯wj − wj+1∣/∣wj − wj+1∣ ≤ 1. Next, +for the summand involving Gj, the argument is completely analogous, apart from the two error terms +in +˚ +A +wj,wj+1 +j+1 += ˚ +A +wj,wj+2 +j+1 ++ O(∣wj − ¯wj+1∣ + ∣wj − sj+1wj+2∣)Esj+1 +(5.77) ++ O(∣wj − ¯wj+1∣ + ∣wj + sj+1 ¯wj+2∣)E−sj+1 , +appearing for an Aj+1 = ˚ +A +wj+1,wj+2 +j+1 +, which has been correctly regularised. Here, we applied Lemma 3.3 +and denoted, as usual, sj+1 = −sgn(Im wj+1Imwj+2). Now, for the error terms, we assume that the + +44 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +second summand in each O(...) is non-zero (otherwise we are back to (5.76)) and argue by induction: +Indeed, using (2.16) and applying a resolvent identity, we find +∣wj − ¯wj+1∣ + ∣wj − sj+1wj+2∣ +wj − wj+1 +GjEsj+1Gj+2 +(5.78) += ∣wj − ¯wj+1∣ + ∣wj − sj+1wj+2∣ +(wj − wj+1)(wj − sj+1wj+2)sj+1(G(wj) − G(sj+1wj+2))Esj+1 , +such that, in the resulting chain we have reduced k by two and a by one, and the prefactor in (5.78) is +bounded by 1/η. The argument for the second error in (5.77) is completely analogous, after realizing +that (∣wj − ¯wj+1∣ + ∣wj + sj+1 ¯wj+2∣)/(∣wj − wj+1∣∣wj + sj+1wj+2∣) ≤ 1/η. +On the contrary, if sj = −sgn(ImwjImwj+1) = −, i.e. the adjacent spectral parameters lie the +same half-plane (without loss of generality the upper one), then we use the integral representation from +Lemma 5.1 to write +Aj−1GjE+Gj+1Aj+1 = +1 +2πi ∫Γ +Aj−1G(z)Aj+1 +(z − wj)(z − wj+1) dz , +(5.79) +where Γ is an appropriately chosen contour. If j − 1,j + 1 /∈ I, i.e. both Aj−1,Aj+1 were not counted +as regularised, then the claim follows by induction and estimating the integral by η−1 (up to log factors) +since k has been reduced by one, and a has been preserved. On the other hand, if both Aj−1,Aj+1 were +counted as regularised, then we use Lemma 3.3 to write them as +˚ +A +wj−1,wj +j−1 += ˚ +A +wj−1,z +j−1 ++ O(∣wj − z∣)E+ + O(∣wj − z∣)E− , +˚ +A +wj+1,wj+2 +j+1 += ˚ +A +z,wj+2 +j+1 ++ O(∣wj+1 − z∣)E+ + +O(∣wj+1 − z∣)E− . +(5.80) +The resulting term with ˚ +A +wj−1,z +j−1 +, ˚ +A +z,wj+2 +j +can be estimated by induction since k has been reduced by +one, a has been preserved and the integral may be estimated by η−1. The other terms with either one +or two E± can also be estimated by induction since the integral is at most logarithmically divergent, k +has been reduced by one and a by at most two. Finally, if in (5.79) one of Aj−1,Aj+1 were counted as +regularised, then we use the relevant expansion from (5.80), so that for the resulting term with ˚ +A, k has +been reduced by one, and a has been preserved, so that the η−1 estimate on the integral is affordable. +The other term with E± can also be estimated by induction with both a,k reduced by one, and the +integral being at most logarithmically divergent. This concludes the proof. +□ +6. Proof of the reduction inequalities, Lemma 4.9 +During the proof of Lemma 4.9, we will heavily rely on the following integral representation for the +absolute value ∣G∣ of a resolvent (see also [28, Lemma 5.1]). +Lemma 6.1. (Integral representation for the absolute value of a resolvent) +Let w = e + iη ∈ C ∖ R. Then the absolute value of the resolvent G(w) can be represented as +∣G(e + iη)∣ = 2 +π ∫ +∞ +0 +ImG(e + i +√ +η2 + s2) +ds +√ +η2 + s2 . +(6.1) +Proof. This immediately follows from the functional calculus for H and the identity +1 +∣x − iη∣ = 1 +iπ ∫ +∞ +0 +( +1 +x − i(η2 + s2)1/2 − +1 +x + i(η2 + s2)1/2 ) +ds +√ +η2 + s2 . +□ +Proof of Lemma 4.9. To keep the notation simpler within this proof we may often denote +Ai = ˚ +Ai = ˚ +Awi,wi+1 +i +, +i.e. sometimes we drop the spectral parameters wi = ei + iηi. +We start with the proof of (4.25), for which, similarly to [28, Lemma 3.6], we get +Ψav +4 ≲ Nη + N 2η2 (⟨∣G1∣A1∣G2∣A∗ +1⟩⟨∣G2∣A2∣G3∣A∗ +2⟩⟨∣G3∣A3∣G4∣A∗ +3⟩⟨∣G4∣A4∣G1∣A∗ +4⟩) +1/2 +, (6.2) + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +45 +by Lemma 4.2, spectral decomposition, and a Schwarz inequality. Next, we use (6.1) to write +⟨∣G1∣A1∣G2∣A∗ +1⟩ = 4 +π2 ∬ +∞ +0 +⟨ImG(w1,s)˚ +Aw1,w2 +1 +ImG(w2,t)(˚ +Aw1,w2 +1 +)∗⟩ +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 , +(6.3) +where we defined wi,s ∶= ei + i +√ +η2 +i + s2. The very large s,t–regimes in (6.3) can be easily shown +to be negligible (e.g. see [28, Proof of Lemma 5.1]), i.e. even if not stated explicitly we assume that +the upper integration limit can be replaced by N 100. Additionally, we can restrict to the case when +η ∶= minj ∣Imwj∣ ≤ 1, when this is not the case we use the local law in the regime η > 1 from +Theorems 4.3–4.4 (see [28, Proof of Lemma 5.1] for a detailed argument). We remark that this argument +is not circular since in the proof of the local law for η > 1 sketched below Remark 4.5 one does not use +the reduction inequalities in (4.25)–(4.26). +In order to estimate the rhs. of (6.3) we write ImG = +1 +2i(G − G∗) for both ImG to obtain four +terms with two resolvents; to keep the presentation concise we only present the estimate for one of +them. From now on we only consider only the term ⟨∣G1∣A1∣G2∣A∗ +1⟩, the bound for all the other terms +in the last line of (6.2) is completely analogous and so omitted. In the following we will often use the +approximations from Lemma 3.3 (omitting the trivial ∧1 in the errors for notational simplicity): +˚ +Aw1,w2 = ˚ +Aw1,s,w2,t + O(∣ +√ +η2 +1 + s2 − η1∣ + ∣ +√ +η2 +2 + t2 − η2∣)E+ ++ O(∣ +√ +η2 +1 + s2 − η1∣ + ∣ +√ +η2 +2 + t2 − η2∣)E− , +(˚ +Aw1,w2)∗ = (˚ +A∗)w2,t,w1,s + O(∣e1 − e2∣ + +√ +η2 +1 + s2 + +√ +η2 +2 + t2)E+ ++ O(∣e1 + e2∣ + +√ +η2 +1 + s2 + +√ +η2 +2 + t2)E− . +(6.4) +We point out that when taking the adjoint of the first formula to arrive at the second we used that for +any w1,w2 it holds (˚ +Aw1,w2)∗ = (˚ +A∗)w2,w1, see Lemma 3.3. Recall that within this proof we always +assume that η ≤ 1. From now on for the error terms we will always use the bounds +∣ +√ +η2 +1 + s2 − η1∣ ≲ s , +√ +η2 +1 + s2 ≤ η1 + s , +(6.5) +and a similar bound with η1,s replaced with η2,t. The first bound is not optimal for small η1, but good +enough for our estimates. Then using (6.4) we write +∬ +∞ +0 +⟨G(w1,s)˚ +Aw1,w2 +1 +G(w2,t)(˚ +Aw1,w2 +1 +)∗⟩ +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 += ∬ +∞ +0 +⟨G(w1,s)˚ +A +w1,s,w2,t +1 +G(w2,t)(˚ +A∗ +1)w2,t,w1,s⟩ +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 ++ +∑ +σ∈{+,−} ∬ +∞ +0 +⟨G(w1,s)EσG(w2,t)(˚ +A∗ +1)w2,t,w1,s⟩O(η1 + η2 + s + t) +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 ++ +∑ +σ∈{+,−} ∬ +∞ +0 +⟨G(w1,s)˚ +A +w1,s,w2,t +1 +G(w2,t)Eσ⟩O(η1 + η2 + s + t) +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 ++ +∑ +σ,τ∈{+,−}∬ +∞ +0 +⟨G(w1,s)EσG(w2,t)Eτ⟩O(η2 +1 + η2 +2 + s2 + t2) +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 ++ ∬ +∞ +0 +⟨G(w1,s)[ ∑ +σ +Oσ(∣e1 − σe2∣)Eσ]G(w2,t)(˚ +A∗ +1)w2,t,w1,s⟩ +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 ++ ∬ +∞ +0 +⟨G(w1,s)˚ +A +w1,s,w2,t +1 +G(w2,t)[O(∣e1 − e2∣)E+ + O(∣e1 + e2∣)E−]⟩ +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 ++ ∬ +∞ +0 +⟨G(w1,s)[ ∑ +σ +Oσ(∣e1 − σe2∣)Eσ]G(w2,t)[∑ +τ +Oτ(∣e1 − τe2∣)Eτ]⟩ +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 . +(6.6) + +46 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +We now estimate the terms in the rhs. of (6.6) one by one. In the following estimates we will always +omit log N-factors. We start with +����������� +∬ +∞ +0 +⟨G(w1,s)˚ +A +w1,s,w2,s +1 +G(w2,t)(˚ +A∗ +1)w2,t,w1,s⟩ +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 +����������� +≺ 1 + ψav +2 +Nη , +which readily follows by the definition of Ψav +2 in (4.13) and from the assumption Ψav +2 +≺ ψav +2 . For the +third to the fifth line in (6.6) we use the bound +����������� +∬ +∞ +0 +⟨G(w1,s)EσG(w2,t)B⟩O(η1 + η2 + s + t) +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 +����������� +≺ ∬ +∞ +0 +⎛ +⎝ +1 +√ +η2 +1 + s2 ∧ +1 +√ +η2 +2 + t2 +⎞ +⎠ [η1 + η2 + s + t] +dsdt +√ +η2 +1 + s2√ +η2 +2 + t2 ≲ 1, +(6.7) +for any deterministic norm bounded matrices B and for σ ∈ {+,−}. For the fifth line of (6.6) we used +the bound (s2 + t2) ∧ 1 ≤ (s + t) ∧ 1 (recall that ∧1 is omitted in the error terms in (6.6) for notational +simplicity). Note that here we used: +∣⟨G(w1,s)EσG(w2,t)B⟩∣ ≺ +1 +√ +η2 +1 + s2 ∧ +1 +√ +η2 +2 + t2 , +(6.8) +which holds uniformly in matrices with ∥B∥ ≲ 1. We point out that to obtain the bound (6.8) we used +spectral decomposition of the resolvents and that ⟨wi,Eσwj⟩ = δi,σj to bound +∣⟨G(w1,s)EσG(w2,t)B⟩∣ = ∣ 1 +2N ∑ +i +⟨wi,Bwσi⟩ +(λi − w1,s)(λi − σw2,t)∣ +≲ 1 +N ∑ +i +1 +∣λi − w1,s∣∣λi − σw2,t∣ +≺ +1 +∣Imw1,s∣ ∨ ∣Imw2,t∣ , +where in the last inequality we used the single resolvent local law. +Finally, for the last three lines in (6.6) we use that for any norm bounded matrix B, by resolvent +identity, we have (recall that E+ = I) +∣⟨G(w1,s)BG(w2,t)⟩∣ ≺ +1 +∣w1,s − w2,t∣ , +∣⟨G(w1,s)BG(w2,t)E−⟩∣ ≺ +1 +∣w1,s + w2,t∣ , +(6.9) +which after the integration in (6.6) gives a bound of order one, as a consequence of +∣e1 ± e2∣ +∣w1,s ± w2,t∣ ≲ 1. +Note that here it is important that the error terms in (6.6) involving ∣e1 −e2∣ are always multiplied with +the matrix E+, while errors of order ∣e1 + e2∣ are in the direction of E−. +Combining the computations in (6.6)–(6.9) we conclude that +∣⟨∣G1∣A1∣G2∣A∗ +1⟩∣ ≺ 1 + ψav +2 +Nη , +(6.10) +which, after plugging it in the rhs. of (6.2), clearly implies (4.25) . +For (4.26) for Ψiso +3 , we find +Ψiso +3 +≲ +√ +Nη + Nη2((G1A1∣G2∣A∗ +1G∗ +1)xx(G∗ +4A∗ +3∣G3∣A3G4)yy⟨∣G2∣A2∣G3∣A∗ +2⟩) +1/2 +, +(6.11) +again by Lemma 4.2, spectral decomposition, and a Schwarz inequality. Then, using again the integral +representation (6.1), we find that +(G1A1∣G2∣A∗ +1G∗ +1)xx = 2 +π ∫ +∞ +0 +(G1A1ImG(w2,s)A∗ +1G∗ +1)xx +ds +√ +η2 +2 + s2 , + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +47 +recalling the notation w2,s = e2 + i +√ +η2 +2 + s2. The estimate for this term is fairly similar to the one in +(6.3), hence we present only the main differences and skip the details; actually the current case is easier +since we now have only one ∣G∣. +After splitting ImG = +1 +2i(G − G∗) and handling both terms separately, we can write, similarly to +(6.6) and using (6.4)–(6.5), the following approximation: +∫ +∞ +0 +(G1A1G(w2,s)A∗ +1G∗ +1)xx +ds +√ +η2 +2 + s2 += ∫ +∞ +0 +(G1 ˚ +A +w1,w2,s +1 +G(w2,s)(˚ +A∗ +1)w2,s,w1G∗ +1)xx +ds +√ +η2 +2 + s2 + E . +(6.12) +Here E is an error coming from all the errors in (6.4). For the first term in the second line of (6.12) we +use the bound +����������� +∫ +∞ +0 +(G1 ˚ +A +w1,w2,s +1 +G(w2,s)(˚ +A∗ +1)w2,s,w1G∗ +1)xx +ds +√ +η2 +2 + s2 +����������� +≺ 1 +η (1 + ψiso +2 +√Nη ) , +(6.13) +which follows by the definition of Ψiso +2 . For the error term we do not write the details, since once we +replace (6.8)–(6.9) with (here B,B1,B2 are deterministic norm bounded matrices) +∣(G1B1G(w2,s)B2G∗ +1)xx∣ ≤ (G1B1B∗ +1G∗ +1) +1/2 +xx (G1B∗ +2G(w2,s)G(w2,s)∗B2G∗ +1) +1/2 +xx ≺ +1 +η +√ +η2 +2 + s2 +∣(G1EσG(w2,s)BG∗ +1)xx∣ ≺ +1 +η∣w1 − w2,s∣ , +(6.14) +respectively, the estimate +∣E∣ ≺ 1 +η +(6.15) +follows completely analogously. The estimates (6.14) follow by repeated applications of the resolvent +identity (after commuting Eσ with G in case of the second formula), the trivial bound ∥G∥ ≤ 1/η and +the single resolvent local law. Combining, (6.13)–(6.15) we conclude +∣(G1A1∣G2∣A∗ +1G∗ +1)xx∣ ≺ 1 +η (1 + ψiso +2 +√Nη ) . +(6.16) +The bound in (6.16), together with (6.10) to estimate the averaged term in (6.11), concludes the proof (4.26) +for Ψiso +3 . +Analogously to (6.11), for Ψiso +4 +we find that +Ψiso +4 +≲ +√ +Nη + Nη5/2((G1A1∣G2∣A∗ +1G∗ +1)xx(G∗ +5A∗ +4∣G4∣A4G5)yy⟨∣G2∣A2G3A3∣G4∣A∗ +3G∗ +3A∗ +2⟩) +1/2 +≲ +√ +Nη + N 3/2η5/2((G1A1∣G2∣A∗ +1G∗ +1)xx(G∗ +5A∗ +4∣G4∣A4G5)yy) +1/2 +× (⟨∣G2∣A2∣G3∣A∗ +2⟩⟨∣G3∣A3∣G4∣A∗ +3⟩⟨∣G4∣A∗ +3∣G3∣A3⟩⟨∣G3∣A∗ +2∣G2∣A2⟩) +1/4 +where in the last inequality we used spectral decomposition and a bound as in [28, Proof of Lemma 3.6] +to bound the trace with four G’s and four A’s in terms of a product of traces containing only two G’s +and two A’s. Finally, using the bounds (6.10), (6.16), we conclude the proof of (4.26) for Ψiso +4 +as well. +□ +Appendix A. Properties of the MDE and the stability operator: Proof of Lemma 3.3 +In the first part of this appendix, we derive several elementary properties of the MDE +− 1 +M = w − ˆΛ + S[M], +w ∈ C ∖ R, +(A.1) + +48 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +(recall (2.19)) and its unique solution M (under the usual constraint Im M⋅Imw > 0) where the operator +S was given in (2.20) and ˆΛ ∈ C2N×2N is from (2.2). Afterwards, in the second part, we turn to the +associated two-body stability operator +B ≡ B(w1,w2) ∶= 1 − M(w1)S[⋅]M(w2) +(A.2) +and its adjoint B∗, understood with respect to the standard (normalised) inner product ⟨S,T ⟩ ∶= ⟨S∗T ⟩ +for S,T ∈ C2N×2N, which is given by +B∗ ≡ B∗(w1,w2) ∶= 1 − S[(M(w1))∗ ⋅ (M(w2))∗]. +(A.3) +Moreover, we also explain the relation between the regularisation from Definition 3.1 and the stability +operator. +Finally, after proving and combining LemmasA.1 and A.4 with Lemma A.6 on M and B, respectively, +we will complete the proof of Lemma 3.3. +A.1. The Matrix Dyson Equation (A.1) and its solution. Existence and uniqueness of the solution to +(A.1) under the constraint ImM ⋅Imw > 0 has already been shown in [42]. By [2, Prop. 2.1], this solution +can also be represented as the Stieltjes transform of a compactly supported semi-definite matrix-valued +probability measure on R, which has the immediate consequence that ∥M(w)∥ ≤ ∣Imw∣−1. +Lemma A.1. Let M be the unique solution to (A.1) and write its 2 × 2-block representation as +M = (M11 +M12 +M21 +M22) . +(A.4) +Then we have the following: +(a) The average trace ⟨M⟩ coincides with the solution m of (2.4), ⟨M(w)⟩ = m(w), and the blocks +in (A.4) are given by (2.17)–(2.18). We have M ∗(w) = M( ¯w). +(b) The solution has a continuous extension to the real line from the upper half plane, denoted by +M(e) ∶= limη↓0 M(e + iη); the limit from the lower half plane is M ∗(e). The self-consistent +density of states of the MDE, defined as ρ(e) = 1 +π ⟨ImM(e)⟩, is identical to the free convolution +of µˆΛ ⊞ µsc from (2.3). Both ρ and its Stieltjes transform m are Hölder continuous with a small +universal exponent c, i.e. +∣ρ(e1) − ρ(e2)∣ ≤ C∣e1 − e2∣c, +e1,e2 ∈ R, +and +∣m(w1) − m(w2)∣ ≤ C′∣w1 − w2∣c, +w1,w2 ∈ C+, +(A.5) +where C,C′ depend only on ∥Λ∥. +(c) We have the chiral symmetry +M(w)E− = −E−M(−w). +(A.6) +In particular, for purely imaginary spectral parameter, w = iImw, it holds that m = iImm as +well as M11 = iImM11 and M22 = iImM22. Moreover, the off-diagonal blocks of ImM are +vanishing on the imaginary axis. +(d) Fix κ > 0. For any spectral parameter in the κ-bulk, w ∈ C ∖ R with Rew ∈ Bκ, we have +∥M(w)∥ ≤ C(κ,∥Λ∥) +(A.7) +for some constant depending only on κ and an upper bound on the norm ∥Λ∥. Moreover, ρ(e) is +real analytic on Bκ with derivatives controlled uniformly +max{∣∂kρ(e)∣ ∶ e ∈ Bκ} ≤ C(k,κ,∥Λ∥) +(A.8) +with a constant C(k,κ,∥Λ∥) for any k ∈ N. +Proof. For part (a), a direct computation shows that M from (A.4) with the blocks given in (2.17)–(2.18) +indeed solves (A.1) if m is replaced with ⟨M⟩ in these formulas. The calculation uses the simple ob- +servation that ⟨M11⟩ = ⟨M22⟩ from (2.18), hence S[M] = ⟨M⟩. Furthermore, the MDE also implies +that ⟨M⟩ solves (2.4), but this equation has a unique solution by the theory of free convolutions with a + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +49 +semicircular density, hence m = ⟨M⟩. Finally M ∗(w) = M( ¯w) follows from ¯m(w) = m( ¯w). This +proves (a). +For part (b), since S[M] = ⟨M⟩, we observe that M solves +− 1 +M = w − ˆΛ + ⟨M⟩, +which is exactly the MDE for a deformed Wigner matrix model.14 The point is that the Hermitised +H from (2.15) does not satisfy the uniform lower bound in the flatness condition on the self-energy +operator, i.e. S[T ] ≥ c⟨T ⟩ does not hold in general. Nevertheless, for the purpose of computing M we +can replace H with the deformed Wigner model W +ˆΛ with self-energy given S[T ] = ⟨T ⟩ and which is +flat. Thus we can use several results from the analysis of the MDE with flatness condition. The Hölder- +continuity of the scDos was proven in [2, Prop. 2.2], which easily extends to the Hölder-continuity of +its Stieltjes transform m, see e.g. [1, Lemma A.7]. In particular ⟨M(w)⟩ extends continuously to the +real line and thus the scDos ρ(e) ∶= 1 +π ⟨ImM(e)⟩ = 1 +π Imm(e) is well defined. Since it has the same +Stieltjes transform as the free convolution (2.3) by part (a), we proved that the scDos defined via MDE is +the same as the free convolution (2.3). +The continuous extension of M (and not only its trace) requires an additional argument. For any +open interval I ∈ R define +∥M∥I ∶= sup{∥M(e + iη)∥ ∶ e ∈ I,η > 0}. +Suppose for some open I ∈ R we have ∥M∥I < ∞, then we have the Lipschitz continuity +∥M(w1) − M(w2)∥ ≤ ∥M∥2 +I∣w1 − w2∣, +Rew1,Re w2 ∈ I +following from the resolvent identity applied to M(w) = (ˆΛ − w − m)−1. Thus M(w) continuously +extends to any e ∈ I. +So the key question for the extension (and for many other results on the MDE) is the boundedness +∥M∥I < ∞. In the bulk spectrum, i.e. for any e ∈ R with ρ(e) > 0, we can use the bound +∥M(w)∥ ≤ ∣Imm(w) + Im w∣−1 +that is obtained by taking the imaginary part of (A.1), yielding +ImM = (Imw + ⟨ImM⟩)MM ∗ , +andusing∥MM ∗∥ = ∥M∥2 and∥Im M∥ ≤ ∥M∥. Bythe Höldercontinuity(A.5) insmallneighborhood +I of e (whose size depend on the lower bound on ρ(e)) we obtain ∥M∥I ≲ ρ(e)−2 < ∞. Thus M +continuously extends to I with the same bound and it is locally Lipschitz continuous with a Lipschitz +constant of order ρ(e)−2. In the entire κ-bulk this extension is controlled by a constant depending +only on κ and ∥Λ∥ (via (A.5)). This proves (A.7). +Near the spectral edges we have only an N-independent upper bound for ∥M∥. Using the spectral +decomposition of ˆΛ with eigenvalues νi and normalised eigenvectors yi, i ∈ ±[N], we have +M(w) = ∑ +i +∣yi⟩⟨yi∣ +νi − w − m(w), +thus +∥M(w)∥ ≤ +2N +mini ∣νi − w − m(w)∣ . +(A.9) +On the other hand the imaginary part of (2.4) implies +Imm = +1 +2N ∑ +i +Imm + Imw +∣νi − w − m∣2 +thus +1 +2N ∑ +i +1 +∣νi − w − m∣2 = +Imm +Im m + Imw ≤ 1 +so ∣νi − w − m∣ ≥ 1/ +√ +2N. From (A.9) this gives the uniform bound +∥M(w)∥ ≤ (2N)3/2, +w ∈ C ∖ R, +14That is, a matrix H = W + ˆΛ, where W is a Hermitian matrix with normalised i.i.d. (up to the symmetry) entries of variance +1/(2N). + +50 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +which guarantees the continuous extension of M to the real line with a uniform Lipschitz constant +(2N)3/2. As we have seen, in the bulk this regularity can be improved. 15 +For part (c), the symmetry ρ(e) = ρ(−e) immediately implies the symmetry m(w) = −m(−w) for +its Stieltjes transform. Then (A.6) is an immediate consequences of the formulas (2.17)–(2.18). +Finally, for part (d), the bound (A.7) was already proven above. The real analyticity of ρ and m in the +bulk with the bounds on the derivative (A.8) follows from taking derivatives in (2.4) and using again the +lower bound on Imm. +□ +Finally, we prove some regularity property of the κ-bulk, see (2.21). +Lemma A.2. Let 0 < κ′ < κ be two small constants, then +dist(∂Bκ′,Bκ) ≥ c(κ − κ′) +(A.10) +with some N-independent constant c = c(∥Λ∥) > 0. Moreover, Bκ is a finite union of disjoint compact +intervals; the number of these components depends only on κ and ∥Λ∥. +Proof. As in the proof of Lemma A.1, we interpret Bκ as the κ-bulk of the deformed Wigner matrix +W + ˆΛ, i.e. a model with the flatness condition. The statement on the number of components directly +follows from the real analyticity of ρ and (A.8). +The same argument would also imply(A.10) with a constant c that depends on κ and an upper bound +on ∥Λ∥. To remove the κ-dependence, we need to use the detailed shape analysis for ρ from [4]. In +particular, the flatness condition and ∥M∥I < C(κ) for any interval I ⊂ Bκ (equivalent to [4, Eq. (4.16)]) +implies that Assumption 4.5 in [4] holds. Therefore Theorem 7.2 in [4] applies to our case. This theorem +says that in the regime where ρ is small, it is approximately given by explicit 1/3-Hölder continuous +functions, moreover ρ itself is 1/3-Hölder continuous with Hölder constant depending only on the so- +called model parameters of the problem, which in our case is just an upper bound on Λ (note that [4] +was written for much more complicated self-energy operators to include the MDE analysis for random +matrices with correlated entries). Noticing the κ1/3 power in the definition of Bκ in (2.6), this means +that the boundaries of Bκ are Lipschitz continuous functions of κ when κ is small with a Lipschitz +constant depending only on an upper bound on ∥Λ∥. +□ +Remark A.3. Note that the proof of the independence of c = c(∥Λ∥) of κ required a much more sophisticated +analysis. However, for our main proof, c = c(κ,∥Λ∥) > 0 in (A.10) is sufficient, note that (A.10) is only +used in choosing δ in (4.20) appropriately. More precisely, for fixed L = L(ǫ) and κ0 > 0, given the +family (ℓκ0)ℓ∈[L] of parameters for the domains D(ǫ0,κ0) +ℓ +, we would have that dist(∂B(ℓ−1)κ0,Bℓκ0) ≥ +c(ℓκ0,∥Λ∥)κ0. Now, the cutoff parameter δ in (4.20) is chosen much smaller than c(ℓκ0,∥Λ∥)κ0 for every +ℓ ≤ L(ǫ). +A.2. The stability operator (A.2) and its spectral properties. Throughout the entire paper, the two- +body stability operator (A.2) and its adjoint (A.3) play a crucial role. These operators depend on two (a +priori) different spectral parameters w1,w2 via the solutions M1 = M(w1) and M2 = M(w2) of the +MDE (A.1). For these solutions, we have the following basic lemma. +Lemma A.4. Let w1,w2 ∈ C ∖ R be two spectral parameters and M1 = M(w1),M2 = M(w2) the +corresponding solutions to (A.1). +(a) Then we have the M-Ward identity, +M1 − M2 = [(w1 − w2) + (⟨M1⟩ − ⟨M2⟩)]M2M1 . +(A.11) +In particular, M1 and M2 commute and it holds that +(1 − ⟨MM ∗⟩) ⟨ImM⟩ = Imw ⟨MM ∗⟩. +(A.12) +15We remark that under some extra condition on Λ further improvements away from the bulk are possible for m but not for +M. For example, if the singular values νi of Λ are 1/2-Hölder continuous in the sense that ∣νi − νj∣ ≤ C0[∣i − j∣/N]1/2, then m +is also uniformly bounded and 1/3-Hölder continuous with a constant depending on C0, see Section 11.4 of [1]. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +51 +(b) Fix κ > 0 and let Rew1,Re w2 ∈ Bκ. Then, for Imw1Imw2 > 0, we have the perturbative +estimate +∥M(w1) − M(w2)∥ = O(∣w1 − w2∣ ∧ 1). +Proof. Part (a) is an immediate consequence of the MDE (A.1) using the fact that +M = (ˆΛ − (w + m)) +−1 +is a resolvent of ˆΛ. The special case (A.12) follows from (A.11) with w1 = w and w2 = ¯w, and taking a +trace. +For part (b), we focus on the case of small imaginary parts for the spectral parameters (the comple- +mentary regime being trivial) and use that M is analytic away from the real axis and differentiate (A.1) +w.r.t. w, such that we find +∂wM = +1 +1 − ⟨M 2⟩M 2 +by means of Lemma A.1 (a). Next, using (A.12), the denominator is lower bounded as +∣1 − ⟨M 2⟩∣ = ∣(1 − ⟨MM ∗⟩) − 2i⟨MIm M⟩∣ ≥ 2∣⟨(ImM)2⟩∣ ≥ 2⟨ImM⟩2 , +(A.13) +which shows that ∥∂wM∥ ≲ 1 in the bulk. Now the claim follows from the fundamental theorem of +calculus together with (A.7). +□ +Armed with this information, we can now turn to the following lemma, collecting several basic +spectral properties stability operator B. Its proof will be given at the end of this section. +Lemma A.5. Let w1,w2 ∈ C ∖ R and M1,M2 be the respective solutions of (A.1). +(a) The associated two-body stability operator +B = 1 − M1S[⋅]M2 +has two non-trivial eigenvalues β± (the other (2N)2 − 2 are equal to one), given by +β± = 1 ∓ ⟨M1E±M2E±⟩. +(A.14) +The corresponding right- and left-eigenvectors +B[R±] = β±R± , +B∗[L∗ +±] = ¯β±L∗ +± , +take the explicit form +R± = M1E±M2 , +L± = E± , +(A.15) +up to a normalisation ensuring that ⟨L±,R±⟩ = 1. +(b) The eigenvalues (A.14) can be lower bounded as +∣β±∣ ≳ (∣Rew1 ∓ Rew2∣ + ∣Imw1∣ + ∣Imw2∣) ∧ 1. +(A.16) +In particular, the inverse stability operator B−1 exists. +(c) Fix κ > 0 and denote s ∶= −sgn(Imw1 Imw2). Then, for Rew1,Re w2 ∈ Bκ, we have that +∣β−s∣ ≳ 1. +By the last item, given s ∶= −sgn(Imw1 Imw2), we will always refer to +(β ∶= 1 − s⟨M1EsM2Es⟩, R ∶= M1EsM2 , L ∶= Es) +(A.17) +as the critical eigentriple (and accordingly β as the critical eigenvalue etc.), consisting of the eigenvalue +and the corresponding right- and left-eigenvector. Moreover, the estimate (A.16) shows that, if we have +(recall (3.5)) +1± +δ(w1,w2) ∶= φδ(Rew1 ∓ Rew2) φδ(Imw1) φδ(Im w2) = 0 +for some δ > 0, then the inverse stability operator B−1 is bounded and none of the eigenvalues β± is +really critical. In the complementary regime, 1± +δ(w1,w2) = 1, and Rew1,Re w2 ∈ Bκ, we shall now +explain the interplay between the critical eigentriple (A.17) and the regularisation (3.6). + +52 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Lemma A.6. Let w1,w2 ∈ C ∖ R with Rew1,Rew2 ∈ Bκ for some fixed κ > 0 and denote the relative +sign of imaginary parts by s ∶= −sgn(Imw1 Im w2). Moreover, let M1 = M(w1),M2 = M(w2) be the +respective solutions of (A.1) and A ∈ C2N×2N a bounded deterministic matrix. +(a) If 1s +δ(w1,w2) = 1 for some δ > 0 small enough, the critical left- and right-eigenvectors (A.17) are +normalised as ⟨L,R⟩ ∼ 1. In particular, if 1± +δ(w1,w2) = 1, the respective denominator in the +regularisation ˚ +Aw1,w2 (see (3.6)) is bounded away from zero. +(b) The operator X12, acting as +X12[B] ∶= ((B∗ +12)−1[B∗]) +∗ = (1 − S[M1 ⋅ M2]) +−1[B], +B ∈ C2N×2N , +where B12 ∶= 1−M1S[⋅]M2, is well defined and bounded on the s-regular component ˚ +As (w.r.t. the +pair of spectral parameters (w1,w2)) of any bounded A. This means, for +˚ +As ∶= A − 1s +δ(w1,w2) ⟨M1AM2Es⟩ +⟨M1EsM2Es⟩Es +(A.18) +it holds that ∥X12[˚ +As]∥ ≲ 1. +In particular, combining Lemma A.4 (b) with Lemma A.6 (a), Lemma A.1 (c), and Lemma A.4 (a), we +conclude the perturbative statements from Lemma 3.3. +Proof of Lemma A.6. For part (a), similarly to the proof of Lemma A.5 (c) given below, we focus on the +extreme case w2 = s ¯w1, where the critical eigentriple is given by +(β = 1 − s⟨M(w1)EsM(s ¯w1)Es⟩, R = M(w1)EsM(s ¯w1), L = Es) . +(A.19) +Now by means of the chiral symmetry M(w1)E− = −E−M(−w1), we readily obtain +⟨L,R⟩ = s⟨M1M ∗ +1 ⟩ = s +⟨ImM1⟩ +Imw1 + ⟨ImM1⟩ ∼ 1, +where we used(A.12)inthe secondstep. Thisprincipalnormalisationof orderpersistsaftersmallpertur- +bation of w2 around the extreme case, but as long as 1s +δ(w1,w2) = 1. Our claim for the denominators +in the regularisation (3.6) follows immediately from the representation in (A.19). +For part (b), we first note that, by means of Lemma A.5, the statement is trivial for constellations of +spectral parameters w1,w2 satisfying 1s +δ(w1,w2) = 0 and we can hence focus on the complementary +extreme case 1s +δ(w1,w2) = 1. Then it follows from the explicit form +X12[B] = B + ∑ +σ +σ +⟨M1BM2Eσ⟩ +1 − σ⟨M1EσM2Eσ⟩Eσ +and Lemma A.5 that +X12[B] = s 1 +β ⟨M1BM2Es⟩Es + O(1)[B], +(A.20) +where O(1) is a shorthand notation for a linear operator E ∶ C2N×2N → C2N×2N satisfying ∥E[B]∥ ≲ +∥B∥. Now, plugging ˚ +As from (A.18) into (A.20) yields the desired. +□ +It remains to give the proof of Lemma A.5. +Proof of Lemma A.5. For (a), we first observe that, due to the simple structure of S[⋅], indeed (2N)2 −2 +of the (2N)2 eigenvalues of B are equal to one. The expressions (A.14) and (A.15) can be verified by +direct computation, invoking Lemma A.4 in combination with Lemma A.1. +For (b) with w1 ≠ ±w2, we first find that +1 +β± += +1 +1 ∓ ⟨M1E±M2E±⟩ = 1 + ⟨M1⟩ ∓ ⟨M2⟩ +w1 ∓ w2 +(A.21) +as a consequence of Lemma A.4 (a) and Lemma A.1. Now, using that ∣⟨M⟩∣ ≤ ⟨MM ∗⟩1/2 < 1, which +follows from MM ∗ = ImM/(Im w + ⟨ImM⟩) (see Lemma A.4 (a)), we conclude that +∣β±∣ ≳ ∣Rew1 ∓ Rew2∣ ∧ 1 +(A.22) + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +53 +by application of a triangle inequality in (A.21). Next, we estimate +min {∣β+∣,∣β−∣} ≥ ∣1 − ⟨M1M ∗ +1 ⟩1/2⟨M2M ∗ +2 ⟩1/2∣ ≳ (∣Imw1∣ + ∣Imw2∣) ∧ 1, +(A.23) +where in the first step we used ⟨MM ∗⟩ < 1 together with a Schwarz inequality, and (A.12) in the second +step. Combining (A.22) and (A.23) yields the claim. +Finally, for (c), we consider the case of small imaginary parts for the spectral parameters (the com- +plementary regime being trivial) and focus on the extreme case w1 = −sw2. Then, using (A.6) and (A.13), +we obtain +∣β−s∣ = ∣1 − ⟨M 2 +1 ⟩∣ ≥ 2⟨Im M1⟩2 ≳ 1. +(A.24) +This principal lower bound persists after small perturbations of w2, and the complementary regime +can be dealt with by (A.16). +□ +Appendix B. Proof of Theorem 2.6 +In this appendix, we give a short proof of the usual single resolvent local law in the bulk given in +Theorem 2.6. In the literature, bulk local laws are established under the usual flatness assumption (see +[36, Assumption E]) on the self-energy operator S, i.e. +c⟨R⟩ ≤ S[R] ≤ C⟨R⟩ +(B.1) +for some constants c,C > 0 and any positive semi-definite matrix R ≥ 0. However, for our model, +the stability operator S[R] = ∑σ σ⟨REσ⟩Eσ violates the lower bound in the flatness condition (B.1), +which is why we need to provide a separate argument. The main idea is that lacking of the lower bound +in (B.1) is compensated by the orthogonality relation ⟨GE−⟩ = ⟨ME−⟩ = 0 as a consequence of (5.5). +The following argument heavily relies on [36, Theorem 4.1], where a general high-moment bound +on the underlined term in +⟨(G − M)B⟩ = −⟨W GX [B]M⟩ + ⟨G − M⟩⟨(G − M)X [B]M⟩ +(B.2) +and its isotropic counterpart (see (B.3) below)has been shown. We stress that this estimate from [36] does +not require the lower bound in (B.1) for the self-energy operator S. As usual, we suppressed the spectral +parameter w ∈ C ∖ R satisfying Rew ∈ Bκ for some fixed κ > 0 from the notation. The expansion +(B.2) for an arbitrary deterministic matrix B ∈ C2N×2N has already been established in (5.15), where we +introduced the linear operator X [B] ∶= (1 − S[M ⋅ M]) +−1[B] acting on matrices. +For given B, we now decompose it into its (−)-regular and (−)-singular component (see (A.18), the +cutoff function being irrelevant here), +B = ˚ +B− + ⟨MBME−⟩ +⟨ME−ME−⟩E− , +respectively. For the second summand, we note that ⟨GE−⟩ = ⟨ME−⟩ = 0, and we can hence focus on +the regular component, i.e. assume that B = ˚ +B− is (−)-regular. +In this case, for a bounded deterministic ∥B∥ ≲ 1 we thus have ∥X [B]∥ ≲ 1 from Lemma A.6. With +the high-moment bound on the underlined term from [36, Theorem 4.1, part (b)] one can conclude the +proof of Theorem 2.6 in the averaged case, ∣⟨(G−M)B⟩∣ ≺ (Nη)−1, by a standard bootstrap argument +(see, e.g., [36, Sections 5.3 and 5.4]). +In the isotropic case, we evaluate (B.2) for B = 2N ∣y⟩ ⟨x∣, where x,y ∈ C2N are deterministic +vectors in with ∥x∥,∥y∥ ≲ 1. More precisely, we subtract its (−)-singular component (which can be +dealt with separately as explained above) and insert +B = ˚ +B− = 2N ∣y⟩ ⟨x∣ − ⟨x,ME−My⟩ +⟨ME−ME−⟩ E− +in the expansion (B.2), which leaves us with +(G − M)xy = − (W G)x(My) + ⟨G − M⟩(G − M)x(My) +(B.3) ++ [⟨x,ME−My⟩ +⟨ME−ME−⟩ + ⟨x,M 2y⟩ +1 − ⟨M 2⟩ ][⟨W GE−M⟩ − ⟨G − M⟩⟨(G − M)E−M⟩]. + +54 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +Afterrealizingthatthe denominatorsin(B.3)are boundedawayfrom zero (see Lemma A.5andLemma A.6), +the proof of Theorem 2.6 in the isotropic case, ∣(G − M)xy∣ ≺ (Nη)−1/2, can be concluded again by +a standard bootstrap argument, now using the high-moment bound from [36, Theorem 4.1, part (a)] and +the already proven averaged law ∣⟨(G − M)B⟩∣ ≺ (Nη)−1 with ∥B∥ ≲ 1 as an input. +Appendix C. Bounds on the deterministic approximations: Proof of Lemma 4.2 +The goal of this appendix is to define the deterministic approximation +M(w1,B1,w2,...,Bk−1,wk) +to a resolvent chain +G(w1)B1G(w2)⋯Bk−1G(wk) +and prove the bounds from Lemma 4.2. While the definition of M(w1,...,wk) is done for any num- +ber k of spectral parameters w1,...,wk, the bounds in Lemma 4.2 are proven for at most five and the +deterministic matrices B1,...,Bk−1 being regular w.r.t. to the surrounding spectral parameters. +Definition C.1. Fix k ∈ N and let w1,...,wk ∈ C∖R be spectral parameters. As usual, the corresponding +solutions to (2.19) (see also Appendix A) are denoted by M(wj), j ∈ [k]. Then, for deterministic matrices +B1,...,Bk−1 we recursively define +M(w1,B1,...Bk−1,wk) = (B1k) +−1[M(w1)B1M(w2,...,wk) +(C.1) ++ ∑ +σ=± +k−1 +∑ +l=2 +σM(w1)⟨M(w1,...,wl)Eσ⟩EσM(wl,...,wk)] , +where we introduced the shorthand notation +Bmn ≡ B(wm,wn) = 1 − M(wm)S[⋅]M(wn) +for the stability operator (A.2). +Note that the recursion (C.1) is well defined, since on the rhs. of (C.1), there are only M(wm,...,wn) +appearing for which the number of spectral parametersis strictly smaller than on the lhs. of (C.1), i.e. n− +m + 1 < k. +As a preparation for the proof of Lemma 4.2, we shall now show that M(w1,...,wk) from (C.1) +satisfies multiple recursive relations,called recursive Dyson equations, by using a so-called meta argument, +that relies on the fact that M(w1,...,wk) actually approximates a chain of products of resolvents. +In fact, we only picked one of the recursive relations (namely (C.2) with j = 1) for actually defining +M(w1,...,wk) in Definition C.1. Although the second recursion relation (C.3) will not be used in the +proof of Lemma 4.2, it is obtained completely analogous to (C.2) and we hence give it for completeness. +A similar meta argument has been done several times, see e.g. [31]. For convenience of the reader we +repeat it in our setup. +Lemma C.2. (Recursive Dyson equations for M(w1,...,wk), see [28, Lemma 4.1]) +Fix k ∈ N. Let w1,...,wk ∈ C ∖ R be spectral parameters and B1,...,Bk−1 ∈ C2N×2N deterministic +matrices. Then for any 1 ≤ j ≤ k we have the relations +M(w1,...,wk) = M(w1,...,wj−1,Bj−1M(wj)Bj,wj+1,...,wk) +(C.2) ++ ∑ +σ=± +j−1 +∑ +l=1 +σM(w1,...,Bl−1,wl,Eσ,wj,Bj,...,wk)⟨M(wl,...,wj−1)Bj−1M(wj)Eσ⟩ ++ ∑ +σ=± +k +∑ +l=j+1 +σM(w1,...,Bj−1M(wj)Eσ,wl,Bl...,wk)⟨M(wj,...,wl)Eσ⟩ +and +M(w1,...,wk) = M(w1,...,wj−1,Bj−1M(wj)Bj,wj+1,...,wk) +(C.3) + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +55 ++ ∑ +σ=± +j−1 +∑ +l=1 +σM(w1,...,Bl−1,wl,EσM(wj)Bj,...,wk)⟨M(wl,...,wj)Eσ⟩ ++ ∑ +σ=± +k +∑ +l=j+1 +σM(w1,...,Bj−1,wj,Eσ,wl,Bl,...,wk)⟨M(wj)BjM(wj+1,...,wl)Eσ⟩. +If j = 1 or j = k, we define B0 = E+ resp. Bk = E+ in (C.2) and (C.3). +The formulas (C.2) and (C.3) shall be derived by expanding the jth resolvent Gj in the resolvent +chain G1B1 ⋯GjBj ⋯ Bk−1Gk corresponding to M(w1,...,wk) in an underlined term, once to the +right (for (C.2), see (C.9)) and once to the left (for (C.3), see (C.11)). Altogether, this yields 2k different +recursions for M(w1,...,wk), which are listed in the above lemma. Moreover, it would be possible +to prove directly that all these different recursions define the same M(w1,...,wk). This strategy has +been used in a much simpler setup [26] dealing with Wigner matrices. Here, we find it simpler to use +the alternative meta argument. +Proof. The principal idea is to derive the respective relations (C.2) and (C.3) on the level of resolvent +chains G1B1⋯Bk−1Gk, which, after taking the expectation and using that Gi ≈ Mi from Theorem +2.6, yields the same relation on the level of the deterministic approximations. For the purpose of proving +identities about M(w1,...,wk), we may use the most convenient distribution for X, namely Gaussian. +For the sake of this proof, we thus assume the single entry distribution χ of X to be a standard complex +Gaussian χ = NC(0,1), i.e. X in Assumption 2.1 is a complex Ginibre matrix, in which case it holds +that (recall the discussion below (5.3)) +E f(W )W g(W ) = 0. +(C.4) +Let w1,...,wk ∈ C ∖ R be arbitrary (but fixed!) spectral parameters. We now conduct the meta argu- +ment, consisting of three steps. +Step 1. We consider the resolvent chain +G1B1 ⋯ Bk−1Gk . +(C.5) +Expanding G1 via the identity +G1 = M1 − M1W G1 + M1S[G1 − M1]G1 +and using S[G1 − M1] = ⟨G1 − M1⟩ from (5.5), we find that +G1B1 ⋯ Bk−1Gk +=M1B1 ⋯ Bk−1Gk − M1W G1B1 ⋯ Bk−1Gk + ⟨G1 − M1⟩M1G1B1 ⋯ Bk−1Gk +=M1B1 ⋯ Bk−1Gk + ∑ +σ=± +k−1 +∑ +l=2 +σM1⟨G1B1 ⋯ Bl−1GlEσ⟩EσGlBl ⋯ Bk−1Gk +(C.6) +− M1W G1B1 ⋯ Bk−1Gk + ⟨G1 − M1⟩M1G1B1 ⋯ Bk−1Gk + M1S[G1B1 ⋯ Bk−1Gk]Mk , +where in the last step we distributed the derivatives coming from the definition of the underline in (5.3) +according to the Leibniz rule. Now, (C.10) can be rewritten as +G1B1 ⋯ Bk−1Gk +=(B1k)−1[M1B1 ⋯ Bk−1Gk + ∑ +σ=± +k−1 +∑ +l=2 +σM1⟨G1B1 ⋯ Bl−1GlEσ⟩EσGlBl ⋯ Bk−1Gk +− M1W G1B1 ⋯ Bk−1Gk + ⟨G1 − M1⟩M1G1B1 ⋯ Bk−1Gk] . +(C.7) +Apart from the last two terms in (C.7), this is the exact same relation on the level of resolvents as in +Definition C.1 for M(w1,...,wk). +Step 2.Letthe originalmatrixsize N be fixed. Foranyd ∈ N, we considerthe dN×dN Ginibre random +matrix X(d) with entries having variance 1/(dN), and the deformation Λ(d) ∶= Λ ⊗ Id ∈ CdN×dN, +where Id ∈ Cd×d is the identity matrix. Analogously to (2.2) and (2.15), we also define the Hermitisations + +56 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +ˆΛ +(d) and W (d), as well as the resolvents G(d) +i += G(d)(wi) ∶= (W (d) + ˆΛ +(d) − wi)−1. It is crucial to +observe that the correspondingly modified MDE +− +1 +M (d) = w − ˆΛ +(d) + S(d)[M (d)] +under the usual Imw ImM (d) > 0 constraint with +S(d)[R] ∶= ̃Ẽ +W (d)R̃ +W (d) = ∑ +σ +σ⟨R E(d) +σ ⟩E(d) +σ +, +where +E(d) +σ +∶= Eσ ⊗ Id , +has the unique solution M (d) = M ⊗Id, where M is the unique solution of the MDE (2.19) on C2N×2N. +In particular, if we define B(d) +i +∶= Bi ⊗ Id for all i ∈ [k], then it holds that (C.1) defined with M (d) +i +and B(d) +i +as inputs, also satisfies M (d)(w1,B(d) +1 +,...,B(d) +k−1,wk) = M(w1,B1,...,Bk−1,wk) ⊗ Id. +We now multiply the analogue of (C.7) in boldface matrices by some B(d) +k += Bk ⊗ Id with Bk ∈ +C2N×2N and take the averaged trace. Next, by means of (C.4), taking the expectation of the resulting +expression removes the underlined term. Hence, using the one-to-one correspondence between the +terms in the second line of (C.7) and the terms on the rhs. of (C.1), mentioned below (C.7), it follows by +telescopic replacement and a simple induction on the length k of the chain, that +lim +d→∞E ⟨G(d) +1 B(d) +1 +⋯ G(d) +k B(d) +k ⟩ = ⟨M(w1,B1,...,wk)Bk⟩ +(C.8) +by means of the usual global law [36, Theorem 2.1] for the last term on the rhs. of (C.7). In fact, due to the +tensorisation, we have that ∣⟨G(d) +1 +− M (d) +1 +⟩∣ ≺ 1/(Nd) since ∣Imw1∣ ≳ 1, where the implicit constant +potentially depends on N but not on d. +We emphasise that the tensorisation by Id is indeed a necessary step, since the matrices Mi and Bi +are N-dependent and hence one cannot take the limit N → ∞ in (C.8) for d = 1. +Step 3. Having (C.8) at hand, the recursive relations in (C.2) and (C.3) can be proven as follows: For (C.2), +let 1 ≤ j ≤ k and expand Gj in (C.5) according to +Gj = Mj − MjW Gj + MjS[Gj − Mj]Gj , +(C.9) +which yields, analogously to (C.6), +G1 ⋯ Bj−1GjBj ⋯ Gk = G1 ⋯ Bj−1MjBj ⋯ Gk +(C.10) ++ ∑ +σ=± +j−1 +∑ +l=1 +σG1 ⋯ Bl−1Gl⟨Gl ⋯ Gj−1Bj−1MjEσ⟩EσGjBj ⋯ Gk ++ ∑ +σ=± +k +∑ +l=j+1 +σG1 ⋯ Bj−1Mj⟨GjBj ⋯ Bl−1GlEσ⟩EσGlBl ⋯ Gk +− G1 ⋯ Bj−1MjW GjBj ⋯ Gk + ⟨Gj − Mj⟩G1 ⋯ Bj−1MjGjBj ⋯ Gk . +Hence, after taking the trace against some arbitrary Bk ∈ C2N×2N , by performing the tensorisation +from Step 2, taking an expectation, and using (C.8), we obtain (C.2), but in a trace against Bk. However, +since Bk was arbitrary, we conclude the desired. +For the second recursion (C.3), the argument is identical except from the fact that we expand Gj in +(C.5) according to +Gj = Mj − GjWMj + GjS[Gj − Mj]Mj . +(C.11) +□ +The recursive relations from Lemma C.2 can be used to show the bounds from Lemma 4.2 on the +deterministic counterparts in the definition of Ψav/iso +k +in (4.13) resp. (4.14) for k ≤ 4. Recall that all +deterministic matrices Ai appearing in the respective averaged or isotropic chain are regular in the +sense of Definition 4.1. +Proof of Lemma 4.2. In the following, we will distinguish the two regimes η ≤ 1 and η > 1 and argue +for each of them separately, iteratively using Lemma C.2. Before going into the iteration, recall that +∥M(w1)∥ ≲ min(1, +1 +∣Im w1∣) from Lemma A.1, which immediately yields (4.9) for k = 1. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +57 +Regime η ≤ 1. Using (C.2) for k = j = 2, we find that +M(w1,A1,w2) = M(w1)X12[A1]M(w2) = B−1 +12 [M(w1)A1M(w2)], +(C.12) +where X12[B] ∶= (1−S[M(w1) ⋅ M(w2)])[B] for B ∈ C2N×2N . Since A1 is regular, we conclude +(4.8) for k = 1 (by means of Lemma A.6 (b)), which immediately translates to (4.9) for k = 2. +Next, for (4.8) and k = 2, we again use (C.2) with j = 2, such that we obtain +M(w1,A1,w2,A2,w3) =M(w1,X12[A1]M(w2)A2,w3) +(C.13) ++ ∑ +σ +σM(w1,X12[A1]M(w2)Eσ,w3)⟨M(w2,A2,w3)Eσ⟩. +Moreover, using (4.9) for k = 2 in combination with (C.12) and the lower bound (A.16) on the eigenvalues +of the stability operator B, (4.8) for k = 2 readily follows. +For (4.9) and k = 3 we need a different representation of M(w1,A1,w2,A2,w3) as +B−1 +13[M(w1)A1M(w2,A2,w3) + ∑ +σ +σM(w1)EσM(w2,A2,w3)⟨M(w1,A1,w2)Eσ⟩], +which follows from (C.2) with j = 1 (or simply by Definition C.1). This implies +⟨B−1 +13[⋯]A3⟩ = ⟨[⋯]X31[A3]⟩ +and thus, since ∥[⋯]∥ ≲ 1 from (4.8) with k = 1 and ∥X31[A3]∥ ≲ 1 (recall Lemma A.6 (b)), we have +proven (4.9) for k = 3. +In order to see (4.8) for k = 3, we first need to show that (4.8) for k = 2 remains valid, if only one of the +two involved matrices A1,A2 is regular. Henceforth, we will assume that A1 = ˚ +A1 and A2 is arbitrary, +the other case being similar and hence omitted. We start with (C.13) and use the lower bound (A.16) on +the eigenvalues of B in the first term in (C.13), such that the remaining terms to be investigated are in the +last line of (C.13), where we study each factor separately. Thereby, we focus on the case Imw1 > 0 and +s1 = s2 = + (recall (3.7)), other constellations being completely analogous. Now, in the second factor in +the last line of (C.13) we use +∣⟨M(w2,A2,w3)E−⟩∣ = ∣⟨M(w2)A2M(w3)X32[E−]⟩∣ ≲ 1 +for σ = −. For σ = +, we find, using cyclicity of the trace, that ∣⟨M(w2,A2,w3)E+⟩∣ equals +∣⟨A2M(w3,E+,w2)⟩∣ = +1 +∣w3 − w2∣∣⟨A2(M(w3) − M(w2))⟩∣ ≲ 1 + +1 +∣w3 − w2∣ . +In the first factor in the last line of (C.13), we use the usual bound (A.16) for σ = − and conclude the +desired estimate together with the bound on the second factor for σ = −. However, for σ = +, the +argument is slightly more involved: Using the usual notations ej = Rewj and ηj = ∣Imwj∣, recall +from the proof of Lemma 5.6 (see the estimate of (5.45)) that +⟨M1X12[A +○1,2 +1 +]M2M ∗ +2 E−⟩ = O(∣e1 + e2∣ + η1 + η2) , +which readily implies that +⟨M1X12[A +○1,2 +1 +]M2M3E−⟩ = O(∣e2 − e3∣ + ∣e1 + e2∣ + η1 + η2 + η3) +(C.14) +by means of Lemma A.4 (b). Employing the associated decomposition in the first factor in the last line +of (C.13) (and using the analogous cτ(...)-notation as in (5.36)), we find it being equal to +M(w1,(X12[A1]M(w2)) +○1,3,w3) + ∑ +τ +cτ(X12[A +○1,2 +1 +]M2)M(w1,Eτ,w3). +The first summand is easily bounded by one, as follows from (4.8) for k = 1. Using (C.12), the term with +τ = + is also bounded by one. The remaining term with τ = − can be estimated with the aid of (C.14) as +∣e2 − e3∣ + ∣e1 + e2∣ + η1 + η2 + η3 +∣w1 + w3∣ +. +Collecting all the estimates from above, we find that ∥M(w1, ˚ +A1,w2,A2,w3)∥ is bounded by +1 +η + (1 + ∣e1 + e3∣ + ∣e2 − e3∣ + η1 + η2 + η3 +∣e1 + e3∣ + η1 + η3 +)(1 + +1 +∣e3 − e2∣ + η2 + η3 +) ≲ 1 +η , + +58 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +which shows that (4.8) remains valid if only one of the two involved matrices A1, A2 is regular. +Having this at hand, we can now turn to the proof of (4.8) for k = 3. In fact, by (C.2) for k = 4, we +find +M(w1,..,w4) =M(w1,X12[A1]M(w2),A2,w3,A3,w4) +(C.15) ++ ∑ +σ +σM(w1,X12[A1]M(w2)Eσ,w3,A3,w4)⟨M(w2,A2,w3)Eσ⟩ ++ ∑ +σ +σM(w1,X12[A1]M(w2)Eσ,w4)⟨M(w2,A2,w3,A3,w4)Eσ⟩, +where the first and second line of (C.15) are bounded by 1 +η and we can thus focus on the last line. Struc- +turally, this term is the analog of the last line in (C.13) and also proving it being bounded by 1 +η is com- +pletely analogous to the arguments above. This concludes the proof of (4.8) for k = 3, from which (4.9) +for k = 4 immediately follows. +Finally, we turn to the proof of (4.8) for k = 4. By (C.2) for j = 1 (or simply by Definition C.1) we +find the different representation +M(w1,...,w5) =B−1 +15[M(w1)A1M(w2,...,w5) ++ ∑ +σ +σM(w1)EσM(w2,...,w5)⟨M(w1,A1,w2)Eσ⟩ ++ ∑ +σ +σM(w1)EσM(w3,...,w5)⟨M(w1,...,w3)Eσ⟩ ++ ∑ +σ +σM(w1)EσM(w4,A4,w5)⟨M(w1,...,w4)Eσ⟩]. +Combining ∥[⋯]∥ ≲ η−1, as follows from (4.8) for k ∈ [3] and (4.9) for k ∈ [4], with the usual bound +(A.16), we conclude the desired. This finishes the proof in the first regime where η ≤ 1. +Regime η > 1. In this second regime, we note that all inverses of stability operators are bounded (see +(A.16)). Moreover, it easily follows from (C.2) that every summand in the definition of M(w1,...,wk) +carries at least k factors of (different) M(wi). Now, as mentioned in the beginning of the proof, we +have ∥M(wi)∥ ≲ 1/η, which implies the desired bound. +□ +Appendix D. Motivating derivation of the regularisation +In this appendix, we shall motivate and derive the regularisation (3.2) introduced in Definition 3.1 by +considering two basic examples. First, in Section D.1, we compute +E ∣⟨W G(iη)A⟩∣ +2, +(D.1) +which is the leading contribution to ⟨(G − M)B⟩ with A = X [B]M, see (5.15). We will show that, +in order to be able to reduce its naive size 1/(Nη)2 to the target 1/(N 2η), we need that ⟨A,V±⟩ = 0, +i.e. we need A ∈ C2N×2N to be orthogonal to two certain directions V± in C2N×2N. Note that, we +chose the spectral parameter w = iη to be on the imaginary axis, assuming that 0 ∈ Bκ for some κ > 0. +In this case, both cutoff functions (4.5) in the actual definition of the regularisation satisfy 1± +δ(iη,iη) = 0 +for η > 0 small enough. Hence, at least a posteriori, we really catch both directions V± and not only one. +This calculation is rather foundational and unambiguously reveals two directions V±, for which we need +that ⟨A,V±⟩ = 0, in order to reduce the naive size of (D.1). +Second, in Section D.2, we consider the averaged chain +⟨GΛ1(w1)A1GΛ2(w2)A2⟩, +(D.2) +where the resolvents are even allowed to have generally different deformations, Λ1 ≠ Λ2. Let M1 ∶= +M Λ1(w1) and M2 ∶= M Λ2(w2). For simplicity, we will assume that the stability operators +Bm(∗)n(∗) ∶= 1 − M (∗) +m S[⋅]M (∗) +n +, +m,n ∈ [2], +(D.3) +for all constellations of adjoints, have at most one critical eigenvalue βm(∗)n(∗) which is not of order +one (with associated right and left eigenvectors Rm(∗)n(∗) and Lm(∗)n(∗), respectively, cf. (A.17)). As + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +59 +shown in Lemma A.5(c), this is the case, e.g., if Λ ≡ Λ1 = Λ2 and Rew1,Re w2 ∈ BΛ +κ. (This remains +true for other more general random matrix models with a flat (see (B.1)) self-energy operator [2].) +Again, the main question is what special property A1,A2 must have so that (D.2) be smaller than its +naive size of order 1/η obtained from a simple Schwarz inequality. Instead of directly computing the +second moment of the corresponding underline term (see, e.g. (E.1)), we will make a pragmatic ansatz +on the regularisation. We then start a proof for a bound on (D.2) and find that certain deterministic +terms are too big for general A1,A2. We shall see that there exist two matrices ˜V± ∈ C2N×2N (which +turn out to be certain right eigenvectors Rm(∗)n(∗) of (D.3), see (D.13) and (D.16) later), such that, if +⟨Ai, ˜V±⟩ = 0, these critical terms are smaller. We observe that, for the situation Λ1 = Λ2 and w1 = +w2 = iη, the expressions for ˜V± in fact coincide with those for V± obtained in Section D.1, showing that +the foundational and the pragmatic approaches lead to the same regularisation. +Finally, in Section D.3, motivated by the previous tandem of foundational and pragmatic computa- +tions in Sections D.1 and D.2, respectively, we list generally valid (i.e. for arbitrary w1,w2 also away +from the imaginary axis) explicit formulas for the directions V± in case that Λ1 = Λ2. These explicit +formulas are identical to those used in the regularisation introduced in Definition 3.1. +D.1. Variance calculation of (D.1). In the following, we simply write G = G(iη) for ease of notation. +Then, using a cumulant expansion and neglecting cumulants of order at least three (or assuming that +X is Ginibre), one gets +E∣⟨W GA⟩∣ +2 = 1 +N ∑ +ab +RabE⟨∆abGA⟩∂ba⟨A∗G∗W⟩ += 1 +N ∑ +ab +RabE⟨∆abGA⟩⟨GA∗G∗∆ba⟩ +(D.4) ++ 1 +N 2 ∑ +abcd +RabRcdE⟨∆abG∆dcGA⟩⟨A∗G∗∆baG∗∆cd⟩ += +1 +N 2 ∑ +σ +σE⟨EσGAEσA∗G∗⟩ + +1 +N 2 ∑ +στ +στE⟨EσG∗EτGA⟩⟨EσGEτ(GA)∗⟩. +The rescaled cumulant Rab ∶= Nκ(ab,ba) has been introduced below (5.9) and ∆ab ∈ C2N×2N con- +tains only one non-zero entry at position (a,b), i.e. (∆ab)cd = δacδbd. +As we will show, the cumulant expansion (D.4) yields that (up to a constant) +E ∣⟨W GA⟩∣ +2 ≈ +E∣⟨ImGA⟩∣ +2 +(Nη)2 ++ +E∣⟨ImGAE−⟩∣ +2 +(Nη)2 ++ O ( +1 +N 2η ) . +(D.5) +Indeed, the first summand in the last line of (D.4) is estimated by 1/(N 2η), the target size, with the aid +of a trivial Schwarz inequality and a Ward identity using Theorem 2.6. By writing out the summation +in the last summand, we get in total four terms. Since their treatment is very similar, we focus on two +exemplary terms with σ = τ = + (analogous to σ = τ = −) and σ = −τ = − (analogous to σ = −τ = +). +For the former, we apply a Ward identity and find it to be given by +E ∣⟨ImGA⟩∣ +2 +(Nη)2 +, +(D.6) +which, without any further information on A, using that ⟨GA⟩ ∼ 1 from Theorem 2.6, is too big, +compared to the targeted 1/(N 2η)-size. However, this drastically improves if ⟨ImM,A⟩ = 0 (recall +that ImM is self adjoint): Since ⟨(G − M)A⟩ and ⟨W GA⟩ are roughly of the same size (see (5.15) and +(B.2)), the contribution (D.6) basically becomes a lower-order correction. We have thus identified the +first of the two directions V±, to which A has to be orthogonal to in order to reduce the naive size of +(D.1), namely +V+ = α+ ImM +for some non-zero +α+ ∈ C. +(D.7) +The latter case, σ = −τ = −, is slightly more involved due to the asymmetry of the two factors in +the last summand in the last line of (D.4): For the first factor, again a Ward identity is sufficient. In the + +60 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +second factor, we use (2.16) together with Lemma 5.1 (with Im G(w) instead of G(w) in the integral) in +the approximate form G∗G∗ ∼ ImG/η, as follows by replacing the Cauchy kernel in the integral +∫ +ImG(x + iη) +x2 + η2 +dx ∼ ImG(iη) +η +by a δ-distribution. Overall, this leaves us (roughly) with +E∣⟨ImGAE−⟩∣ +2 +(Nη)2 +(D.8) +for the second case. Hence, arguing for (D.8) completely analogous as done for (D.6), we find the second +direction V−, to which A has to be orthogonal to, in order to reduce the naive size of (D.1), namely +V− = α− ImME− +for some non-zero +α− ∈ C. +(D.9) +We point out that the first term in (D.5) would have worked in the exact same way also for spectral +parameters w = e + iη with e ≠ 0. However, the second direction V− would not have been visible in +this scenario, since the second term in (D.5) would have been replaced by (at least for an upper bound) +E∣⟨ImG(e + iη)AE−⟩∣ +2 +N 2η (∣e∣ + η) ++ +E∣⟨ImG(e + iη)AE−⟩⟨ImG(−e + iη)E−A∗⟩∣ +N 2η (∣e∣ + η) +. +D.2. General structural regularisation in (D.2). We begin with the general rather structural regular- +izing decomposition of a matrix A (recall (3.2)), which shall be conducted as (dropping the tilde, which +has been temporarily introduced below (D.3)) +A○ ≡ ˚ +A ∶= A − ⟨V+,A⟩U+ − ⟨V−,A⟩U− +(D.10) +for some Uσ,Vσ ∈ C2N×2N to be determined but subject to the conditions ⟨Vσ,Uτ⟩ = δσ,τ and +⟨Uσ,Uσ⟩ = 1. We point out, that the following calculations are largely insensitive to the form of the +self-energy operator S[⋅] (but see Footnote 16) and hence the conclusions for Uσ and Vσ derived in this +section are valid beyond our concrete model (up to the fact that, due to the chiral symmetry (2.16), the +regularisation involves a two-dimensional projection). +The goal of the present subsection is to show that V± must be chosen as certain right eigenvectors +Rm(∗)n(∗) of (D.3). This follows by expanding (D.2) and identifying several terms, whose size is too big +for general deterministic matrices. Now, these terms can be neutralised, if ⟨Ai,Rm(∗)n(∗)⟩ = 0 for +certain right eigenvectors. However, as already mentioned in Section 3, for the directions U± there are +a priori no further constraints or conditions (apart from orthogonality and normalisation). Hence, as it +turns out to be convenient for our proofs, we will choose the matrices Uσ in such a way, that a resolvent +identity, i.e. the transformation of a product into a difference, +GΛ1(w1)UσGΛ2(w2) ≈ (GΛ1(w1) − GΛ2(σw2))Uσ , +can be applied (here, the symbol ‘≈’ neglects lower order terms). Finally, the condition ⟨Vσ,Uτ⟩ = δσ,τ +will guarantee that the regularisation is idempotent, i.e. (˚ +A)○ = ˚ +A. Note that our general ansatz (D.10) +is restricted to the non-degenerate situation, where Uσ and Vσ are non-orthogonal, ⟨Vσ,Uσ⟩ ∼ 1. This +is guaranteed for our concrete model with deformations Λ1 = Λ2 (see Section D.3) but requires some +non-trivial arguments in more general cases. +Although the regularisation is inherently two-dimensional (at least for our model), we also define +˚ +Aσ = A○σ ∶= A − ⟨Vσ,A⟩Uσ , +σ ∈ {+,−}, +and refer to A○σ as the σ-regular component (or σ-regularisation) of A and to ⟨Vσ,A⟩Uσ as its σ-singular +component. Note that (A○+)○− = (A○−)○+ = ˚ +A, since ⟨Vσ,Uτ⟩ = δσ,τ. +As usual, we use the common notation ηi ∶= ∣Imwi∣ for i ∈ [2] and abbreviate (see (3.7)) +si ∶= −sgn(ImwiImwi+1), +i ∈ [2], +(D.11) +where the indices are understood cyclically modulo 2 (cf. Definition 4.1). This means that, in particular, +s1 = s2 due to the short length of the chain (D.2). In the following, we will drop the arguments by + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +61 +writing, e.g., M1 = M Λ1(w1) and G2 = GΛ2(w2). Moreover, we take A1 = ˚ +A1 and A2 = ˚ +A2 to be +regular, i.e. orthogonal to some yet to be specified V±. +Now, by means of +G1 = M1 − M1W G1 + M1S[G1 − M1]G1 , +we immediately find +G1A1G2 = M1A1G2 − M1W G1A1G2 + M1S[G1 − M1]G1A1G2 , +from which we conclude that +B12[G1A1G2] = M1A1M2 + M1A1(G2 − M2) − M1W G1A1G2 ++ M1S[G1 − M1]G1A1G2 + M1S[G1A1G2](G2 − M2). +This implies +⟨(G1A1G2 − M A1 +12 )A2⟩ = ⟨M1A1(G2 − M2)X21[A2]⟩ − ⟨M1W G1A1G2X21[A2]⟩ ++ ⟨M1S[G1 − M1]G1A1G2X21[A2]⟩ ++ ⟨M1S[G1A1G2](G2 − M2)X21[A2]⟩ +where we defined +M A1 +12 ∶= B−1 +12 [M1A1M2] = M1X12[A1]M2 = M(w1,A1,w2) +(D.12) +(recall (4.2) and see Appendix C) and used the shorthand notation +Xmn[B] = ((B∗ +nm)−1[B∗]) +∗ = (B−1 +m∗n∗)∗[B], +B ∈ C2N×2N , +where the adjoint of Bnm is understood like in (A.3). +So far, the regularisation of A1 and A2 has been rather structural. To make it more concrete, we +must allow Vσ and Uσ to be potentially different depending on which of the Ai is regularised. In order +to do so, we also temporarily introduce the additional index i, referring to the considered Ai. That is, +we will write Vσ,i instead of Vσ. +The matrices Vsi,i (recall (D.11) for the definition of si) shall be determined by requiring that +∥M A1 +12 ∥ = ∥M1X 12[A1]M2∥ ≲ ∥A1∥ +for i = 1 +and +∥X21[A2]∥ ≲ ∥A2∥ +for i = 2, +meaningthatthe (adjointof the)stabilityoperatorhasa boundedinverse onregularobservables(i.e.sub- +tracting the si-singular component amounts to removing the ‘bad direction’ of the stability operators +X12 and X12, respectively). From this condition, we find the characterisation of Vs1,1 and Vs2,2, namely +Vs1,1 = R1∗2∗ = (R21)∗ +and +Vs2,2 = R2∗1∗ = (R12)∗ , +(D.13) +up to a normalisation constant, which can be specified only after determining Uσ (recall that ⟨Vσ,Uτ⟩ = +δσ,τ and ⟨Uσ,Uσ⟩ = 1). Recall from (D.3), that we denote by Rm(∗)n(∗) and Lm(∗)n(∗) the (normalised) +right and left eigenvectors of Bm(∗)n(∗) corresponding to the (potentially) critical eigenvalue βm(∗)n(∗). +Indeed, in order to verify that (D.13) is the right choice for Vsi,i, we use the decomposition +Xmn = (B−1 +m∗n∗)∗ = +1 +¯βm∗n∗ ∣Lm∗n∗⟩ ⟨Rm∗n∗∣ + O(1), +(D.14) +where O(1) is a shorthand notation for a linear operator E ∶ C2N×2N → C2N×2N satisfying ∥E[B]∥ ≲ +∥B∥. This linear operator is represented by a contour integration of the form +1 +2πi ∮ +dz +z − B∗ +m∗n∗ +where the contour encircles all non-critical eigenvalues of B∗ +m∗n∗ and remains at an order one distance +from the entire spectrum. Note that for general non-Hermitian operators the resolvent (z −B∗ +m∗n∗)−1 +wouldnotnecessarilybe bounded(independentlyofN)justbecause z iswellawayfrom the eigenvalues. + +62 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +However, the explicit form of S (see (2.20)) implies 16that B∗ +m∗n∗ = 1+T where T is a rank-two operator. +For such operators elementary linear algebra shows that +∥ +1 +z − B∗ +m∗n∗ +∥ ≲ [dist(z,Spec(B∗ +m∗n∗))] +−2, +i.e. the non-Hermitian instability only affects a two-dimensional subspace. +Using (D.14) we find +X12[˚ +As1 +1 ] = +1 +¯β1∗2∗ (⟨R1∗2∗,A1⟩ − ⟨Vs1,1,A1⟩⟨R1∗2∗,Us1,1⟩)L1∗2∗ + O(1)[A1] +for the decomposition of A1 and +X21[˚ +As2 +2 ] = +1 +¯β2∗1∗ (⟨R2∗1∗,A2⟩ − ⟨Vs2,2,A2⟩⟨R2∗1∗,Us2,2⟩)L2∗1∗ + O(1)[A2], +for the decomposition of A2. This implies that for (⋯) to be vanishing for every ˚ +Asi +i , the matrix Vsi,i +has to be chosen according to (D.13) (recall ⟨Vσ,i,Uτ,i⟩ = δσ,τ).17 Overall, subtracting the si-singular +component already accounts for removing the ‘bad direction’ of a involved stability operator and thus +– in particular – reduces the naive size of the deterministic approximation (D.12). +However, removing the si-singular component is not sufficient: Although ⟨Vsi,i,U−si,i⟩ = 0 and +thus U−si,i is si-regular, we observe that +⟨G1U−s1,1G2U−s2,2⟩ +(D.15) +still (potentially) has large fluctuations: In our concrete i.i.d. model, take z ≡ z1 = z2 (to be suppressed +from the notation) and w ≡ w1 = −w2 with e = Rew1 and η = Im w1 > 0 w.l.o.g., which implies that +s1 = s2 = + and Uσ = Eσ for σ = ± (see the discussion below (3.3)). In this situation, we use (2.16) and +thus (D.15) takes the form +⟨G(e + iη)E−G(−e − iη)E−⟩ = −⟨G(e + iη)G(e + iη)⟩. +By construction of Vsi,i, the corresponding deterministic approximation (D.12) is bounded by one, but +this is dominated by the fluctuation of order 1/(Nη2) in the relevant small regime η ∼ N −1+ǫ. This +example shows again, what we have already established in Section D.1: For our concrete model, at least +close to the imaginary axis, the regularisation (3.2) is necessarily a two-dimensional operation. +For determining the other directions V−si,i, we note that the regularisation should be designed +in such a way, that it covers also the cases where one (or both) of the resolvents G1,G2 are taken +as an adjoint (see, e.g., (5.10) and (6.10)). Hence, requiring that the same arguments leading to (D.13) +should also be followed for (i) ⟨G1A1G∗ +2A2⟩ and (ii) ⟨G∗ +1A1G2A2⟩ (considering ⟨G∗ +1A1G∗ +2A2⟩ would +again lead to a conclusion for Vsi,i as the relative sign of imaginary parts is preserved), we find that +V−s1,1 = (R2∗1)∗ and V−s2,2 = (R12∗)∗ in case (i), and V−s1,1 = (R21∗)∗ and V−s2,2 = (R1∗2)∗ in +case (ii). In general, the right eigenvectors for these two cases are not the same. However, as pointed +out in Footnote 17, there is a certain tolerance in choosing the V±. Therefore, within this tolerance and +in order to have a consistent and conceptually simple choice, we take V−s1,1 from case (i) and V−s2,2 +from case (ii), i.e. +V−s1,1 = R1∗2 = (R2∗1)∗ +and +V−s2,2 = R2∗1 = (R1∗2)∗ . +(D.16) +Here, in both situations the spectral parameter being the right neighbor of Ai receives a complex con- +jugate. In comparison, if we took V−s1,1 from case (ii) and V−s2,2 from case (i), we would have ended up +with the alternative regularisation from Footnote 10, where the left neighbor of Ai received a complex +conjugate. Again, the relations in (D.16) are understood up to a normalizing constant, which can be +specified only after determining Uσ. +16This is the only place in Section D.2 where the special form of S is currently used. For more general S operator an appropriate +generalisation of the symmetrised (saturated) self-energy operator [2, Def. 4.5] to two different spectral parameters is needed, see [45, +Eq. (2.30)] in the commutative case. +17In case that Λ1 = Λ2, by the lower bound (A.16), the choices in (D.13) not necessarily have to be made exact, but tolerate an +error of the order given in the rhs. of (A.16). Having such a tolerance might be important if one treats the Λ1 ≠ Λ2 case (contrary to +Λ1 = Λ2 as done in this paper) and still has to satisfy the constraints ⟨Vσ, Uτ ⟩ = δσ,τ and ⟨Uσ, Uσ⟩ = 1. + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +63 +Now, it is very important to observe that,for our concrete model with Λ1 = Λ2 and w1 = w2 = iη (in +particular, s1 = s2 = −), our choices for V± in (D.13) and (D.16) agree with those in (D.7)and (D.9)obtained +from a variance calculation with only a single resolvent. This follows from the explicit formulas for the +critical right eigenvector in (A.17), Lemma A.4 (a), and Lemma A.1 (c). +D.3. Explicit formulas for our concrete model and Λ1 = Λ2. In this subsection, we will give ex- +plicitformulasforV± andU± forourconcrete modelwithone fixeddeformationΛ. Infact, forΛ1 = Λ2, +the so far unspecified matrices Uσ can be characterised by requiring that, jointly with the symmetry re- +lationE−Gz(−w)E− = −Gz(w), a resolventidentity can be appliedto G2UσG1. This yields, together +with the normalisation ⟨Uσ,Uσ⟩ = 1, that18 +U+ = E+ +and +U− = E− . +The singular (or critical) eigenvectors of the stability operators characterizing Vsi,i can also be ex- +plicitlycalculated. Using(D.13) and(D.16), we infer, bymeansof (A.17)andthe normalisation/orthogonality +condition ⟨Vσ,i,Uτ,i⟩ = δσ,τ, that +Vs1,1 = +M2Es1M1 +⟨M2Es1M1Es1⟩ , +V−s1,1 = +M ∗ +2 E−s1M1 +⟨M ∗ +2 E−s1M1E−s1⟩ , +Vs2,2 = +M1Es2M2 +⟨M1Es2M2Es2⟩ , +V−s2,2 = +M ∗ +1 E−s2M2 +⟨M ∗ +1 E−s2M2E−s2⟩ , +(D.17) +matching the definition of the regularisation given in (4.6) and (3.6). The normalisation is obvious and +the orthogonality readily follows from Lemma A.1 in combination with Lemma A.4. +Finally, we remark that in order to define the regularisation (3.6) and work with (D.13) and (D.16), +it is not necessary to have the explicit forms for Vσ,i at hand. Instead, the single instance of relevant +explicit formulas is the proof of Theorem 2.7, more precisely, the bound in Proposition 3.4, where one +needs that for ∣Imw1∣ ∼ N −1+ǫ, e.g., (R1∗1)∗ is close to ImM1 (up to a normalisation). But this is +true beyond our model, as easily follows after taking the imaginary part of the general matrix Dyson +equation (see [37]) +− 1 +M = w − A + S[M], +Imw ⋅ Im M > 0 +with self-adjoint matrix of expectations A = A∗ and (flat, see (B.1)) self-energy operator S[⋅]. In fact, this +yields +(1 − MS[⋅]M ∗)(Im M) = (Imw)MM ∗ , +i.e. for ∣Imw∣ ≪ 1 very small, ImM is an approximate right eigenvector of the stability operator +1 − MS[⋅]M ∗ corresponding to the critical eigenvalue (recall the discussion below (D.3)). +Appendix E. Proof of Lemmas 5.8 and 5.9 +In this appendix, we carry out the proofs of the two Lemmas 5.8 and 5.9. +Proof of Lemma 5.8. Similarly to the proof of Lemma 5.6, we get from Appendix D and (4.2) that +⟨(G1A1G2 − M1X12[A1]M2)A2⟩ +(E.1) += ⟨M1A1(G2 − M2)X21[A2]⟩ − ⟨M1W G1A1G2X21[A2]⟩ ++ ⟨M1S[G1 − M1]G1A1G2X21[A2]⟩ + ⟨M1S[G1A1G2](G2 − M2)X21[A2]⟩. +We note that ∥X12[˚ +A1]∥ ≲ 1 and ∥X21[˚ +A2]∥ ≲ 1 by means of Lemma A.6. +Then, analogously to (5.35), we need to further decompose X21[A2]M1 in the last three terms in +(5.34) as +X21[˚ +A2]M1 = (X21[˚ +A2]M1)○ + ∑ +σ +1σ +δ cσ(X21[˚ +A2]M1)Eσ , +18Note that the assignment of ± is a priori not determined, but we chose it in that way. This is also reflected in (D.13) and (D.16). + +64 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +where we again suppressedthe spectral parameters(and the relative sign of their imaginary parts, which +has been fixed by Imw1 > 0 and Imw2 < 0) in the notation for the linear functionals cσ(⋅) on C2N×2N +defined as +c+(B) ∶= ⟨M2BM1⟩ +⟨M2M1⟩ +and +c−(B) ∶= ⟨M2BM ∗ +1 E−⟩ +⟨M2E−M ∗ +1 E−⟩ . +(E.2) +Continuing the expansion of (E.1), we arrive at +⟨M1 ˚ +A1(G2 − M2)X21[˚ +A2]⟩ − ⟨W G1 ˚ +A1G2(X21[˚ +A2]M1)○⟩ ++ ⟨S[G1 − M1]G1 ˚ +A1G2(X21[˚ +A2]M1)○⟩ + ⟨S[G1 ˚ +A1G2](G2 − M2)(X21[˚ +A2]M1)○⟩ ++ ∑ +σ +1σ +δ cσ(X21[˚ +A2]M1)[ − ⟨W G1 ˚ +A1G2Uσ⟩ + ⟨S[G1 − M1]G1 ˚ +A1G2Eσ⟩ ++ ⟨S[G1 ˚ +A1G2](G2 − M2)Eσ⟩]. +We emphasise that, in case of ˚ +A2 and its linear dependents, the regular component is defined w.r.t. the +pair of spectral parameters (w2,w1). +Next, analogously to the proof of Lemma 5.6, we undo the underline in [⋯], such that our expansion +of (E.1) becomes +⟨(G1 ˚ +A1G2 − M1X12[˚ +A1]M2)˚ +A2⟩ += ⟨M1 ˚ +A1(G2 − M2)X21[˚ +A2]⟩ − ⟨W G1 ˚ +A1G2(X21[˚ +A2]M1)○⟩ +(E.3) ++ ⟨S[G1 − M1]G1 ˚ +A1G2(X21[˚ +A2]M1)○⟩ + ⟨S[G1 ˚ +A1G2](G2 − M2)(X21[˚ +A2]M1)○⟩ ++ ∑ +σ +1σ +δ cσ(X21[˚ +A2]M1)[ − ⟨˚ +A1G2Eσ⟩ + ⟨G1 ˚ +A1G2˚Φσ⟩ + cσ(Φσ)⟨G1 ˚ +A1G2Eσ⟩], +where +Φσ ∶= Eσ 1 +M1 +− S[M2Eσ] +(E.4) +was further decomposed with the aid of cσ(Φτ) ∼ δσ,τ and we used the notation (E.2). +We can now write (E.3) for both, ˚ +A2 = ˚Φ+ and ˚ +A2 = ˚Φ−, and solve the two resulting equation for +⟨G1 ˚ +A1G2˚Φσ⟩ and ⟨G1 ˚ +A1G2˚Φ−⟩. Observe that by means of +cτ(X21[˚Φσ]M1) ∼ δσ,τ , +the original system of linear equations boils down to two separate ones. Thus, plugging the solutions +for ⟨G1 ˚ +A1G2˚Φ±⟩ back into (E.3) we arrive at +⟨(G1 ˚ +A1G2 − M1X12[˚ +A1]M2)˚ +A2⟩ += − ⟨W G1 ˚ +A1G2(X21[˚ +A2]M1)○⟩ + ⟨G1 − M1⟩⟨G1 ˚ +A1G2(X21[˚ +A2]M1)○⟩ ++ ⟨M1 ˚ +A1(G2 − M2)X21[˚ +A2]⟩ + ⟨S[G1 ˚ +A1G2](G2 − M2)(X21[˚ +A2]M1)○⟩ +(E.5) ++ ∑ +σ +1σ +δ cσ(X21[˚ +A2]M1) +1 − 1σ +δ cσ(X21[˚Φσ]M1) +[ − ⟨W G1 ˚ +A1G2(X21[˚Φσ]M1)○⟩ +(E.6) ++ ⟨G1 − M1⟩⟨G1 ˚ +A1G2(X21[˚Φσ]M1)○⟩ + ⟨M1 ˚ +A1(G2 − M2)X21[˚Φσ]⟩ ++ ⟨S[G1 ˚ +A1G2](G2 − M2)(X21[˚Φσ]M1)○⟩ +(E.7) +− ⟨˚ +A1(G2 − M2)Eσ⟩ + cσ(Φσ)⟨(G1 ˚ +A1G2 − M +˚ +A1 +12 )Eσ⟩] . +(E.8) +We now need to check that the denominators in (E.6) are bounded away from zero. +Lemma E.1. For small enough δ > 0, we have that +∣1 − 1σ +δ (w2,w1)cσ(X21[˚Φσ]M1)∣ ≳ 1 +for +σ = ±. +Proof. Completely analogous to Lemma 5.7. +□ + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +65 +Next, there are two particular terms, namely the ones of the form +⟨S[G1 ˚ +A1,2 +1 G2](G2 − M2)˚ +A2,1 +2 ⟩, +(E.9) +appearing in (E.5) and (E.7), and +cσ(X21[˚ +A2,1 +2 ]M1)cσ(Φσ)⟨(G1 ˚ +A1,2 +1 G2 − M1X12[˚ +A1,2 +1 ]M2)Eσ⟩, +(E.10) +appearing in (E.8), whose naive size 1/(Nη2) does not match the target. Hence, they have to be dis- +cussed in more detail. In (E.9) and (E.10), we emphasised the pair of spectral parameters with respect +to which the regularisation has been conducted. Moreover, for the following estimates, we recall the a +priori bounds (4.23). +Estimating (E.9). We begin by expanding +⟨S[G1 ˚ +A1,2 +1 G2](G2 − M2)˚ +A2,1 +2 ⟩ = ∑ +σ +σ ⟨G1 ˚ +A1,2 +1 G2Eσ⟩⟨(G2 − M2)˚ +A2,1 +2 Eσ⟩ +(E.11) +and note that, analogously to (5.47), +˚ +Ai,j +i Eσ = (˚ +Ai,j +i Eσ) +○i,i + O(∣ei − σej∣ + ∣ηi − ηj∣)E+ + O(∣ei − σej∣ + ∣ηi − ηj∣)E− +(E.12) +as well as +˚ +Ai,j +i Eσ = (˚ +Ai,j +i Eσ) +○j,j + O(∣ei − σej∣ + ∣ηi − ηj∣)E+ + O(∣ei − σej∣ + ∣ηi − ηj∣)E− +(E.13) +for i ≠ j ∈ [2] and σ = ±. +In the first term in (E.11), for σ = + and Eσ = E+, we use a resolvent identity and the usual averaged +local law (4.15) in combination with (E.12), (E.13) and (4.6), in order to bound it as +∣⟨G1 ˚ +A1,2 +1 G2⟩∣ ≺ 1 + +1 +∣e1 − e2∣ + η1 + η2 +max +i∈[2] ∣⟨(Gi − Mi)(˚ +A1,2 +1 )○i,i⟩∣. +(E.14) +For σ = − and Eσ = E−, we use (2.16) and employ the integral representation from Lemma 5.1 with +τ = +, +J = Bℓκ0 , +and +˜η = +ℓ +ℓ + 1η , +for which we recall that wj ∈ D(ǫ0,κ0) +ℓ+1 +, i.e. in particular η ≥ (ℓ + 1)N −1+ǫ0 and hence ˜η ≥ ℓN −1+ǫ0. +After splitting the contour integral and bounding the individual contributions as described in (5.11), we +obtain, with the aid of Lemma 4.2, +∣⟨G1A +○1,2 +1 +G2E−⟩∣ ≺ 1 + ∫Bℓκ0 +∣⟨G(x + i˜η)A +○1,2 +1 +E−⟩∣ +∣(x − e1 − i(η1 − ˜η)) (x + e2 − i(η2 − ˜η))∣dx +≺1 + ∫Bℓκ0 +∣⟨(G(x + i˜η) − M(x + i˜η))(A +○1,2 +1 +E−) +○x+i˜ +η,x+i˜ +η⟩∣ +∣(x − e1 − i(η1 − ˜η)) (x + e2 − i(η2 − ˜η))∣ +dx , +where in the second step, we freely added and subtracted M(x − i˜η) by residue calculus, used (E.12) +and (E.13), and absorbed logarithmic corrections from the integral into ‘≺’. This finally yields that +∣⟨G1A +○1,2 +1 +G2E−⟩∣ ≺ 1 + +1 +∣e1 + e2∣ + η1 + η2 +⋅ +ψav +1 +Nη1/2 . +(E.15) +Combining (E.14) and (E.15) with the estimate +∣⟨(G2 − M2)A +○2,1 +2 +Eσ⟩∣ ≺ ∣e1 − σe2∣ + ∣η1 − η2∣ +Nη ++ +ψav +1 +Nη1/2 +(E.16) +for the second term in (E.11),whichreadily followsfrom (E.12) and (4.15), we find that (E.9) can be bounded +as +∣⟨S[G1A +○1,2 +1 +G2](G2 − M2)A +○2,1 +2 +⟩∣ ≺ +1 +Nη + (ψav +1 )2 +(Nη)2 , +(E.17) +where we used the trivial estimate ψav +1 ≺ η−1/2. +Estimating (E.10). For the term (E.10), we first note that the two prefactors cσ(X21[A +○2,1 +2 +]M1) and +cσ(Φσ) are bounded. However, completely analogous to the proof of Lemma 5.6, in each of the two + +66 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +cases σ = ±, the bound on one of the prefactors can be improved: In the first case, σ = +, we use (A.12) +and compute +c+(Φ+) = +⟨M1⟩(1 − ⟨M1M2⟩) +⟨M1M2⟩ += O(∣e1 − e2∣ + η1 + η2) . +∣⟨G1 ˚ +A1G2 − M(w1, ˚ +A1,w2)⟩∣ ≺ +1 +Nη + +1 +∣e1 − e2∣ + η1 + η2 +max +i∈[2] ∣⟨(Gi − Mi)(A +○1,2 +1 +)○i,i⟩∣ +which is obtained completely analogous to (E.14), we conclude that (E.10) for σ = + can be estimated by +1/(Nη). Similarly, in the second case, σ = −, we perform a computation similar to the one leading to +(5.16) and use (A.12) in order to obtain that c−(X12[A +○1,2 +1 +]M2) equals +i +2 +⟨M1A +○1,2 +1 +M ∗ +2 E−⟩ +⟨M1E−M ∗ +2 E−⟩ ++ 1 +2i +⟨M1A +○1,2 +1 +M2E−⟩ +⟨M1E−M ∗ +2 E−⟩ +1 + ⟨M1E−M ∗ +2 E−⟩ +1 + ⟨M1E−M2E−⟩ = O(∣e1 + e2∣ + η1 + η2) +Combining this with the bound +∣⟨(G1A +○1,2 +1 +G2 − M(w1,A +○1,2 +1 +,w2))E−⟩∣ ≺ +1 +Nη + +1 +∣e1 + e2∣ + η1 + η2 +⋅ +ψav +1 +Nη1/2 +which is obtained completely analogous to (E.15), we conclude that (E.10) can be estimated by 1/(Nη) +– now in both cases σ = ±. +Conclusion. Summarizing our investigations, we have shown that +⟨(G1 ˚ +A1G2 − M(w1, ˚ +A1,w2))˚ +A2⟩ = −⟨W G1 ˚ +A1G2 ˚ +A′ +2⟩ + O≺(E av +2 ) , +where we used the shorthand notation +˚ +A′ +2 ∶= (X21[˚ +A2]M1) +○ + ∑ +σ +1σ +δ cσ(X21[˚ +A2]M1) +1 − 1σ +δ cσ(X21[˚Φσ]M1) +(X21[˚Φσ]M1) +○ +(E.18) +in the underlined term. Combining (E.17) and the bound on (E.10) establishedabove with the usual single +resolvent local laws (4.15) and the bounds on deterministic approximations in Lemma 4.2, we collected +all the error terms from the expansion around (E.5)–(E.8) in (5.53). +□ +Proof of Lemma 5.9. We denote Ai ≡ ˚ +Ai, except we wish to emphasise Ai being regular. As usual, we +use the customary shorthand notations and start with +G2 = M2 − M2W G2 + M2S[G2 − M2]G2 , +such that we get +G1 ˜A1G2 ˚ +A2G3 = G1 ˜A1M2 ˚ +A2G3 − G1 ˜A1M2W G2 ˚ +A2G3 + G1 ˜A1M2S[G2 − M2]G2 ˚ +A2G3 +for ˜A1 = X12[A1] with A1 = ˚ +A1 (note that ∥X12[˚ +A1]∥ ≲ 1 by Lemma A.6) and the linear operator +X12 has been introduced in (5.33). The definition of X23 is completely analogous. +Extending the underline to the whole product, we obtain +G1( ˜A1−S[M1 ˜A1M2])G2 ˚ +A2G3 +=G1 ˜A1M2 ˚ +A2G3 − G1 ˜A1M2W G2 ˚ +A2G3 + G1 ˜A1M2S[G2 ˚ +A2G3]G3 ++ G1 ˜A1M2S[G2 − M2]G2 ˚ +A2G3 + G1S[(G1 − M1) ˜A1M2]G2 ˚ +A2G3 , +which leaves us with +G1 ˚ +A1G2 ˚ +A2G3 − M(w1,A1,w2,A2,w3) +(E.19) += (G1 [X12[˚ +A1]M2(˚ +A2 + S[M2X23[˚ +A2]M3])]G3 − M(w1,[⋯],w3)) +− G1X12[˚ +A1]M2W G2 ˚ +A2G3 + G1X12[˚ +A1]M2S[G2 − M2]G2 ˚ +A2G3 ++ G1S[(G1 − M1)X12[˚ +A1]M2]G2 ˚ +A2G3 + G1X12[˚ +A1]M2S[G2 ˚ +A2G3 − M2X23[˚ +A2]M3]G3 , +where we used Lemma C.2 for assembling the purely deterministic terms on the l.h.s. To continue, +we first note that ∥X12[˚ +A1]∥ ≲ 1 and ∥X23[˚ +A2]∥ ≲ 1 (again, the matrices being regular removes the +potentially ‘bad direction’ of the stability operators X12 and X23). + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +67 +Then, we need to further decompose X12[A1]M2 in the last four terms in (E.19) as +X12[A1]M2 = (X12[A1]M2) +○ + ∑ +σ +1σ +δ cσ(X12[A1]M2)Eσ , +(E.20) +where, similarly as for ⋅○, we suppressed the spectral parameters w1,w2 in the notation for the linear +functionals cσ(...), which have been defined in see (5.36). Now, plugging (E.20) into (E.19) we find +G1 ˚ +A1G2 ˚ +A2G3 − M(w1, ˚ +A1,w2, ˚ +A2,w3) +(E.21) += (G1 [X12[˚ +A1]M2(˚ +A2 + S[M2X23[˚ +A2]M3])]G3 − M(w1,[⋯],w3)) +− G1(X12[˚ +A1]M2) +○W G2 ˚ +A2G3 + G1(X12[˚ +A1]M2) +○S[G2 − M2]G2 ˚ +A2G3 ++ G1S[(G1 − M1)(X12[˚ +A1]M2) +○]G2 ˚ +A2G3 + G1(X12[˚ +A1]M2) +○S[G2 ˚ +A2G3 − M2X23[˚ +A2]M3]G3 ++ ∑ +σ +1σ +δ cσ(X12[˚ +A1]M2)[ − G1EσW G2 ˚ +A2G3 + G1EσS[G2 − M2]G2 ˚ +A2G3 ++ G1S[(G1 − M1)Eσ]G2 ˚ +A2G3 + G1EσS[G2 ˚ +A2G3 − M2X23[˚ +A2]M3]G3]. +Next, as in the earlier sections (see, e.g., the display above (E.4)), in the last line of (E.21) we now undo +the underline and find the bracket [⋯] to equal (the negative of) +G1Eσ( ˚ +A2 + S[M(w2, ˚ +A2,w3)])G3 − G1ΦσG2 ˚ +A2G3 , +where we denoted +Φσ ∶= Eσ 1 +M2 +− S[M1Eσ]. +It is apparent from the expansion (E.21) (and it can also be checked by hand) that +M(w1,Eσ ˚ +A2 + EσS[M(w2, ˚ +A2,w3)],w3) = M(w1,Φσ,w2, ˚ +A2,w3), +which finally yields +G1 ˚ +A1G2 ˚ +A2G3 − M(w1, ˚ +A1,w2, ˚ +A2,w3) +(E.22) += (G1 [X12[˚ +A1]M2(˚ +A2 + S[M2X23[˚ +A2]M3])]G3 − M(w1,[⋯],w3)) +− G1(X12[˚ +A1]M2) +○W G2 ˚ +A2G3 + G1(X12[˚ +A1]M2) +○S[G2 − M2]G2 ˚ +A2G3 ++ G1S[(G1 − M1)(X12[˚ +A1]M2) +○]G2 ˚ +A2G3 + G1(X12[˚ +A1]M2) +○S[G2 ˚ +A2G3 − M2X23[˚ +A2]M3]G3 ++ ∑ +σ +1σ +δ cσ(X12[˚ +A1]M2)[ − (G1Eσ(˚ +A2 + S[M(w2, ˚ +A2,w3)])G3 − M(w1,[⋯]w3)) ++ (G1˚ΦσG2 ˚ +A2G3 − M(w1,˚Φσ,w2, ˚ +A2,w3)) + ∑ +σ +cσ(Φσ)(G1EσG2 ˚ +A2G3 − M(w1,Eσ,w2, ˚ +A2,w3))], +where we further decomposed Φσ in the last line of (E.22) (while using the first relation in (5.40)) just as +X12[A1]M2 in (E.20). +Next, we write (E.22) for both, A1 = ˚ +A1 = ˚Φ+ and A1 = ˚ +A1 = ˚Φ−, and solve the two resulting linear +equations for G1˚Φ±G2 − M(w1,˚Φ±,w2). Observe that by means of the second relation in (5.40) the +original system of linear equations boils down to two separate ones. Thus, plugging the solutions for +G1˚Φ±G2 ˚ +A2G3 − M(w1,˚Φ±,w2, ˚ +A2,w3) back into (E.22), we arrive at +G1 ˚ +A1G2 ˚ +A2G3 − M(w1, ˚ +A1,w2, ˚ +A2,w3) +(E.23) += (G1 [X12[˚ +A1]M2(˚ +A2 + S[M2X23[˚ +A2]M3])]G3 − M(w1,[⋯],w3)) +− G1(X12[˚ +A1]M2) +○W G2 ˚ +A2G3 + G1(X12[˚ +A1]M2) +○S[G2 − M2]G2 ˚ +A2G3 ++ G1S[(G1 − M1)(X12[˚ +A1]M2) +○]G2 ˚ +A2G3 + G1(X12[˚ +A1]M2) +○S[G2 ˚ +A2G3 − M2X23[˚ +A2]M3]G3 ++ ∑ +σ +1σ +δ cσ(X12[˚ +A1]M2) +1 − 1σ +δ cσ(X12[˚Φσ]M2) +[ − (G1[Eσ(˚ +A2 + S[M(w2, ˚ +A2,w3)])]G3 − M(w1,[⋯]w3)) + +68 +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES ++ (G1 [X12[˚Φσ]M2(˚ +A2 + S[M2X23[˚ +A2]M3])]G3 − M(w1,[⋯],w3)) +− G1(X12[˚Φσ]M2) +○W G2 ˚ +A2G3 + G1(X12[˚Φσ]M2) +○S[G2 − M2]G2 ˚ +A2G3 ++ G1S[(G1 − M1)(X12[˚Φσ]M2) +○]G2 ˚ +A2G3 + G1(X12[˚Φσ]M2) +○S[G2 ˚ +A2G3 − M2X23[˚ +A2]M3]G3 ++ cσ(Φσ)(G1EσG2 ˚ +A2G3 − M(w1,Eσ,w2, ˚ +A2,w3))]. +It has been shown in Lemma 5.7 that the denominators are bounded away from zero. +Next, we take the scalar product of (E.23) with two deterministic vectors x,y satisfying ∥x∥,∥y∥ ≤ +1. In the resulting expression, in case that 1σ +δ (w1,w2) = 1 (as we assumed in (??)), there are three +particular terms, namely the ones of the form +(G1S[(G1 − M1)A +○1,2 +1 +]G2 ˚ +A2G3)xy , +(E.24) +as appearing twice, in the fourth and second to last line, +(G1A +○1,2 +1 +S[G2 ˚ +A2G3 − M(w2, ˚ +A2,w3)]G3)xy , +(E.25) +as appearing, again twice, in the fourth and second to last line, +cσ(X12[˚ +A1]M2)cσ(Φσ)(G1EσG2 ˚ +A2G3 − M(w1,Eσ,w2, ˚ +A2,w3))xy , +(E.26) +as appearing in the last line, whose naive sizes 1/(Nη3), 1/(Nη3), and 1/ +√ +Nη4 do not match the +target. Hence, they have to be discussed in more detail. +Estimating (E.24). For the terms of the first type, we begin by expanding +(G1S[(G1 − M1)A +○1,2 +1 +]G2 ˚ +A2G3)xy = ∑ +σ +σ⟨(G1 − M1)A +○1,2 +1 +Eσ⟩(G1EσG2 ˚ +A2G3)xy +and recall from (E.16) that first factor can be estimated by +∣⟨(G1 − M1)A +○1,2 +1 +Eσ⟩∣ ≺ ∣e1 − σe2∣ + ∣η1 − η2∣ +Nη ++ +ψav +1 +Nη1/2 . +(E.27) +In the second factor, we distinguish the two cases σ = ±. For σ = +, we find +G1G2A +○2,3 +2 +G3 = G1A +○2,3 +2 +G3 − G2A +○2,3 +2 +G3 +(e1 − e2) + i(η1 + η2) +by a simple resolvent identity, which together with +˚ +Aw2,w3 +2 += ˚ +Aw1,w3 +2 ++ O(∣e1 − e2∣ + ∣η1 − η2∣ + ∣e1 − e3∣ + ∣η1 − η3∣)E+ ++ O(∣e1 − e2∣ + ∣η1 − η2∣ + ∣e1 + e3∣ + ∣η1 − η3∣)E− +from Lemma 3.3 (note the difference between the E+-error and the E−-error!) and the usual isotropic +law (4.15) yields the estimate +∣(G1G2A +○2,3 +2 +G3)xy∣ ≺ 1 +η + +1 +∣e1 − e2∣ + η1 + η2 +⎛ +⎝1 + +ψiso +1 +√ +Nη2 +⎞ +⎠ , +(E.28) +where we again used the a priori bound (4.23). For σ = − we employ the integral representation from +Lemma 5.1 and argue similarly as for (E.15) such that we finally obtain +∣(G1E−G2A +○2,3 +2 +G3)xy∣ ≺ 1 +η + +1 +∣e1 + e2∣ + η1 + η2 +⎛ +⎝1 + +ψiso +1 +√ +Nη2 +⎞ +⎠ . +(E.29) +Now, combining (E.27) with (E.28) and (E.29), we find +∣(G1S[(G1 − M1)A +○1,2 +1 +]G2 ˚ +A2G3)xy∣ ≺ +1 +√ +Nη3 (1 + ψav +1 ψiso +1 +Nη +) , +(E.30) +where we used that ψav +1 ≺ η−1/2 trivially by (4.15). + +EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES +69 +Estimating (E.25). For terms of the second type, we again start by expanding +(G1A +○1,2 +1 +S[G2 ˚ +A2G3 − M(w2, ˚ +A2,w3)]G3)xy += ∑ +σ +σ⟨(G2 ˚ +A2G3 − M(w2, ˚ +A2,w3))Eσ⟩(G1A +○1,2 +1 +EσG3)xy . +Then, for the first factor, we recall from the estimate of (E.9) that +∣⟨(G2A +○2,3 +2 +G3 − M(w2,A +○2,3 +2 +,w3))Eσ⟩∣ ≺ +1 +Nη + +1 +∣e2 − σe3∣ + η2 + η3 +⋅ +ψav +1 +Nη1/2 . +Treating the second factor analogously to (E.28) and (E.29) above, we find +∣(G1A +○1,2 +1 +EσG3)xy∣ ≺ ∣e2 − σe3∣ + ∣η2 − η3∣ +η ++ ⎛ +⎝1 + +ψiso +1 +√ +Nη2 +⎞ +⎠ . +Combining the two estimates, we have shown that +∣(G1A +○1,2 +1 +S[G2 ˚ +A2G3 − M(w2, ˚ +A2,w3)]G3)xy∣ ≺ +1 +√ +Nη3 (1 + ψiso +1 +Nη + ψav +1 ψiso +1 +Nη +) +(E.31) +where we again used that ψav +1 ≺ η−1/2 trivially by (4.15). +Estimating (E.26). For the third term, we recall the (improved) estimates +c+(Φ+) = O(∣e1 − e2∣ + η1 + η2) +c−(X12[˚ +A1]M2) = O(∣e1 + e2∣ + η1 + η2) +on the anyway bounded prefactors, which have been shown in the course of estimating (5.45). By arguing +analogously to (E.28) and (E.29), we also find +∣(G1EσG2 ˚ +A2G3 − M(w1,Eσ,w2, ˚ +A2,w3))xy∣ ≺ +1 +√ +Nη3 + +1 +∣e1 − σe2∣ + η2 + η3 +ψiso +1 +√ +Nη2 . +Now, combining these estimates, we conclude +∣(E.26)∣ ≺ +1 +√ +Nη3 (1 + ψiso +1 ) . +(E.32) +Conclusion. Summarizing our investigations, we have shown that +(G1 ˚ +A1G2 ˚ +A2G3 − M(w1, ˚ +A1,w2, ˚ +A2,w3))xy = −(G1 ˚ +A′ +1W G2 ˚ +A2G3)xy + O≺(E iso +2 ) , +where we used the shorthand notation +˚ +A′ +1 = (X12[A1]M2) +○ + ∑ +σ +1σ +δ cσ(X12[A1]M2) +1 − 1σ +δ cσ(X12[˚Φσ]M2) +(X12[˚Φσ]M2) +○ +(E.33) +in the underlined term. Combining (E.30), (E.31), and (E.32) with the usual single resolvent local laws +(4.15) and the bounds on deterministic approximations in Lemma 4.2, we collected all the error terms +from (E.23) in (5.61). +□ +References +[1] O. H. Ajanki, L. Erdős, T. Krüger. Quadratic vector equations on complex upper half-plane. American MathematicalSociety Vol. 261. +No. 1261 (2019) +[2] O. H. Ajanki, L. Erdős, T. 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Duke Mathematical Journal 55, 919–941 (1987) + diff --git a/_tE1T4oBgHgl3EQf8wWT/content/tmp_files/load_file.txt b/_tE1T4oBgHgl3EQf8wWT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3977df013c67672c39bdd09f8954264557ad360b --- /dev/null +++ b/_tE1T4oBgHgl3EQf8wWT/content/tmp_files/load_file.txt @@ -0,0 +1,3780 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf,len=3779 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='03549v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='PR] 9 Jan 2023 OPTIMAL LOWER BOUND ON EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES GIORGIO CIPOLLONI Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA LÁSZLÓ ERDŐS# AND JOSCHA HENHEIK# IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria DOMINIK SCHRÖDER∗ ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We consider large non-Hermitian N × N matrices with an additive independent, identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=') noise for each matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We show that already a small noise of variance 1/N completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by Deutsch [33] and proven for Wigner ma- trices in [24], to arbitrary non-Hermitian matrices with an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly com- putable Ginibre ensemble apart from the very recent upper bounds given in [7] and [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Introduction Traditional random matrix theory focuses on statistics of eigenvalues, where spectacular univer- sality phenomena arise: the local spectral statistics tend to become universal as the dimension goes to infinity with new distributions arising;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' most importantly the celebrated Wigner-Dyson-Mehta bulk statistics and the Tracy-Widom edge statistics in the Hermitian spectrum and the Ginibre statistics in the non-Hermitian spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' More recently eigenvectors of Hermitian ensembles received considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' They also become universal, albeit in a more conventional way: they tend to be entirely ran- domised, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Haar distributed [16, 17, 47, 11, 27, 29, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this paper we study two related questions: how do eigenvectors and singular vectors of a typical non-Hermitian random matrix in high dimension look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To answer them, we introduce a new decomposition of general observables that identifies correlations of the Hermitised resolvents as entire matrices at different spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This cap- tures correlations of the singular well beyond correlations of traces of resolvents that govern only the E-mail addresses: gc4233@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='edu, lerdos@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='at, joscha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='henheik@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='at, dschroeder@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 60B20, 15B52, 62F22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Eigenvalue condition number, Non-Hermitian perturbation theory, Quantum unique ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' #Supported by ERC Advanced Grant “RMTBeyond” No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 101020331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ∗Supported by the SNSF Ambizione Grant PZ00P2_209089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 1 2 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Somewhat surprisingly, we are then able to transfer information on singular vectors to the non-Hermitian eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Non-Hermitian eigenvector overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To be specific, we consider non-Hermitian N × N ma- trices of the form Λ + X, where Λ is an arbitrary deterministic matrix and X is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We assume that the norm of Λ is bounded independently of N and X has independent, identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=') centred matrix elements with variance E∣xij∣2 = 1 N with some further moment conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This normalisation guarantees that ∥X∥ ≤ 2 + o(1) and the spectrum of X lies essentially in the unit disk (circular law) with very high probability, hence Λ and X remain of comparable size as N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that X perturbs each matrix elements of Λ by a small random amount of order 1/ √ N, however the spectra of Λ and Λ + X substantially differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The analysis of non-Hermitian random matrices is typically much harder than that of the Hermit- ian ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Non-Hermitian matrices have two different sets of spectral data: eigenvalues/vectors and singular values/vectors which cannot be directly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, the study of singular vectors and eigenvectors substantially differ: while singular vectors can still be understood from a Hermitian theory, there is no such route for eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Unlike for non-Hermitian eigenvalues, where Girko’s formula translates their linear statistics into a Hermitian problem, no similar "Hermitisation" relation is known for non-Hermitian eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Furthermore, left and right eigenvectors differ and their rela- tion is very delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Assuming that each eigenvalue µi of Λ+X is simple, we denote the corresponding left and right eigenvectors by li, ri, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Λ + X)ri = µiri , lt i(Λ + X) = µilt i , under the standard bi-orthogonality relation ⟨¯lj,ri⟩ = lt jri = δi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that this relation leaves a large freedomin choosing the normalisationof each eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The key invariant quantity is the eigenvector overlap Oij ∶= ⟨rj,ri⟩⟨lj,li⟩, which emerges in many problems where non-Hermitian eigenvectors are concerned, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' [3, 19, 20, 8, 13, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Two prominent examples are (i) in numerical linear algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' where √Oii is the eigenvalue condition number determining how fast µi moves under small perturbation in the worst case using the formula √ Oii = lim t→0 sup {∣µi(Λ + X + tE) − µi(Λ + X) t ∣ ∶ E ∈ CN×N,∥E∥ = 1} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' [7]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (ii) in the theory of the Dyson Brownian motion for non-Hermitian matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' where Oij gives the correlation of the martingale increments for the stochastic evolution of the eigenvalues µi and µj as the matrix evolves by the natural Ornstein-Uhlenbeck flow (see [40], [13, Appendix A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The main result of this paper is an almost optimal lower bound of order N on the diagonal over- lap Oii, with very high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the context of numerical linear algebra this means that non- Hermitian eigenvalues of Λ + X still move at a speed of order √ N under the "worst" perturbation E in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), despite having added a random smoothing component X to Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that in numerics one typically views the random smoothing as a tool to reduce the overlap of Λ in order to enhance the stability of its eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' our result shows a natural limitation for such reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Complementary upper bounds on Oii have recently been proven in [7] and [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' These hold only in expectation sense, as Oii has a fat-tail, and they are off by a factor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We remark, however, that N is the most relevant parameter of the problem only from our random matrix theory point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Works motivated by numerical analysis, such as [7, 43] and references therein, often focus on tracking the γ-dependence for the problem Λ + γX in the small noise regime γ ≪ 1 in order to reduce the effect of the random perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this setup the non-optimality of the N-power may be considered less relevant1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the context of the Dyson Brownian motion, our lower bound on Oii implies a diffusive lower bound on the eigenvalues of the Ornstein-Uhlenbeck (OU) matrix flow, generalizing the analogous 1As long as γ is N independent, one may set γ = 1 by a simple rescaling so we refrain from carrying this extra factor in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We remark that our methods would allow to trace the polynomial γ-dependence in all our main estimates as well, albeit not with an optimal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 3 result of Bourgade and Dubach [13, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6] from Ginibre ensemble to arbitrary i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ensemble (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thermalisation of singular vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The key step to our lower bound on Oii is a thermalisation result on the singular vectors that is of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Namely, we show that singular vectors of Λ + X are fully randomised in the large N limit in the sense that their quadratic forms with arbitrary test matrices have a deterministic limit with an optimal N −1/2 speed of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This holds with very high probability which enables us to make such statement for matrices of the form (Λ − z) + X simultaneously for any shift parameterz, even for random ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We will use this for z = µ, an eigenvalue of Λ + X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This allows us to gain access to eigenvectors of Λ + X, by noticing that singular vectors and eigenvectors are unrelated in general with an obvious exception: if µ is an eigenvalue of Λ + X, then any vector in the kernel of Λ + X − µ is an eigenvector of Λ + X with eigenvalue µ, and a singular vector of Λ + X − µ with singular value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence high probability statements for singular vectors can be converted into similar statements for eigenvectors – this key idea may be viewed as the eigenvector version of the transfer principle between eigenvalues and singular values encoded in Girko’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Our thermalisation result for singular vectors may be viewed as the non-Hermitian analogue of the Quantum Unique Ergodicity (QUE) for Hermitian Wigner matrices proven in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now briefly explain the QUE phenomenon and its physics background in the simplest Hermitian context before we consider the singular vectors of Λ+X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, via a standard Hermitisation procedure we will turn the singular vector problem to a Hermitian eigenvector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For Hermitian random matrices H, that can be considered as the Hamilton operator of a disordered quantum system, a major motivation comes from physics, where the randomisation of the eigenvectors is interpreted as a thermalisation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The Eigenstate Thermalisation Hypothesis (ETH) by Deutsch [33] and Srednicki [50] (see also [32, 34]) asserts that any deterministic Hermitian matrix A (observable), be- comes essentially diagonal in the eigenbasis of a "sufficiently chaotic" Hamiltonian, where chaos may come from an additional randomness or from the ergodicity of the underlying classical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In other words, ⟨ui,Auj⟩ − δij⟨⟨A⟩⟩i → 0, as N → ∞ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) where {ui} is a orthonormal eigenbasis of H and the deterministic "averaged" coefficient ⟨⟨A⟩⟩i is to be computed from the statistics of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the mathematicsliterature the same problem is known as the Quantum (Unique) Ergodicity, orig- inally formulated for the Laplace-Beltrami operator on surfaces with ergodic geodesic flow, see [49, 30, 56], on regular graphs [5] and on special arithmetic surfaces [48, 18, 46, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In [24] we proved QUE in the strongest form with an optimal speed of convergence for the eigenvectors of Wigner matrices that, by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Wigner’s vision, can be viewed as the "most random" Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this case, the diagonal limit ⟨⟨A⟩⟩i in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) is independent of i and given by the normalised trace ⟨A⟩ ∶= 1 N TrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, in subse- quent papers [27, 29] (see also [11]) even the normal fluctuation of √ N[⟨ui,Aui⟩ − ⟨A⟩] was proven, followed by the proof of joint Gaussianity of finite many overlaps in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Previously QUE results were proven for rank one observables (see [44, 53] under four moment matching and [16] in general) and finite rank observables [47], see also [9] for deformed Wigner matrices and [17] for band matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The proofs crucially used that H is Hermitian, heavily relying on sophisticated Hermitian techniques (such as local laws and Dyson Brownian Motion) developed in the last decade for eigenvalue universality questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Back to our non-Hermitian context, we consider the singular vectors {ui,vi}N i=1 of Λ + X, (X + Λ)(X + Λ)∗ui = σ2 i ui , (X + Λ)∗(X + Λ)vi = σ2 i vi , belonging to the singular value σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We view them as the two N-dimensional components of the eigen- vectors wi = (ui,vi) of the 2N-dimensional Hermitisation of Λ + X, defined as H = HΛ ∶= W + ˆΛ , W ∶= ( 0 X X∗ 0 ) , ˆΛ ∶= ( 0 Λ Λ∗ 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) In particular, from the overlaps ⟨wi,Awj⟩ of eigenvectors for the Hermitised problem with a gen- eral (2N) × (2N) matrix A one may read off all the singular vector overlaps of the form ⟨ui,Buj⟩, 4 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES ⟨vi,Bvj⟩ and ⟨ui,Bvj⟩ with any N × N matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Therefore our goal is to show the general ther- malisation phenomenon, the convergence of ⟨wi,Awj⟩ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2)), for the Hermitised matrix HΛ thus generalizing the ETH proven in [24] beyond Wigner matrices and with an additional arbitrary matrix Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Unlike in the Wigner case, the limit ⟨⟨A⟩⟩i genuinely depends on the index i and part of the task is to determine its precise form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that due to the large zero blocks, W has about half as many random degrees of freedom as a Wigner matrix of the same dimension has, moreover the block structure gives rise to potential instabilities, thus the ETH for HΛ is considerably more involved than for Wigner ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the next section we explain the main new method of this paper that systematically handles all these instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Structural decomposition of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We introduce a new concept for splitting general observables into "regular" and "singular" components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' where the singular component gives the leading contribution and the regular component is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the case of Wigner matrices H in [24, 25] we used the decomposition A = ⟨A⟩ + ˚ A, where the traceless part of A, ˚ A ∶= A − ⟨A⟩, is the regular component and the projection2 of A onto the one dimensional space spanned by the identity matrix is the singular component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This gave rise to the following decomposition of resolvent G = G(w) = (H − w)−1 for any w ∈ C ∖ R: ⟨GA⟩ = m⟨A⟩ + ⟨A⟩⟨G − m⟩ + ⟨G˚ A⟩, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) where m = m(w) is the Stieltjes transform of the semicircle law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The second term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) is asymptot- ically Gaussian of size ⟨G − m⟩ ∼ (Nη)−1 [41] and the last term is also Gaussian, but of much smaller size ⟨G˚ A⟩ ∼ ⟨˚ A˚ A∗⟩1/2/(Nη1/2) in the interesting regime of small η ∶= ∣Imw∣ ≪ 1 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Similar decomposition governs the traces of longer resolvent chains of Wigner matrices,for example ⟨GAG∗B⟩ = ⟨GG∗⟩ = 1 η ⟨ImG⟩ ∼ 1 η ≫ 1 if A = B = I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' both observable matrices are purely singular, while for regular (and bounded) observ- ables A = ˚ A, B = ˚ B we have ⟨GAG∗B⟩ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) Both examples indicate the √η-rule (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5 later), informally asserting that each regu- lar observable renders the size of a resolvent chain smaller by a factor √η than its singular counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In [28, 29] we obtained the deterministic leading terms and optimal error estimates on the fluctuation for resolvent chains of arbitrary length ⟨G(w1)A1G(w2)A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='⟩ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) with arbitrary observables in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The answer followed the √η-rule hence it heavily depended on the Ai = ⟨Ai⟩ + ˚ Ai decomposition for each observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, in order to estimate ⟨ui,Auj⟩ − δij⟨A⟩ = ⟨ui, ˚ Auj⟩ for ETH in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2), we had N∣⟨ui, ˚ Auj⟩∣2 ≲ ⟨ImG(w1)˚ AImG(w2)˚ A⟩ ≲ 1, where we first used spectral decompositionof both G’s and then used a version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Here the spectral parameters wk = ek+iη are chosen such that e1 and e2 be close to the eigenvalues corresponding to ui and uj, respectively, and η ∼ N −1 in order to resolve the spectrum on the fine scale of the individual eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 The key point in all these analyses for Wigner matrices was that the regular/singular concept was independent of the spectral parameter: the same universal decomposition into tracial and traceless parts worked in every instance along the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' One consequence is the i-independence of the limiting overlap ⟨⟨A⟩⟩i ∶= ⟨A⟩ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 2We equip the space of matrices with the usual normalised Hilbert-Schmidt scalar product, ⟨A, B⟩ ∶= 1 N Tr A∗B = ⟨A∗B⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 3Strictly speaking we used η = N −1+ξ with any small ξ > 0, and all estimates held up to an N ξ factor but we ignore these technicalities in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 4A quick direct way to see this independence is the special case of Gaussian Wigner matrices (GUE or GOE), where the eigen- vectors are Haar distributed, independently of their eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 5 For more complicated ensembles, like HΛ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3), especially if an arbitrary matrix Λ is involved, the correct decomposition depends on the location in the spectrum of H where we work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To guess it, first we recall the single resolvent local law (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) for the resolvent G = GΛ(w) = (HΛ − w)−1, asserting that ⟨GA⟩ ≈ ⟨MA⟩, where M = M Λ(w) solves a nonlinear deterministic equation, the Matrix Dyson Equation (MDE), see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then a heuristic calculation (see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) shows that for w = e + iη ∈ C+ we have E ∣⟨(G − M)A⟩∣ 2 ≈ ∣⟨ImMA⟩∣ 2 (Nη)2 + ∣⟨ImMAE−⟩∣ 2 N 2η(∣e∣ + η) + O( 1 N 2η ) , E− ∶= (1 0 0 −1) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) indicating that the singular component of A is two dimensional, depends on w, and for any A orthogonal to the two singular directions ImM and E−Im M the size of ⟨(G − M)A⟩ is smaller by a factor √η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The first singular direction is always present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The second singular direction is a consequence of the block structure of H and it is manifested only for w near the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For energies ∣e∣ ∼ 1, only the first singular direction, namely the one involving ImM plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' What about longer chains (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Each matrix Ai is sandwiched between two resolvents with different spectral parameters wi, wi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We find that the correct decomposition of any A between two resolvents in a chain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' G(w)AG(w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' depends only on w,w′ and it has the form A = ⟨V+,A⟩U+ + ⟨V−,A⟩U− + ˚ A , V± = V w,w′ ± , ˚ A = ˚ Aw,w′ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) where the first two terms form the singular component of A, and ˚ A, defined by this equation, is the regular component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We will establish that both V+ and V− are the right eigenvectors of a certain stability operator B acting on C2N×2N that corresponds to the Dyson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For example, if Imw and Imw′ have opposite signs then V+ is the right eigenvector of B[⋅] = 1 − M( ¯w)S[⋅]M( ¯w′), where S is covariance operator for the matrix W in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' V± with other sign combinations are defined very similarly (in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 we present all cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, the special directions ImM and E−ImM that we found by direct variance calculation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) emerge canonically as eigenvectors of a certain stability operator!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Similar variance calculation for longer chains would reveal the same consistency: the variance of the chain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) is the smallest if each Ai is regular with respect to the two neighboring spectral parameters wi,wi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that the choice of V± is basically dictated by variance calculations like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, the matrices U± in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) can still be chosen freely up to their linear independence and the normalisation re- quirement ⟨Vσ,Uτ⟩ = δσ,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The latter guarantees that the sum of the singular terms in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) is actually a (non-orthogonal) projection ∣U+⟩⟨V+∣ + ∣U−⟩⟨V−∣ acting on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since V± are the right eigenvectors of a stability operator, one may be tempted to choose U± as certain left eigenvectors but we did not find this guiding principle helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Instead, we use this freedom to simplify the calculation of the singular terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Substituting the singular part of A into .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' G(w)AG(w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', we need to compute G(w)U±G(w′) and quite pragmatically we choose U± such that the resolvent identity could be applied and thus re- duce the length of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thanks to the spectral symmetry of H = HΛ, for its resolvent we have E−G(−w)E− = −G(w), and we find that U+ = I, U− = E− do the job, which accidentally coincide with the left eigenvectors of the stability operator for the special case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 we present the canonical choices of V± and U± in a more general situation and explain at which stage of the proof their correct choice emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In our current application only V± are nontrivial (in particular energy dependent), while U± are very simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This is due to the fact that the chain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) consists of resolvents of the same operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In more general problemsone maytake resolvents with two different Λ’s in the chain, in which case U± are also nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This decomposition scheme is the really novel ingredient of our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Several other tools we use, such as recursive Dyson equations, hierarchy of master inequalities and reduction inequalities have been introduced before (especially in our related works on Wigner matrices [24, 25]), but the dependence of the decomposition on the spectralparametersin the current setup requires quite different new estimates along the arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We informally explain the prototype of such an estimate at the beginning of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 6 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We define the 2N × 2N matrices E± ∶= E1 ± E2, where E1 ∶= (1 0 0 0) and E2 ∶= (0 0 0 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Each entry of the matrix is understood as a multiple of the N × N–identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By ⌈⋅⌉, ⌊⋅⌋ we denote the upper and lower integer part, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for x ∈ R we define ⌈x⌉ ∶= min{m ∈ Z∶m ≥ x} and ⌊x⌋ ∶= max{m ∈ Z∶m ≤ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We denote [k] ∶= {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',k} for k ∈ N and ⟨A⟩ ∶= d−1Tr(A), d ∈ N, is the normalised trace of a d × d-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For positive quantities A,B we write A ≲ B resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A ≳ B and mean that A ≤ CB resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A ≥ cB for some N-independent constants c,C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We denote vectors by bold-faced lower case Roman letters x,y ∈ C2N, for some N ∈ N, and define ⟨x,y⟩ ∶= ∑ i ¯xiyi , Axy ∶= ⟨x,Ay⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Matrix entries are indexed by lower case Roman letters a,b,c,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' from the beginning of the alpha- bet and unrestricted sums over a,b,c,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' are always understood to be over {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',N,N + 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',2N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Analogously, unrestricted sums over lower case Roman letters i,j,k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' from the middle of the alphabet are always understood to be over {−N,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',−1,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, the lower case Greek letters σ and τ as indices indicate a sign, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' σ,τ ∈ {+,−}, and unrestricted sums over σ,τ are understood to be over {+,−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We will use the concept of ‘with very high probability’, meaning that any fixed D > 0, the probability of an N-dependent event is bigger than 1 − N −D for all N ≥ N0(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Also, we will use the conven- tion that ξ > 0 denotes an arbitrarily small constant, independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we introduce the common notion of stochastic domination (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', [35]): For two families X = (X(N)(u) ∣ N ∈ N,u ∈ U (N)) and Y = (Y (N)(u) ∣ N ∈ N,u ∈ U (N)) of non-negative random variables indexed by N, and possibly a parameter u, then we say that X is stochastically dominated by Y , if for all ε,D > 0 we have sup u∈U(N) P[X(N)(u) > N ǫY (N)(u)] ≤ N −D for large enough N ≥ N0(ǫ,D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this case we write X ≺ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' If for some complex family of random variables we have ∣X∣ ≺ Y , we also write X = O≺(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Acknowledgement: The authors are grateful to Oleksii Kolupaiev for valuable discussions, especially about the choice of contours in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Main results We consider real or complex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' matrices X , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' N × N matrices whose entries are independent and identically distributed as xab d= N −1/2χ for some real or complex random variable χ satisfying the following assumptions: Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We assume that Eχ = 0 and E∣χ∣2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Furthermore, we assume the existence of high moments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', that there exist constants Cp > 0, for any p ∈ N, such that E ∣χ∣p ≤ Cp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Additionally, in the complex case, we assume that E χ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For definiteness, in the sequel we perform our entire analysis for the complex case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the real case being completely analogous and hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Non-Hermitian singular vectors and eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Fix a deterministic matrix Λ ∈ CN×N, with N-independent norm bound, ∥Λ∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let {σi}i∈[N] be the singular values of X + Λ with corresponding (normalised) left- and right-singular vectors {ui}i∈[N] and {vi}i∈[N], respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (X + Λ)vi = σiui and (X + Λ)∗ui = σivi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) All these objects naturally depend on Λ, but we will omit this fact from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let νi, i ∈ [N], be the increasingly ordered singular values of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Define the Hermitisation of Λ as ˆΛ ∶= ( 0 Λ Λ∗ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) Due to its block structure, the spectrum of ˆΛ is symmetric with respect to zero, with eigenvalues {νi}0≠∣i∣≤N such that ν−i = −νi for all i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The empirical density of states of ˆΛ is denoted by µˆΛ ∶= 1 2N ∑ 0≠∣i∣≤N δνi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let µsc be the Wigner semicircle distribution with density ρsc(x) ∶= (2π)−1√ [4 − x2]+, where [⋯]+ is the positive part of a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Define the free additive convolution µ = µΛ ∶= µsc ⊞ µˆΛ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) which is a probability distribution on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now briefly recall basic facts about the free convolution with the semicircle density (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the classical paper by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Biane [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Most conveniently µ may be defined by inverting its Stieltjes transform m(w) = mΛ(w) ∶= ∫R µ(de) e − w , w ∈ C ∖ R, where m satisfies the implicit equation m(w) = ∫R µˆΛ(de) e − (w + m(w)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) With the additional constraint Imm(w) ⋅ Im w > 0 this equation has a unique solution that is analytic away from the real axis with m(w) = m(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since µˆΛ is symmetric to zero with bounded support, µ is also symmetric with support bounded independently of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover µ is absolutely continuous with respect to Lebesgue measure with density denoted by ρ = ρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The density ρ may be obtained5 as the boundary value of Im m at any e on the real line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ρ(e) ∶= lim η↓0 ρ(e + iη), ρ(w) ∶= 1 π ∣Imm(w)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) Infactm itself hasa continuousextensionto the realaxisfrom the upperhalf plane m(e) ∶= limη↓0 m(e+ iη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proving the existence of these limits is standard from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, for any (small) κ > 0, we define the κ-bulk of the density ρ as Bκ = BΛ κ ∶= {x ∈ R ∶ ρ(x) ≥ κ1/3} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) which is symmetric to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, we denote a (modified) ith quantile of the density ρ by γi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' i + N 2N = ∫ γi −∞ ρ(e)de , ∣i∣ ≤ N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) and we immediately conclude by symmetry of ρ that γi = −γ−i for every ∣i∣ ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Our first main result establishes the thermalisation of singular vectors of X + Λ in the bulk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for indices i,j with quantiles γi,γj uniformly in the bulk of the density ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 5For orientation of the reader we mention that ρ is the deterministic approximation, the so-called self-consistent density of states (scDos), for the empirical eigenvalue density of the Hermitisation of X +Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This connection will be explainedin the next Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 8 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Thermalisation of Singular Vectors) Fix a bounded Λ ∈ CN×N and κ > 0 independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let {ui}i∈[N] and {vi}i∈[N] be the (normalised) left- and right-singular vectors of X + Λ, respectively, where X is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' matrix satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for any deterministic matrix B ∈ CN×N with ∥B∥ ≲ 1 it holds that6 max i,j �������������� ⟨ui,Buj⟩ − δj,i ⟨Im [ γj+m(γj ) ΛΛ∗−(γj +m(γj ))2 ] B⟩ πρ(γj) �������������� ≺ 1 √ N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8a) max i,j �������������� ⟨vi,Bvj⟩ − δj,i ⟨Im [ γj+m(γj ) Λ∗Λ−(γj +m(γj ))2 ] B⟩ πρ(γj) �������������� ≺ 1 √ N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8b) max i,j �������������� ⟨ui,Bvj⟩ − δj,i ⟨Im [Λ∗(ΛΛ∗ − (γj + m(γj))2) −1] B⟩ πρ(γj) �������������� ≺ 1 √ N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8c) where the maximum is taken over all i,j ∈ [N] such that the quantiles γi,γj ∈ Bκ are in the κ-bulk of the density ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The thermalisation of singular vectors will be a simple corollary to the Eigenstate Thermalisation Hypothesis (ETH) for the Hermitisation HΛ of X + Λ, which is formulated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 will be given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Our second main result concerns the bi-orthonormal left and right eigenvectors {li}i∈[N] and {ri}i∈[N], respectively, of X + Λ, with corresponding eigenvalues {µi}i∈[N], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (X + Λ)ri = µiri , lt i(X + Λ) = µilt i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) where t denotes the transpose of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' More precisely, the following theorem provides a lower bound on the diagonal part of the overlaps matrix Oij ∶= ⟨rj,ri⟩⟨lj,li⟩, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) defined subject to the standard normalisation ⟨¯lj,ri⟩ = lt jri = δi,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) We restrict our results to eigenvalues µi in the bulk of X + Λ in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We say that z ∈ C is in the bulk of X + Λ if and only if there exists an N-independent κ > 0 for which 0 ∈ BΛ−z κ = {x ∈ R ∶ ρΛ−z(x) ≥ κ1/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' There is no simple characterisation of the bulk of X + Λ in terms of the spectrum of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, taking the imaginary part of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) at w = 0 + i0 shows that 0 ∈ BΛ−z κ is equivalent to 1 N N ∑ i=1 1 νi(Λ − z)2 + κ2/3 ≥ 1, where νi(Λ − z) are the singular values of Λ − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Consider X + Λ, with Λ being a deterministic matrix as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and with X being an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' matrix satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 and the single-entry distribution χ have a density with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then Oii ≻ N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) where the index i ∈ [N] is such that µi is in the bulk of X + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 6The deterministic terms following the Kronecker symbol δj,i in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) will be shown to be bounded in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 9 In the introduction we already mentioned the consequence of this result on the sensitivity of an eigenvalue of X+Λ under smallperturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now we explain its other consequence on the diffusivity of the Dyson-type eigenvalue dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let each entry of X = X(t) evolve as an independent complex OU process, dXij = dBij √ N − 1 2Xijdt, where Bij are independent standard complex Brownian motions and the initial condition X(0) sat- isfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A direct calculation [13, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1] shows that the eigenvalues µi = µi(t) follow the Dyson-type stochastic dynamics dµi = dMi − 1 2µidt, {µi(0)} = SpecX(0), 1 ≤ i ≤ N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) where the martingales Mi have brackets ⟨Mi,Mj⟩ = 0 and d⟨Mi,Mj⟩t = 1 N Oij(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, we immediately obtain, for any ǫ > 0 that E[∣µi(t) − µi(0)∣21(µi(0) ∈ Bκ)] ≥ tN −ǫ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) up to some time t ≤ T (κ), where Bκ denotes the κ-bulk of X(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For Ginibre initial condition X(0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) was established in [13, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6], we now generalise it to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We remark that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) is similar to its Hermitian counterpart, the standard Dyson Brownian motion (DBM) on the real line, with some notable differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, in the latter process the eigenvalues cannot cross each other, hence they are quite rigid and confined to an interval of size essential 1/N, so they are not diffusive beyond a time-scale 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Along the evolution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) the non-Hermitian eigenvalues still repel each other (encoded in the typically negative off-diagonal overlaps, see [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3] in the Gaussian case), but they still can pass by each other and not hindering the diffusive behavior (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The most prominently and extensively studied [39, 6, 52, 14, 15, 55, 54, 21, 22, 23] deformation is Λ = −z with z ∈ C, since it plays a key role in Girko’s formula [39] expressing linear statistics of non- Hermitian eigenvalues of X in terms of the Hermitisation of X − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this case, the self-consistent equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) reduces to the well-known cubic relation − 1 m = w + m − ∣z∣2 w + m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As a consequence, the deterministic terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) drastically simplify (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', the fractions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8b) are simply replaced by ⟨B⟩) and one also has explicit formulas for the bulk (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) in terms of solution of a cubic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, for ∣z∣ < 1 − ǫκ, the κ-bulk Bκ consists of a single interval, while for ∣z∣ ≥ 1 − ǫκ it consists of two intervals, where ǫκ ∼ κ2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the former case 0 ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Consequently, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 gives the lower bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) for all the diagonal overlaps Oii of eigenvectors of X whose eigenvalue µi lies in a disk of radius 1 − ǫ with some ǫ > 0 independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ∼ κ2/3 ∼ κ ∼ κ1/3 2 2 Rew ρ(Rew) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Depicted is the density ρ for the deformation Λ = −z with ∣z∣ slightly less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' On the horizontal axis, we indicated the two components of the bulk Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The distance between Bκ and a regular edge scales like κ2/3, while near an (approximate) cusp the distance between the two components scales linearly (see also (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 10 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES In the next section we explain the key technical result behind our main theorems, the eigenstate thermalisation for the Hermitisation of X + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Eigenstate Thermalisation Hypothesis for the Hermitisation of X + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The key to access the non-Hermitian singular vectors of X + Λ is to study its Hermitisation [39], which is defined as H = HΛ ∶= ( 0 X + Λ (X + Λ)∗ 0 ) =∶ W + ˆΛ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) where ˆΛ∗ = ˆΛ was defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and can also be viewed as the matrix of expectation ˆΛ = EHΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We denote by {wi}∣i∣≤N the (normalised) eigenvectors of H and by {λi}∣i∣≤N the corresponding eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7 By means of the singular value decomposition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), the eigenvalues and eigenvectors of H are related to the singular values and singular vectors of X + Λ as follows: wi = (ui,vi)t and λi = σi for i ∈ [N], up to a normalisation, since now ∥ui∥2 = ∥vi∥2 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, the block structure of H induces a symmetric spectrum around zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' λ−i = −λi for any i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This symmetry for the eigen- values is also reflected in the eigenvectors, which satisfy w−i = E−wi for any i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By spectral decomposition, this immediately shows the chiral symmetry E−G(w) = −G(−w)E−, with E− = (1 0 0 −1) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) for the resolvent G(w) = GΛ(w) ∶= (HΛ − w)−1, with spectral parameter w ∈ C ∖ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We also have ⟨G(w)E−⟩ = 0 for any w since ⟨wi,E−wi⟩ = ∥ui∥2 − ∥vi∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A basic feature of a very large class of random matrices is that their resolvent becomes approximately deterministic in the large N limit, often even for any spectral parameter with ∣Imw∣ ≥ N −1+ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' these statements are called local laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In our case the deterministic approximation of the resolvent G(w) is given by M(w) = M Λ(w) ∶= ⎛ ⎝ M11(w) ΛM22(w) w+m(w) Λ∗M11(w) w+m(w) M22(w) ⎞ ⎠ ∈ C2N×2N , w ∈ C ∖ R, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) with each block being understood as a matrix in CN×N, where the diagonal entries are defined via M11(w) ∶= w + m(w) ΛΛ∗ − (w + m(w))2 , M22(w) ∶= w + m(w) Λ∗Λ − (w + m(w))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) Here we require m(w) = ⟨M(w)⟩, which is an implicit equation for the function m(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Simple calculation shows that this implicit equation is exactly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To derive these formulas systematically, we recall that the deterministic approximation to G(w) is obtained as the unique solution to the matrix Dyson equation (MDE) (extensively studied in [2, 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The MDE corresponding to the random matrix H is given by − 1 M(w) = w − ˆΛ + S[M(w)] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) under the constraint ImM(w) ⋅ Imw > 0, where Im M(w) ∶= 1 2i[M(w) − (M(w))∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Here S[⋅], the self-energy operator, is defined via S[T ] ∶= ̃E( ̃ H − EH)T ( ̃ H − EH) for any T ∈ C2N×2N, where ̃ H denotes an independent copy of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In our case we have S[T ] = 2E1⟨T E2⟩ + 2⟨E1T ⟩E2 = ∑ σ=± σ⟨T Eσ⟩Eσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) Using ⟨M11(w)⟩ = ⟨M22(w)⟩ that directly follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18), it is straightforward to check that M(w) as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) satisfies the MDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since the MDE has a unique solution, we see that the density ρ defined via free convolution in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 coincides with the self-consistent density of states 7In the definition of the eigenvectors and eigenvalues, we omitted 0 in the set of indices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ∣i∣ ≤ N really means i ∈ {−N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', −1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 11 (scDos) corresponding to the MDE, defined as the boundary value of 1 π ⟨ImM⟩ on the real axis in the theory of MDE [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the reader’s convenience in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 we will collect a few facts about M, in particular we will show that it has a continuous extension as a matrix valued function to the real axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the limit M(e) ∶= limη↓0 M(e +iη) exists for any e ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This extends the similar result on its trace mentioned in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we will also show that for spectral parameters w ∈ C ∖ R with Rew ∈ Bκ, we have ∥M(w)∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, we will also prove an important regularity property of the κ-bulk, namely that dist(∂Bκ′,Bκ) ≥ c(κ − κ′) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) for any small 0 < κ′ < κ and some N-independent constant c = c(∥Λ∥) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, for our proof it is sufficient if c = c(κ,∥Λ∥), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' an additional κ dependence is allowed – in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 we will explain that this weaker result is considerably easier to obtain (see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We will also show that Bκ is a finite disjoint union of compact intervals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the number of these components depends only on κ and ∥Λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The above mentioned concentration of G around M is the content of the following single resolvent local law, both in averaged and isotropic form, which we prove in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Single resolvent local law for the Hermitisation H) Fix a bounded deterministic Λ ∈ CN×N and κ > 0 independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for any w ∈ C ∖ R with ∣w∣ ≤ N 100 and Rew ∈ Bκ, we have ∣⟨(G(w) − M(w))B⟩∣ ≺ 1 Nη , ∣⟨x,(G(w) − M(w))y⟩∣ ≺ 1 √Nη , where η ∶= ∣Imw∣, for any bounded deterministic matrix ∥B∥ ≲ 1 and vectors ∥x∥,∥y∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Our main result for the Hermitised random matrix H is the Eigenstate Thermalisation Hypothesis (ETH), that in mathematical terms is the proof of an optimal convergence rate of the strong Quantum Unique Ergodicity (QUE) for general observables A, uniformly in the bulk (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) of the spectrum of H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' in the bulk of the scDos ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Eigenstate Thermalisation Hypothesis for the Hermitisation H) Fix some bounded Λ ∈ CN×N and κ > 0 independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let {wi}∣i∣≤N be the orthogonal eigenvectors of the Hermitisation H of X + Λ, where X is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' matrix satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for any deterministic matrix A ∈ C2N×2N with ∥A∥ ≲ 1 it holds that max i,j ∣⟨wi,Awj⟩ − δj,i ⟨ImM(γj)A⟩ ⟨ImM(γj)⟩ − δj,−i ⟨ImM(γj)E−A⟩ ⟨ImM(γj)⟩ ∣ ≺ 1 √ N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) where the maximum is taken over all ∣i∣,∣j∣ ≤ N, such that the quantiles γi,γj ∈ Bκ defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) are in the bulk of the scDos ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The main technical result underlying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7 is an averaged local law for two resolvents with different spectral parameters, which we will formulate in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Given the optimal bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22), following a Dyson Brownian Motion (DBM) analysis similar to [27, 29], it is possible to prove a CLT for single diagonal overlaps ⟨wi,Awi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, for the sake of brevity, we do not present this argument here and defer the CLT analysis to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the following Section 3 we precisely define the regularisation and we will prove our main results formulated above assuming the key technical Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This proposition is obtained from a local law, which we prove in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Local laws are proved by a hierarchy of master and reduction inequali- ties, that are derived in Sections 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Several technical and auxiliary results are deferred to the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of the main results The key to understanding the eigenvector overlaps and showing our main results is an improved bound on the averaged trace of two resolvents with regular (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 below) deterministic matrices A1,A2 in between, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for ⟨G(w1)A1G(w2)A2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) 12 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Naively, for arbitrary A1,A2, estimating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) via a trivial Schwarz inequality and Ward identity yields the upper bound ∣⟨G(w1)A1G(w2)A2⟩∣ ≺ 1/η, where η ∶= minj ∣Imwj∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, this bound dras- tically improves, whenever the matrices A1,A2 are regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' orthogonal to certain critical eigenvec- tors V± of the associated two-body stability operators (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2), which is denoted as Aj = ˚ Aj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this case, in our key Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 we will show that ∣⟨G(w1)˚ A1G(w2)˚ A2⟩∣ ≺ 1 even for very small η ∼ N −1+ǫ as a consequence of a more precise local law for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), which we present in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We find that (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) both the size of its deterministic approximation and the fluctuation around this mean heavily depend on whether (one or both of) the matrices A1,A2 are regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' satisfy ⟨V±,Aj⟩ = 0, or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Therefore, the general rather structural regularizing decomposition (or regularisation) of a matrix A shall be conducted as A○ ≡ ˚ A ∶= A − ⟨V+,A⟩U+ − ⟨V−,A⟩U− (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) for Uσ,Vσ ∈ C2N×2N satisfying ⟨Vσ,Uτ⟩ = δσ,τ and the normalisation ⟨Uσ,Uσ⟩ = 1, where recall that ⟨R,T ⟩ ∶= ⟨R∗T ⟩ denotes the (normalised) Hilbert-Schmidt scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The regularisation map (1 − Π) ∶ C2N×2N → C2N×2N , A ↦ ˚ A , where Π is a two-dimensional (non-orthogonal) projection,8 is closely related to the built-in chiral sym- metry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Indeed, for other ensembles without this special structure only one of the terms ⟨Vσ,A⟩Uσ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) would be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As mentioned above, the matrices V± are determined as critical eigenvectors (with corresponding small eigenvalue) of naturally associated two-body stability operators with their precise form worked out in Appendix D and given in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, for the directions U± there are a priori no further constraints (apart from orthogonality and normalisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, as it turns out to be convenient for our proofs, we will choose the matrices Uσ (in principle, even allowing for two different deformations Λ1 ≠ Λ2) in such a way, that a resolvent identity GΛ1(w1)UσGΛ2(w2) ≈ (GΛ1(w1) − GΛ2(σw2))Uσ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) can be applied (here, the symbol ‘≈’ neglects lower order terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This is used to reduce the number of resolvents in a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that, again due to the eminent chiral symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) for the resolvents, there are in fact two matrices Uσ for which a resolvent identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Although the regularisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) shall be motivated for arbitrary deformations Λ1,Λ2 in Appen- dix D, we will henceforth choose a single bounded deformation Λ ∈ CN×N, which remains fixed with the just mentioned exception in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For a single deformation Λ, this restricts the matrices U± satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) to be given by E±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In case that the spectral parameters (w1,w2) appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) (with a single fixed deformation Λ) are such that none of the eigenvectors of the stability operator is critical (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Appendix A), we consider every matrix A as regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The distinction between these two scenarios is regulated by cutoff functions 1± δ introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Regular observables: A bound on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As already mentioned above, our main result for the Hermitised random matrix, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7, shall be derived from a bound on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), where we assume the (real parts of the) spectral parameters w1,w2 to be in the bulk of the scDos ρ (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now specify the concept of regularisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The eigenvectors V± will be com- puted in Appendix D, the matrices U± are simply chose as E±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Regular observables) Given κ > 0, let9 δ = δ(κ,∥Λ∥) > 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) 8The condition ⟨Vσ, Uτ ⟩ = δσ,τ guarantees that the regularisation is idempotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ( ˚ A)○ = ˚ A and Π2 = Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 9Note that the parameter δ > 0 is independent of the matrix size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 13 be sufficiently small (to be chosen below, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20)) and let w,w′ ∈ C ∖ R with Rew,Re w′ ∈ Bκ be spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Furthermore, we introduce the (symmetric) cutoff functions 1± δ(w,w′) ∶= φδ(Rew ∓ Rew′) φδ(Imw) φδ(Imw′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) where 0 ≤ φδ ≤ 1 is a smooth symmetric bump function on R satisfying φδ(x) = 1 for ∣x∣ ≤ δ/2 and φδ(x) = 0 for ∣x∣ ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (a) We define the (w,w′)-regular component or (w,w′)-regularisation ˚ Aw,w′ of a matrix A as10 ˚ Aw,w′ ∶= A − ∑ τ=± 1τs δ (w,w′) ⟨M(Rew + iIm w)AM(Rew′ + τiImw′)Eτs⟩ ⟨M(Re w + iIm w)EτsM(Rew′ + τiImw′)Eτs⟩Eτs , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) where the relative sign of the imaginary parts is defined as s ≡ sw,w′ ∶= −sgn(Imw Im w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) (b) We say that A is (w,w′)-regular if and only if A = ˚ Aw,w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The regularisation shall be revisited in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, where we tailor it to certain averaged (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) or isotropic (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) resolvent chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We have several comments concerning the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (i) In case that at least one of the spectral parameters is away from the imaginary axis, say ∣Re w∣ > δ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', then the regularisation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) contains at most one summand: If 1+ δ(w,w′) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Rew is close to Rew′, then we have that ˚ Aw,w′ ∶= A − ⟨M(w)AM(Rew′ + siImw′)⟩ ⟨M(w)M(Re w′ + siImw′)⟩ E+ , whereas if 1− δ(w,w′) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' if Rew is close to −Rew′, then we have that ˚ Aw,w′ ∶= A − ⟨M(w)AE−M(−Rew′ + siIm w′)⟩ ⟨M(w)M(−Re w′ + siIm w′)⟩ E− , where we used that M(w)E− = −E−M(−w) as shown in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, ultimately as a consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (ii) The cutoff functions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) satisfy the basic symmetry properties 1± δ(w,w′) = 1± δ( ¯w,w′) = 1± δ(w, ¯w′) = 1± δ( ¯w, ¯w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, ˚ A is not symmetric in its two spectral parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ˚ Aw,w′ ≠ ˚ Aw′,w in general (iii) For spectral parameters satisfying 1± δ(w,w′) > 0, it will be shown in Appendix A that the respective denominators in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) are bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, the linear map A ↦ ˚ A is bounded with a bound depending only on δ and ∥Λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (iv) Whenever it holds that 1± δ(w,w′) = 0 then also 1± δ′(w,w′) = 0 for every 0 < δ′ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Conversely, whenever it holds that 1± δ(w,w′) = 1 then also 1± δ′(w,w′) = 1 for every 0 < δ < δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (v) We point out that the notion of regularity implicitly depends on κ and δ and hence also on the (norm of the) deformation Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, the regularisation defined above satisfies the following elementary properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The iden- tities in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) are immediate from the definition, the perturbative statements are proven in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Fix a bounded deterministic deformation Λ ∈ CN×N and let A ∈ C2N×2N be an arbitrary bounded deterministic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 10Putting the summation parameter τ at the second spectral parameter w′ (and not at w) is simply a free choice, which we made here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' More precisely, defining the regularisation as ˜˚ Aw,w′ ∶= A − ∑ τ=± 1τs δ (w, w′) ⟨M(Re w + τiIm w)AM(Rew′ + iIm w′)Eτs⟩ ⟨M(Re w + τiImw)EτsM(Re w′ + iIm w′)Eτs⟩ Eτs would equally work in our proofs (see Appendices A and D for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 14 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES (i) Let w,w′ ∈ C ∖ R with Re w,Rew′ ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, we have the identities (˚ Aw,w′) ∗ = ˚ (A∗) ¯ w′, ¯ w , ˚ Aw,w′E− = ˚ (AE−) w,−w′ , E− ˚ Aw,w′ = ˚ (E−A) −w,w′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) (ii) Moreover, by definition it holds that ˚ Aw, ¯ w′ = ˚ Aw,w′ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) and setting s ∶= −sgn(ImwImw′), we have the perturbative estimate11 ˚ A ¯ w,w′ = ˚ Aw,w′ + O(∣w − s ¯w′∣ ∧ 1)Es + O(∣w + sw′∣ ∧ 1)E−s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) (iii) Let w1,w′ 1,w2,w′ 2 ∈ C∖R with Rew1,Re w′ 1,Re w2,Rew′ 2 ∈ Bκ as well as Imw1⋅Im w2 > 0 and Imw′ 1 ⋅ Imw′ 2 > 0 be spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then we have that ˚ Aw2,w′ 1 = ˚ Aw1,w′ 1 + O(∣w1 − w2∣ ∧ 1)E+ + O(∣w1 − w2∣ ∧ 1)E− , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) ˚ Aw1,w′ 2 = ˚ Aw1,w′ 1 + O(∣w′ 1 − w′ 2∣ ∧ 1)E+ + O(∣w′ 1 − w′ 2∣ ∧ 1)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) We can now state the bound on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) for regular observables, which shall be proven in Section 4 as an immediate corollary to a local for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and the bound from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Fix a bounded deterministic Λ ∈ CN×N, ǫ > 0, κ > 0, and let w1,w2 ∈ C with ∣w1∣,∣w2∣ ≤ N 100, Rew1,Rew2 ∈ Bκ, and ∣Imw1∣,∣Imw2∣ ≥ N −1+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, let A1 ∈ C2N×2N be a (w1,w2)-regular and A2 ∈ C2N×2N a (w2,w1)-regular deterministic matrix, both satisfying ∥A1∥,∥A2∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then we have ∣⟨G(w1)˚ Aw1,w2 1 G(w2)˚ Aw2,w1 2 ⟩∣ ≺ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Estimating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) for general observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Armed with the correct regularisation, we can now present a systematic analysis of ⟨G(w1)A1G(w2)A2⟩ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) for arbitrary bounded deterministic matrices A1,A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Decomposing A1,A2 according to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 as A1 = ˚ Aw1,w2 1 + ⟨⟨A1⟩⟩+ w1,w2E+ + ⟨⟨A1⟩⟩− w1,w2E− , A2 = ˚ Aw2,w1 2 + ⟨⟨A2⟩⟩+ w2,w1E+ + ⟨⟨A2⟩⟩− w2,w1E− , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) (where ⟨⟨⋅⟩⟩σ w,w′ can be read off as the coefficients in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6)) and plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), we find that ⟨G(w1)A1G(w2)A2⟩ = ∑ σ,τ ⟨⟨A1⟩⟩σ w1,w2⟨⟨A2⟩⟩τ w2,w1⟨G(w1)EσG(w2)Eτ⟩ + ∑ σ ⟨⟨A1⟩⟩σ w1,w2⟨G(w1)EσG(w2)˚ Aw2,w1 2 ⟩ + ∑ τ ⟨⟨A2⟩⟩τ w2,w1⟨G(w1)˚ Aw1,w2 1 G(w2)Eτ⟩ + ⟨G(w1)˚ Aw1,w2 1 G(w2)˚ Aw2,w1 2 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) Which terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) are effectively present depends on the coefficients ⟨⟨⋅⟩⟩σ w,w′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' on the singular components of A1,A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For terms with nonzero coefficients the following rule of thumb applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' De- noting η ∶= min (∣Imw1∣,∣Im w2∣) ≥ N −1+ǫ, the terms ⟨GEGE⟩ in the first line of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) are bounded by 1/η, the terms ⟨GEG˚ A⟩ in the two middle lines of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) are bounded by 1/√η, and ⟨G˚ AG˚ A⟩ in the last line is of order one (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This is in perfect agreement with the √η-rule mentioned in the Introduction (see also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Some of these bounds are actually sharp for special values of w1,w2, for example ⟨G(w)E+G( ¯w)E+⟩ = ⟨ImG(w)⟩ η ∼ 1 η , or ⟨G(w)E−G(− ¯w)E−⟩ = −⟨ImG(w)⟩ η , 11Note that the asymmetry between (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) stems from the asymmetry imposed in the definition of the regularisation, namely by placing the summation index τ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) at the second spectral parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 15 where we used the chiral symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, two terms with στ = −1 in the first line of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) are identically zero by applying the chiral symmetry, followed by the resolvent identity and ⟨GE−⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For a middle term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) we have ⟨G(w)E+G( ¯w)˚ A ¯ w,w⟩ = 1 η ⟨ImG(w)˚ A ¯ w,w⟩ ≲ 1 + 1 Nη 1 √η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the very last relation we treated ⟨G(w)˚ A ¯ w,w⟩ and ⟨G( ¯w)˚ A ¯ w,w⟩ separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In both cases we first used Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 to adjust the regularisation to ˚ Aw,w and ˚ A ¯ w, ¯ w, respectively, to match the new single- resolvent setup and then we applied the corresponding single-resolvent local law with regular observ- able (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that the most critical estimate concerns the last line of(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the regular part for both observ- able matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) is obtained from a local law with two resolvents and two regular matrices, while the first and the middle terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) can be understood already from an improved local law for one resolvent and one regular matrix (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 below) after applying resolvent identities and adjusting the regularisation by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Furthermore, observe that the sizes of the first three lines in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) are sensitive to w1,w2 via the resolvent identities, for example ⟨G(w1)E+G(w2)E+⟩ = ⟨G(w1) − G(w2)⟩ w1 − w2 ≲ 1 ∣w1 − w2∣, or ⟨G(w1)E−G(w2)E−⟩ ≲ 1 ∣w1 + w2∣, while the last line in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) is typically order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Summarizing, the singular parts of ⟨G(w1)A1G(w2)A2⟩ can be explicitly computed (using single- resolvent local laws) as explicit functions of w1,w2, while the regular part remains of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A combinationof our decomposition(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), the perturbation formulasfrom Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, and our single- and two-resolvent local laws together with their explicit deterministic terms from the subsequent Section 4 provide an effective recipe to compute ⟨G(w1)A1G(w2)A2⟩ with high precision in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We refrain from formulating it as a comprehensive theorem due to the large number of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We will first focus on the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7 and turn to the proofs of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As a first step towards the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7, we show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) indeed follows from a bound similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13), where G is replaced by ImG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The proof of the following simple lemma is given after completion of the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Fix a bounded deterministic Λ ∈ CN×N, ǫ > 0, κ > 0, and let B ∈ C2N×2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for any bulk indices ∣i∣,∣j∣ ≤ N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' with γi,γj ∈ Bκ, and η ≥ N −1+ǫ, we have N ∣⟨wi,Bwj⟩∣2 ≺ (Nη)2⟨ImG(γi + iη)BImG(γj + 2iη)B∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) The same bound still holds without the factor of two in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, we chose to have it, in order to ensure that the spectral parameters of the two resolvents are always forced to be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Having Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5 at hand, we are left with estimating the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) for B = A − ⟨ImM(γj)A⟩ ⟨ImM(γj)⟩ E+ − ⟨ImM(γj)E−A⟩ ⟨ImM(γj)⟩ E− (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) using Proposition3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that the two terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) carrying a δ-symbol arise from the orthogonality relations ⟨wi,E±wj⟩ = δj,±i, following from the spectral symmetry described around (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now write out ImG(w) = (G(w) − G( ¯w))/(2i), such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) leaves us with four different terms, each of which can be bounded individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since their treatment is completely analogous, we focus on the exemplary term ⟨G(γi + iη)BG(γj − 2iη)B∗⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) with the deterministic matrix B being defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We rely on the following simple perturbative lemma, which follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 by invoking Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 16 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using the notation introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), the matrix B ∈ C2N×2N from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) satisfies B = ˚ Aγi+iη,γj −2iη + O(∣γi − γj∣ + η)E+ + O(∣γi + γj∣ + η)E− , B∗ = ˚ (A∗) γj −2iη,γi+iη + O(∣γi − γj∣ + η)E+ + O(∣γi + γj∣ + η)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) Hence, plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18), we get a sum of several terms, which can all be estimated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the ‘leading term’, we use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 to get that ∣⟨G(γi + iη)˚ Aγi+iη,γj−2iηG(γj − 2iη) ˚ (A∗) γj−2iη,γi+iη⟩∣ ≺ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Two further representative terms are given by O(∣γi ∓ γj∣ + η) ⟨G(γi + iη)E±G(γj − 2iη)C⟩, where C ∈ C2N×2N is some generic bounded matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), these terms can be rewritten as O(∣γi ∓ γj∣ + η) ⟨G(γi + iη)G(±(γj − 2iη))E±C⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Foreithersignchoice (due to the factortwo), we cannow employa simple resolventidentityG(w1)G(w2) = [G(w1) − G(w2)]/(w1 − w2), leaving us with O(∣γi − γj∣ + η) (γi ∓ γj) + (1 ± 2)iη ⟨[G(γi + iη) − G(±(γj − 2iη))]C⟩, which is surely stochastically dominatedby one by means of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thus, collecting all the terms, we find that ∣(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18)∣ ≺ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, we choose η = N −1+ξ for an arbitrarily small ξ > 0, such that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5 with B as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) yields Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ We conclude with giving a proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By spectral decomposition we write ⟨ImG(γi + iη)BImG(γj + 2iη)B∗⟩ = 1 2N ∑ k,l 2η2∣⟨wk,Bwl⟩∣2 [(λk − γi)2 + η2][(λl − γj)2 + 4η2] ≳ η2∣⟨wi,Bwj⟩∣2 N[(λi − γi)2 + η2][(λj − γj)2 + 4η2] ≻ ∣⟨wi,Bwj⟩∣2 Nη2 , which proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We point out that in the last inequality we used rigidity of the eigenvalues [2, 36]: ∣λi − γi∣ ≺ 1 N , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) which holds for bulk indices as a standard consequence of the single-resolvent local law, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The bounds in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8a), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8b), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8c) follow from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7 by choosing A = (B 0 0 0) , A = (0 0 0 B) , and A = (0 0 B 0) , respectively, and invoking Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By the definition Hz ∶= ( 0 X + Λ − z (X + Λ − z)∗ 0 ) it follows that µ ∈ Spec(X + Λ) if and only if λµ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Here by λz i we denoted the eigenvalues of Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We remark that Λ is omitted by the notation since it is fixed throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, using the bound for products of two resolvents and two regular matrices in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13), we will now prove the lower bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) for the overlap of left and right eigenvectors corresponding to eigenvalues µ which lies in the bulk of the spectrum of X + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Define F ∶= (0 1 0 0), then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) we conclude sup z∈bulk ⟨ImGz(iη)FImGz(iη)F ∗⟩ ≺ 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) where the supremum is taken over the bulk as given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Here we used that F is regular in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' this immediately follows from the fact that F is off–diagonal and ImM(iη) is diagonal (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now want to show that if we choose z = µi to be a bulk eigenvalue of X + Λ the upper bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) implies a lower bound on Oii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To make the notation simpler, from now on we denote µ = µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Consider the non-Hermitian left/right–eigenvectors l,r, with corresponding eigenvalue µ, defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Without loss of generality we can assume that µ is a simple eigenvalue, since the spectrum of X + Λ is simple with probability one owing to the continuous distribution of the entries of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, we define P ∶= ⎛ ⎝ l l∗ ∥l∥2 0 0 rr∗ ∥r∥2 ⎞ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Clearly P is a rank two orthogonal projection whose range is the kernel of Hµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Recall that Ker(Hµ) has dimension two since µ is simple and l,r are u1,v1, respectively (up to scalar multiples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, almost surely, by spectral decomposition (and by the spectral symmetry of Hµ) ImGµ(iη) = P η + ∑ ∣i∣≥2 η (λµ i )2 + η2 (uµ i vµ i )(uµ i vµ i ) ∗ ≥ P η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) we thus obtain 1 ≻ sup z∈bulk ⟨ImGz(iη)FImGz(iη)F ∗⟩ ≻ 1 η2 ⟨PFPF ∗⟩ = ∣⟨l,r⟩∣ 2 Nη2∥r∥2∥l∥2 , which, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), implies Oii =∥r∥2∥l∥2 ≻ 1 Nη2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Choosing η = N −1+ǫ/2, this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Local laws with regular observables The goal of the present section is to establish the key Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 by proving an averaged local law for a product of two resolvents (of the Hermitisation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15)) in the bulk of the scDos ρ with regular (recall Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 and see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 below) deterministic matrices A1,A2 in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Throughout the rest of this paper, we consider the case of several spectral parameters w1,w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' and fixed bounded deformations Λ1 = Λ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ≡ Λ ∈ CN×N, which we continue to omit from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using the abbreviations Gi ∶= G(wi) ∶= GΛ(wi) (and analogously for Mi), the deterministic approximation to the resolvent chain G1B1G2 ⋯GkBkGk+1 18 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES for arbitrary deterministic B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bk12 is denoted by M(w1,B1,w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk,Bk,wk+1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) and defined recursively in the length k of the chain in Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We may call these formulas recursive Dyson equations as they provide us with the correct deterministic quantity for longer resolvent chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As an example, we have that M(w1,B1,w2) = B−1 12 [M1B1M2] = M1X12[B1]M2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) where B−1 12 is the inverse stability operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and X12 = (1 − S[M1 ⋅ M2]) −1 (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='33) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As already mentioned above, we are aiming at local laws for expressions of the form ⟨G1A1 ⋯GkAk⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) in the averaged case, or (G1A1 ⋯AkGk+1)xy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) in the isotropic case, where the deterministic matrices A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Ak are assumed to be regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The general concept of regularity depending on two spectral parameters w and w’ has already been introduced in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the following definition we tailor this concept to observables in chains (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' It basically says that observable Aj, located between Gj = G(wj) and Gj+1 = G(wj+1) in these chains will naturally be regularised using the spectral parameters wj and wj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Regular observables in chains) Fix κ > 0 and let δ = δ(κ,∥Λ∥) > 0 be small enough (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Consider one of the two expressions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) for some fixed length k ∈ N and bounded matrices ∥Ai∥ ≲ 1 and let w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk+1 ∈ C ∖ R be spectral parameters with Rewi ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For any j ∈ [k], analogously to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5), we denote 1± δ(wj,wj+1) ∶= φδ(Re wj ∓ Rewj+1) φδ(Imwj) φδ(Imwj+1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) and sj ∶= −sgn(ImwjImwj+1), where, here and in the following, in case of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3), the indices are understood cyclically modulo k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (a) For i ∈ [k] we define the regular component or regularisation of Ai from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the pair of spectral parameters (wi,wi+1)) as ˚ Ai ∶= ˚ Awi,wi+1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) (b) Moreover, we call Ai regular (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (wi,wi+1)) if and only if ˚ Ai = Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For example, in case of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) for k = 1 with spectral parameter w1 ∈ C ∖ R satisfying Rew1 ∈ Bκ, ∣Rew1∣ ≤ δ/4 and ∣Imw1∣ ≤ δ/2 (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5)), the regular component of A1 is given by ˚ A1 ∶= A1 − ⟨ImM1A1⟩ ⟨ImM1⟩ E+ − ⟨M1A1M1E−⟩ ⟨M1E−M1E−⟩E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) We emphasise, that our notation˚⋅ for the regular component of Ai does not have an overall fixed meaning but depends on the spectral parameters of the resolvents ‘surrounding’ the deterministic ma- trix Ai under consideration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ⟨ ⋯ GiAiGi+1 ⋯ ⟩ or ( ⋯ GiAiGi+1 ⋯ )xy , or in case of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) for k = 1 the single spectral parameter involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, if we aim at specifying the spectral parameters defining the operation˚⋅ , we add them (or their indices) as a subscript, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' write ˚ Awi,wi+1 i ≡ ˚ Ai,i+1 i ≡ ˚ Ai ≡ A○ i ≡ A i,i+1 i ≡ A wi,wi+1 i , as done in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, and do not use imprecise notation ˚ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The just explained caveats are in stark contrast to the case of Wigner matrices [24, 28, 29], where the regular component of a matrix A is simply its traceless part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ˚ A = A − ⟨A⟩, irrespective of the spectral parameters involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Apart from this independence of the location in the spectrum, there is a one further important difference to our case, which we already mentioned in Section 3: For Wigner 12We will use the the notational convention, that the letter B denotes arbitrary (generic) matrices, while A is reserved for regular matrices, in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 19 matrices, the condition for A being regular is one-dimensional and hence restricts A to a (N 2 − 1)- dimensional subspace of CN×N (the traceless matrices), whereas in our case, the regularity condition is two-dimensional (if 1σ δ (⋅,⋅) = 1) and hence restricts a regular matrix A to a ((2N)2 −2)-dimensional subspace of C2N×2N, which depends on the ‘surrounding’ spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now give bounds on the size of the deterministic term M(w1,B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk,Bk,wk+1) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), where all Bi are regular in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The proof of this lemma is presented in Appen- dix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Bounds on M, see [28, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4]) Fix κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let k ∈ [4] and w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk+1 ∈ C ∖ R be spectral parameters with Rewj ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for bounded regular deterministic matrices A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Ak (in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), we have the bounds ∥M(w1,A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Ak,wk+1)∥ ≲ ⎧⎪⎪⎨⎪⎪⎩ 1 η⌊k/2⌋ if η ≤ 1 1 ηk+1 if η > 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) ∣⟨M(w1,A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Ak−1,wk)Ak⟩∣ ≲ ⎧⎪⎪⎨⎪⎪⎩ 1 η⌊k/2⌋−1 ∨ 1 if η ≤ 1 1 ηk if η > 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for the deterministic approximation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) of a resolvent chain, where η ∶= minj ∣Imwj∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the presentation of our main results, we would only need (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for k ∈ [2] from the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, the remaining bounds covered by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 will be instrumental in our proofs of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 below (see Sections 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The main result of the present section and most important input for our proofs in Section 3 is the following averaged local law in the bulk of the spectrum for two resolvents and regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Local laws with two regular matrices) Fix a bounded deterministic Λ ∈ CN×N, ǫ > 0 and κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for spectral parameters w1,w2,w3 ∈ C satisfying maxj ∣wj∣ ≤ N 100, Rewj ∈ Bκ and η ∶= minj ∣Im wj∣ ≥ N −1+ǫ, deterministic vectors x,y with ∥x∥,∥y∥ ≲ 1, and any regular deterministic matrices A1,A2 ∈ C2N×2N (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), we have the averaged local law ∣⟨G1A1G2A2 − M(w1,A1,w2)A2⟩∣ ≺ ⎧⎪⎪⎨⎪⎪⎩ 1 √Nη if η ≤ 1 1 Nη3 if η > 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10a) and the isotropic law ∣⟨x,(G1A1G2A2G3 − M(w1,A1,w2,A2,w3))y⟩∣ ≺ ⎧⎪⎪⎨⎪⎪⎩ 1 η if η ≤ 1 1 √ Nη4 if η > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10b) Together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for k = 2, this proves Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, as a byproduct of our proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, we obtain the following optimal local laws with a single regular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Optimal local laws with one regular matrix) Fix a bounded deterministic Λ ∈ CN×N, ǫ > 0 and κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for spectral parameters w1,w2 ∈ C satisfying maxj ∣wj∣ ≤ N 100, Rewj ∈ Bκ and η ∶= minj ∣Im wj∣ ≥ N −1+ǫ, deterministic vectors x,y with ∥x∥,∥y∥ ≲ 1, and any regular deterministic matrix A1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), we have the optimal averaged local law ∣⟨(G1 − M1)A1⟩∣ ≺ ⎧⎪⎪⎨⎪⎪⎩ 1 Nη1/2 if η ≤ 1 1 Nη2 if η > 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11a) and the optimal isotropic local law ∣⟨x,(G1A1G2 − M(w1,A1,w2))y⟩∣ ≺ ⎧⎪⎪⎨⎪⎪⎩ 1 √ Nη2 if η ≤ 1 1 √ Nη3 if η > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11b) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We have several comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 20 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES (i) The above local laws are in agreement with the √η-rule first established for Wigner matrices in [28]: Every regular deterministic matrix Ai reduces both the size of the deterministic approximation and the error term by a factor √η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (ii) The error terms in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 dealing with two regular matrices can still be improved by a factor 1/√Nη, as shown in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A similar analysis could have been conducted here, but we refrain from doing so, as it is not needed for our main results from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, the error bounds in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) with one regular matrix are in fact optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (iii) Given Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, and Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4, it is possible to deduce similar bounds for averaged and isotropic chains as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10), where not both matrices A1,A2 are regular (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the rest of this paper, we give a detailed proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 in the much more involved η ≤ 1 regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For η > 1, the bound simply follows by induction on the number of resolvents in chain by in- voking the trivial ∥M(w)∥ ≲ 1/∣Im w∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The detailed argument has been carried out in [28, Appendix B] for the case of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, at a certain technical point (within the proof of the master inequalities in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 and the reduction inequalities in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9), the proof uses Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 (and even its analogues for longer chains) for the η > 1 regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' But the master and reduction inequalities are not needed for proving the above estimates in the η > 1 regime, hence the argument is not circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' With partial exception in Appendix C, where we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, throughout the rest of this paper we assume that minj ∣Imwj∣ =∶ η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Basic control quantities and proof of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Our strategy for proving Theo- rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 (and thereby Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 as a byproduct) is to derive a system of master inequalities (Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for the errors in the local laws by cumulant expansion, then use an iterative scheme to gradually improve their estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The cumulant expansion introduces longer resolvent chains, potentially lead- ing to an uncontrollable hierarchy, so our master inequalities are complemented by a set of reduction inequalities (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) to estimate longer chain in terms of shorter ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We have used a similar strat- egy in [28, 29] for Wigner matrices, but now many new error terms due to regularisations need to be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Before entering the detailed proof, we explain the main mechanism of the new type of error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Cumulant expansions applied to chains .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' GiAiGi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' with regular Ai’s introduce more resolvent factors, for example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' GiGiAiGi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' GiE−GiAiGi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' without introducing more A’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Multiple G factors without intermediate A’s appear which we wish to reduce to fewer G factors using resolvent identities or contour integral representations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' in the example above we will use GiGi = G(wi)2 = 1 2πi ∫Γ G(z) (z − wi)2 dz, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) where Γ is an appropriate contour (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' When this formula is inserted into the chain, we have .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' G(z)AiGi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Ai is not regular any more with respect to the neighboring spectral pa- rameters (z,wi+1) since wi has been changed to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We need to regularise Ai to the new situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Fortunately, the regularisation is Lipschitz continuous by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, so roughly speaking we make an error of order ∣z − wi∣ when we regularise Ai from (wi,wi+1) to (z,wi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This error exactly compensates the higher power of z − wi in the denominator in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12), making eventually the adjust- ment of regularisations harmless in the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We need to meticulously implement this strategy for longer chains and also taking into account the chiral symmetry to reduce GiE−Gi in chains like .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' GiE−GiAiGi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='. The precise form of the error terms in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' It is remarkable that the signs appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) exactly match those that arise in the denominators of the contour integral formulas like (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now start the actual proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As the basic control quantities in the sequel of the proof, we introduce the normalised differences Ψav k (wk,Ak) ∶= Nηk/2∣⟨G1A1⋯GkAk − M(w1,A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk)Ak⟩∣, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) Ψiso k (wk+1,Ak,x,y) ∶= √ Nηk+1 ∣(G1A1⋯AkGk+1 − M(w1,A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Ak,wk+1))xy∣ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) for k ∈ N, where we used the short hand notations Gi ∶= G(wi), η ∶= min i ∣Imwi∣, wk ∶= (w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk), Ak ∶= (A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 21 The deterministic matrices ∥Ai∥ ≤ 1, i ∈ [k], are assumed to be regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', Ai = ˚ Ai, see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) and the deterministic counterparts M(w1,A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Ak−1,wk) used in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) (see also (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1)) are defined recursively in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For convenience, we extend the above definitions to k = 0 by Ψav 0 (w) ∶= Nη∣⟨G(w) − M(w)⟩∣, Ψiso 0 (w,x,y) ∶= √ Nη∣(G(w) − M(w))xy∣ and observe that Ψav 0 + Ψiso 0 ≺ 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) is the usual single-resolvent local law (in the bulk) from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, where here and in the following the arguments of Ψav/iso k shall occasionally be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We remark that the index k counts the number of regular matrices in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Throughout the entire argument, let ǫ > 0 and κ > 0 be arbitrary but fixed, and let D(ǫ,κ) ∶= {w ∈ C ∶ Rew ∈ Bκ , N 100 ≥ ∣Imw∣ ≥ N −1+ǫ} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) be the target spectral domain, where the κ-bulk Bκ has been introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This target spectral domain D(ǫ,κ) will be reached by shrinking a larger initial spectral domain D(ǫ0,κ0) ∶= {w ∈ C ∶ Rew ∈ Bκ0 , N 100 ≥ ∣Im w∣ ≥ N −1+ǫ0} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) many times, say (L − 1) times, during our whole argument, where L = L(ǫ) is an N-independent positive integer to be determined below (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17), we set ǫ0 ∶= ǫ/2 and chose the initial bulk parameter κ0 = κ0(ǫ,κ) = κ L(ǫ) > 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) The justmentionedshrinkingof domainswillbe conductedalongside the decreasingfamily(D(ǫ0,κ0) ℓ )ℓ∈[L] of spectral domains, defined via D(ǫ0,κ0) ℓ ∶= {w ∈ C ∶ Rew ∈ Bℓκ0 , N 100 ≥ ∣Imw∣ ≥ ℓN −1+ǫ0} ⊂ D(ǫ,κ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) D(ǫ,κ) N −1+ǫ ∼ N −1+ǫ0 D(ǫ0,κ0) D(ǫ0,κ0) 2 D(ǫ0,κ0) 3 ∼ κ2/3 ∼ κ 2 2 Rew Imw Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Depicted are the target spectral domain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), the initial spectral domain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) and four intermediate domains from the family (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The solid black curve represents the symmetric scDos ρ for the perturbation Λ = −z with ∣z∣ slightly less than one (see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Close to a regular edge of the scDos, the horizontal distance between two domains scales like κ2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Near an (approximate) cusp, the scaling agrees with the linear lower bound given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, the cut-off parameter δ > 0 used in the definition of the regular component of an observable (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) shall be chosen by the following two requirements: First, it has to be much smaller than the initial bulk-parameter κ0 from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 0 < δ ≪ cκ0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) 22 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES where c > 0 is the same constant as introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Second, it has to be small enough such that the denominators in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) (see also Appendix A) as well as in Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 are uniformly bounded away from zero – in case that 1σ δ (wi,wi+1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that these requirements also depend on the deformation Λ ∈ CN×N (but only via the norm ∥Λ∥ ≲ 1) as it determines the scDos ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Uniform bounds in domains) Let ǫ > 0 and κ > 0 as above and let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We say that the bounds ∣⟨G(w1)B1 ⋯ G(wk)Bk − M(w1,B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk)Bk⟩∣ ≺ E av , ∣(G(w1)B1 ⋯ BkG(wk+1) − M(w1,B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bk,wk+1))xy∣ ≺ E iso (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) hold (ǫ,κ)-uniformly for some deterministic control parameters E av/iso = E av/iso(N,η), depending only on N and η ∶= mini ∣Imwi∣, if the implicit constant in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) are uniform in bounded deterministic matrices ∥Bj∥ ≤ 1, deterministic vectors ∥x∥,∥y∥ ≤ 1, and admissible spectral parameters wj ∈ D(ǫ,κ) satisfying 1 ≥ η ∶= minj ∣Im wj∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Similarly, we use the phrase that a bound holds (ǫ0,κ0,ℓ)-uniformly (or simply ℓ-uniformly), if the above statement is true with D(ǫ0,κ0) ℓ in place of D(ǫ,κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we may allow for additional restrictions on the deterministic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For example, we may talk about uniformity under the additional assumption that some (or all) of the matrices are regular (in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) is stated for a fixed choice of spectral parameters wj in the left hand side, but it is in fact equivalent to an apparently stronger statement, when the same bound holds with a supremum over the spectral parameters (with the same constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' More precisely, if E iso ≥ N −C for some constant C > 0, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) implies sup w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk+1 ∣(G(w1)B1 ⋯ BkG(wk+1) − M(w1,B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bk,wk+1))xy∣ ≺ E iso (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) (and analogously for the averaged bound), where the supremum is taken over all choices of wj’s in the admissible spectral domain D(ǫ,κ) or D(ǫ0,κ0) ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This bound follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) by a standard grid argument (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', the discussion after [28, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Throughout the entire paper, we will frequently use the equivalence between (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' when integrating such bounds over some spectral parameters as done in Sections 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We can now formulate our main results of the present section, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4, in the language of our basic control quantities Ψav/iso k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Estimates on Ψav/iso 1 and Ψav/iso 2 ) For any ǫ > 0 and κ > 0 we have Ψav 1 + Ψiso 1 ≺ 1 and Ψav 2 + Ψiso 2 ≺ √ Nη (ǫ,κ)-uniformly in regular matrices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for spectral parameters wj ∈ D(ǫ,κ) with 1 ≥ η ∶= minj ∣Im wj∣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' These immediately follow from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ The rest of the proof is structured as follows: First, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we state the master inequalities and corresponding reduction inequalities on the Ψav/iso k parameters, which we then use in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 to prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Afterwards, in Section 5, we prove the master inequalities and, finally, the proof of the reduction inequalities is presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Master inequalities and reduction lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now state the relevant part of a non-linear infi- nite hierarchy of coupled master inequalities for Ψav k and Ψiso k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, for our purposes, it is sufficient to have only the inequalities for k ∈ [2], but with fairly more effort (despite closely following the argu- ments in Section 5) it is possible to obtain analogous estimates for general k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Master inequalities) Assume that Ψav/iso j ≺ ψav/iso j , j ∈ [4], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 23 ℓ-uniformly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for spectral parameters wj ∈ D(ǫ0,κ0) ℓ and 1 ≥ minj ∣Imwj∣) in regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then it holds that Ψav 1 ≺ 1 + ψav 1 Nη + ψiso 1 + (ψav 2 )1/2 (Nη)1/2 + (ψiso 2 )1/2 (Nη)1/4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24a) Ψiso 1 ≺ 1 + ψiso 1 + ψav 1 (Nη)1/2 + (ψiso 2 )1/2 (Nη)1/4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24b) Ψav 2 ≺ 1 + (ψav 1 )2 + (ψiso 1 )2 + ψav 2 Nη + ψiso 2 + (ψav 4 )1/2 (Nη)1/2 + (ψiso 3 )1/2 + (ψiso 4 )1/2 (Nη)1/4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24c) Ψiso 2 ≺ 1 + ψiso 1 + ψav 1 ψiso 1 + (ψiso 1 )2 Nη + ψiso 2 + (ψiso 1 ψiso 3 )1/2 (Nη)1/2 + (ψiso 3 )1/2 + (ψiso 4 )1/2 (Nη)1/4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24d) now (ℓ + 1)-uniformly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for spectral parameters wj ∈ D(ǫ0,κ0) ℓ+1 with 1 ≥ η ∶= minj ∣Imwj∣) in regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As shown in the above proposition, resolvent chains of length k are estimated by resolvent chains up to length 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In order to avoid the indicated infinite hierarchy of master inequalities with higher and higher k indices, we will need the following reduction lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Reduction inequalities) Assume that Ψav/iso n ≺ ψav/iso n holds for 1 ≤ n ≤ 4, ℓ-uniformly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for spectral parameters wj ∈ D(ǫ0,κ0) ℓ with 1 ≥ η ∶= minj ∣Imwj∣) in regular matrices (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Defini- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then we have Ψav 4 ≺ (Nη)2 + (ψav 2 )2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25) on the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Furthermore, we have Ψiso 3 ≺ Nη (1 + ψiso 2 √Nη ) (1 + ψav 2 Nη ) 1/2 , Ψiso 4 ≺ (Nη)3/2 (1 + ψiso 2 √Nη )(1 + ψav 2 Nη ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26) again uniformly in wj ∈ D(ǫ0,κ0) ℓ and in regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Within the proof, we repeatedly use a simple argument, which we call iter- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Iteration) For every D > 0, ν > 0, and α ∈ (0,1), there exists some K = K(D,ν,α), such that whenever (i) X ≺ N D on D(ǫ0,κ0) 1 and (ii) X ≺ x on D(ǫ0,κ0) ℓ for some ℓ ∈ N, implies that X ≺ A + x B + x1−αCα on D(ǫ0,κ0) ℓ+1 for some constants B ≥ N ν and A,C > 0, it also holds that X ≺ A + C on D(ǫ0,κ0) ℓ+K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We can now turn to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Assume that Ψav/iso j ≺ ψav/iso j , j ∈ [4], ℓ-uniformly, for some fixed ℓ > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' it holds on the domain D(ǫ0,κ0) ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24a)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24d), we immediately obtain Ψav 1 + Ψiso 1 ≺ 1 + ψav 1 + ψiso 1 (Nη)1/2 + (ψav 2 )1/2 + (ψiso 2 )1/2 (Nη)1/4 Ψav 2 + Ψiso 2 ≺ 1 + ψiso 1 + (ψav 1 )2 + (ψiso 1 )2 Nη + ψav 2 + ψiso 2 (Nη)1/2 + (ψav 4 )1/2 (Nη)1/2 + (ψiso 1 ψiso 3 )1/2 (Nη)1/2 + (ψiso 3 )1/2 + (ψiso 4 )1/2 (Nη)1/4 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='27) 24 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES on the domain D(ǫ0,κ0) ℓ+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, plugging the first line of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='27) into the second line and using iteration in both lines, we get Ψav 1 + Ψiso 1 ≺ 1 + (ψav 2 )1/2 + (ψiso 2 )1/2 (Nη)1/4 , Ψav 2 + Ψiso 2 ≺ 1 + (ψav 4 )1/2 √Nη + (ψav 2 )1/4 + (ψiso 2 )1/4 (Nη)1/8 ⋅ (ψiso 3 )1/2 (Nη)1/2 + (ψiso 3 )1/2 + (ψiso 4 )1/2 (Nη)1/4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='28) on the domain D(ǫ0,κ0) ℓ+K , for some K as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now use the reduction inequalities from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9 in the second line of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='28): Ψav 1 + Ψiso 1 ≺ 1 + (ψav 2 )1/2 + (ψiso 2 )1/2 (Nη)1/4 Ψav 2 + Ψiso 2 ≺ (Nη)1/2 + ψav 2 √Nη + (Nη)1/4(ψiso 2 )1/2 + (ψav 2 )1/2 + (ψav 2 ψiso 2 )1/2 (Nη)1/4 , + ((Nη)1/4 + (ψav 2 )1/4 + (ψiso 2 )1/4 (Nη)1/8 ) (1 + (ψiso 2 )1/2 (Nη)1/4 + (ψav 2 )1/4 (Nη)1/8 + (ψiso 2 )1/2(ψav 2 )1/4 (Nη)3/8 ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='29) on the domain D(ǫ0,κ0) ℓ+K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, using iteration once again in the second line of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='29), we obtain Ψav 1 + Ψiso 1 ≺ 1 + (ψav 2 )1/2 + (ψiso 2 )1/2 (Nη)1/4 , Ψav 2 + Ψiso 2 ≺ (Nη)1/2 on the domain D(ǫ0,κ0) ℓ+K+K′, for some K′ as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We point out that here we used Schwarz and Young inequalities for several terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, using iteration one last time we conclude Ψav 1 + Ψiso 1 ≺ 1, Ψav 2 + Ψiso 2 ≺ (Nη)1/2 on the domain D(ǫ0,κ0) ℓ+K+K′+K′′, for some K′′ as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We observe that in every application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10 during the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7, the parameter D is uniformly bounded by, say, D ≤ 10, as follows by estimating every resolvent in Ψav/iso k by norm and using the trivial 1/η-bound on inverse of the stability operator in the iterative definition of M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) given in Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A further quick inspection of the above proof shows, that α can be chosen as fixed α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, the parameter ν is lower bounded by (some universal positive constant times) ǫ, since Nη ≥ N ǫ/2 by construction of the initial domain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, the constants K, K′, and K′′ only depend on ǫ and therefore also the maximal number L = L(ǫ) of domain shrinkings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of the master inequalities, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 Before going into the proofs of the master inequalities, we state a simple lemma, which will fre- quently be used in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Recall that the deformation Λ ∈ CN×N is fixed and hence omitted from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Integral representations for products of resolvents) Let k ∈ N and w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk ∈ C ∖ R be spectral parameters, whose imaginary parts have equal sign, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' sgn(Imw1) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' = sgn(Imwk) =∶ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for any J ⊂ R being a union of compact intervals such that Rewi ∈ ˚ J (the interior) for all i ∈ [k] and 0 < ˜η < η ∶= minj ∣Imwj∣, we have the integral representation k ∏ j=1 G(wj) = 1 2πi ∫Γ G(z) k ∏ j=1 1 z − wj dz , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) where the contour Γ from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) is defined as (see Figure 3) Γ ≡ Γτ ˜η(J) ∶= ⎧⎪⎪⎨⎪⎪⎩ ∂(J × [i˜η,i∞)) if τ = + ∂(J × (−i∞,−i˜η]) if τ = − (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 25 and the boundary is parameterised in counter-clockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) remains valid if G(z) is replaced by ImG(z) in the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This easily follows from residue calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Rez Imz Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Depicted is the scenario from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 with five spectral parameters represented as dots in the upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we indicated the union of compact intervals J on the real axis and the contour Γ as described in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that one of the three intervals constituting J does not contain any Rewj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We recall the definition of the second order renormalisation, denoted by underline, from [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For functions f(W ),g(W ) of the random matrix W (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15)), we define f(W )W g(W ) ∶= g(W )W f(W ) − ̃E[(∂̃ W f)(W )̃ Wg(W ) + f(W )̃ W(∂̃ W g)(W )], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) where ∂̃ W denotes the directional derivative in the direction of ̃ W ∶= ( 0 ̃ X ̃ X∗ 0 ) , where ˜ X is a complex Ginibre matrix that is independent of W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The expectation is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the matrix ̃ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that, if W itself consists of a complex Ginibre matrix X, then Ef(W )W g(W ) = 0, while for X with a general distribution this expectation is independent of the first two moments of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In other words, the underline renormalises the product f(W )W g(W ) to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We remark that underline (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) is a well-defined notation, if the ‘middle’ W to which the renormalisation refers is unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This is always be the case in all our proofs, since the functions f,g will be products of resolvents, never involving explicitly monomials in W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We note that ̃Ẽ WR̃ W = 2⟨RE2⟩E1 + 2⟨RE1⟩E2 = ∑ σ σ⟨REσ⟩Eσ = S[R] and furthermore, that the directional derivative of the resolvent is given by ∂̃ W G = −G̃ WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For example, in the special case f(W ) = 1 and g(W ) = (W + ˆΛ − w)−1 = G, we thus have W G = W G + S[G]G by definition of the underline in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using this underline notation in combination with the identity G(W+ˆΛ−w) = E+ and the defining equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) for M, we have G = M − MW G + MS[G − M]G = M − GWM + GS[G − M]M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) Recall that ⟨GE−⟩ = 0 (see below (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16)) which immediatelyyields that S[G] = ∑σ σ⟨GEσ⟩Eσ = ⟨G⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, as shown in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, we have that S[M] = ⟨M⟩ and hence S[⋅] effectively acts like a trace on G and M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' S[G − M] = ⟨G − M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) 26 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Now, similarly to [28], the key idea of the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 is using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) for some Gj in a chain G1A1 ⋯AkGk+1 and extending the renormalisation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) to the whole product at the expense of adding resolvent products of shorter length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This will be done for each of the four estimates from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 separately and presented in an underlined lemma in the beginning of each of the follow- ing subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Afterwards, the renormalisation of the whole product will be handled by cumulant expansion, exploiting that its expectation vanishes up to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' While the proofs of the un- derlined lemmas for Ψav/iso 1 are presented in detail, we defer the analogous arguments for Ψav/iso 2 to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of the first master inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let w ≡ w1 be a spectral parameter in D(ǫ0,κ0) ℓ+1 (in particular in the bulk of the scDos, recall (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19)) and A ≡ A1 a (w,w)-regular matrix (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We use the notation w = e + iη and we assume without loss of generality (by conjugation with E−, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16)) that 1 ≥ η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We also assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) holds (in this subsection we will need it only for Ψav 1 and Ψav 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Representation as full underlined) For any regular matrix A = ˚ A we have that ⟨(G − M)˚ A⟩ = −⟨W G˚ A′⟩ + O≺(E av 1 ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) for some other regular matrix A′ = ˚ A′, which linearly depends on A (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) for the precise formula for A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the error term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), we used the shorthand notation E av 1 ∶= 1 Nη1/2 (1 + ψav 1 Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) Having this approximate representation of ⟨(G − M)˚ A⟩ as a full underlined term at hand, we turn to the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24a) via a (minimalistic) cumulant expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, starting from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), we obtain E∣⟨(G − M)A⟩∣2p =∣−E⟨W GA′⟩⟨(G − M)A⟩p−1⟨(G − M)∗A∗⟩p∣ + O≺((E av 1 )2p) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) ≲E ∣∑σ σ⟨GEσGA′EσGA⟩∣ + ∣∑σ σ⟨G∗EσGA′EσG∗A∗⟩∣ N 2 ∣⟨(G − M)A⟩∣ 2p−2 + ∑ ∣l∣+∑(J∪J∗)≥2 EΞav 1 (l,J,J∗)∣⟨(G − M)A⟩∣ 2p−1−∣J∪J∗∣ + O≺((E av 1 )2p) , where Ξav 1 (l,J,J∗) is defined as Ξav 1 ∶= N −(∣l∣+∑(J∪J∗)+3)/2 ∑ ab Rab∣∂l(GA′)ba∣ ∏ j∈J ∣∂j⟨GA⟩∣ ∏ j∈J∗ ∣∂j⟨G∗A∗⟩∣ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) and the summation in the last line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) is taken over tuples l ∈ Z2 ≥0 and multisets of tuples J,J∗ ⊂ Z2 ≥0 ∖ {(0,0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we set ∂(l1,l2) ∶= ∂l1 ab∂l2 ba, ∣(l1,l2)∣ = l1 + l2, ∑ J = ∑j∈J ∣j∣, and used the shorthand notation Rab ∶= 1(a ≤ N,b ≥ N + 1 or b ≤ N,a ≥ N + 1) for a rescaled cumulant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the remainder of the proof, we need to analyze the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of the inequality derived in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We begin with the third line and study the terms involving Ξav 1 from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Before going into the proof, we note that, due to the cumulant expansion in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8), there are chains of resolvents G and deterministic matrices A appearing, where some of the A’s are not necessarily regular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the spectral parameters of the surrounding G’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The principal idea is to decompose such A with the aid of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and carefully track the resulting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As a rule of thumb, potentially small denominators resulting from resolvent identities or the integral representation in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 are balanced with the linear perturbative estimates from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' See also Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 27 Gaussian contribution: third line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In order to do so, we need to analyze in total four terms, each of which carries a factor of ⟨GEσGA′EσGA⟩ or ⟨G∗EσGA′EσG∗A∗⟩, for σ = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since their treatment is very similar, we focus on the two exemplary terms (i) ⟨GGA′GA⟩ and (ii) ⟨G∗GA′G∗A∗⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) In the analysis of the Gaussian contribution in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we will discuss the analogs of the other two terms in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' First term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10), we apply the integral representation from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 to GG with τ = +, J = Bℓκ0 , and ˜η = ℓ ℓ + 1η , for which we recall that w ∈ D(ǫ0,κ0) ℓ+1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' in particular η ≥ (ℓ + 1)N −1+ǫ0 and hence ˜η ≥ ℓN −1+ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, Γ ≡ Γτ ˜η(J) ⊂ D(ǫ0,κ0) ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, we split the contour Γ in three parts,13 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Γ = Γ1 + Γ2 + Γ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) As depicted in Figure 4, the first part of the contour consists of the entire horizontal part of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The sec- 1 1 2 3 D(ǫ0,κ0) ℓ+1 D(ǫ0,κ0) ℓ Γ w i iN 100 Rez Imz Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The contour Γ is splitinto three parts (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In case of multiple spec- tral parameters, the second part might require a further decomposition at the level indicated by the dashed horizontal line (see Footnote 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Depicted is the situation, where the bulk Bℓκ0 consists of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ond part, Γ2, covers the vertical components up to ∣Im z∣ ≤ N 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, Γ3 consists of the remaining part with ∣Im z∣ > N 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, the contribution coming from Γ3 can be estimated with a trivial norm bound on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For z ∈ Γ2, we use that 1± δ(z,w) = 0 for every w ∈ D(ǫ0,κ0) ℓ+1 (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20)) and hence every matrix is (z,w)-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, after splitting the contour integral and bounding each contribution as just described, we find, with the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, ∣⟨GGA′GA⟩∣ ≺ (1 + ψav 2 Nη ) + ∫Bℓκ0 ∣⟨G(x + i˜η)A′G(e + iη)A⟩∣ (x − e)2 + η2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) Next, we decompose A = ˚ A = ˚ Ae+iη,e+iη and A′ = ˚ A′ = ˚ (A′) e+iη,e+iη according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 as ˚ Ae+iη,e+iη = ˚ Ae+iη,x+i˜η + O(∣x − e∣ + η)E+ + O(∣x − e∣ + η)E− , ˚ (A′) e+iη,e+iη = ˚ (A′) x+i˜η,e+iη, + O(∣x − e∣ + η)E+ + O(∣x − e∣ + η)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 13In the case of several w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', wk, the second part might require a further decomposition: If the spectral parameters of the resolvents which are not involved in such an integral representation have spectral parameters with imaginary parts of absolute value greater than one, we need to split Γ2 according to ∣Im z∣ ≤ 1 and ∣Im z∣ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' While the former will be treated exactly as Γ2 here, the latter shall be estimated by means of the η > 1-laws, which we discussed after Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 28 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Plugging this into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12), we obtain several terms contributing to the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By means of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, the leading term accounts for ∫Bℓκ0 ∣⟨G(x + i˜η) ˚ (A′) x+i˜η,e+iηG(e + iη)˚ Ae+iη,x+i˜η⟩∣ (x − e)2 + η2 dx ≺ 1 η (1 + ψav 2 Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The error terms can be dealt with by simple resolvent identities in combination with the usual single- resolvent local law, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, proving them to be bounded by η−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Indeed, for a generic B ∈ C2N×2N, we consider the exemplary term ∫Bℓκ0 ∣⟨G(x + i˜η)E+G(e + iη)B⟩∣ ∣x − e∣ + η (x − e)2 + η2 dx ≲∫Bℓκ0 ∣⟨(G(x + i˜η) − G(e + iη))B⟩∣ (x − e)2 + η2 dx ≺ 1 η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The second term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) is much simpler than the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' After writing GG∗ = ImG/η, it suffices to realise that, by means of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, A′ = ˚ (A′) e+iη,e−iη , ˚ (A′) e−iη,e−iη = A′ + O(∣e∣)E− , and A∗ = ˚ (A∗) e−iη,e±iη in order to bound ∣⟨G∗GA′G∗A∗⟩∣ ≺ 1 η (1 + ψav 2 Nη ) + ∣e∣ η ∣⟨[G(−e + iη) − G(e − iη)]A∗E−⟩∣ ∣e∣ + η ≺ 1 η (1 + ψav 2 Nη ) with the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, the chiral symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), a resolvent identity and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This finishes the estimate for the Gaussian contribution from the third line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8), for which we have shown that 1 N 2 ∑ σ (∣⟨GEσGA′EσGA⟩∣ + ∣⟨G∗EσGA′EσG∗A∗⟩∣) ≺ 1 N 2η (1 + ψav 2 Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) We are now left with the terms from the last line (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) resulting from higher order cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Higher order cumulants and conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The terms stemming from higher order cumulants are es- timated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5, the precise bound being given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Indeed, plugging (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68a) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) we obtain E∣⟨(G − M)A⟩∣2p ≺ (E av 1 )2p + p ∑ m=1 [ 1 Nη1/2 (1 + ψiso 1 + (ψav 2 )1/2 (Nη)1/2 + (ψiso 2 )1/8 (Nη)1/8 )] m (E∣⟨(G − M)A⟩∣2p) 1−m/2p and get the appropriate estimate E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ∣2p using Young inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since p was arbitrary, it follows that Ψav 1 ≺ 1 + ψav 1 Nη + ψiso 1 + (ψav 2 )1/2 (Nη)1/2 + (ψiso 2 )1/4 (Nη)1/8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The bound given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 is an immediate consequence after a further trivial Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Although the proof of the first master inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24a) is rather short, it already revels a general strategy for dealing with a generic (not strictly) alternating chain ⋯GGAGAGE−AGE−GA ⋯ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) of resolvents G and deterministic matrices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (i) Apply resolvent identites and the integral representation from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 in order to reduce a product of resolvents to a linear combination (discrete or continuous, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For terms of the form GE−G instead of GG this additionally requires an application of the chiral symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 29 (ii) In the resulting strictly alternating chain, decompose every deterministic A according to the regu- larisation from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the spectral parameters of its surrounding resolvents by using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (iii) Estimate the regular parts coming from this decomposition in terms of Ψav/iso k ≺ ψav/iso k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Carefully track the resulting errors stemming from the other parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' These steps shall be applied repeatedly until the entire chain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) has been examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The first two items in the above list a purely mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, the third step is non-trivial and requires careful analysis on a case-by-case basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We have already mentioned that, as a rule of thumb, potentially small denominators resulting from Step (i) are balanced with the linear perturbative numerators from Step (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' It remains to give a proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Similarly as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), we suppress the indices of G ≡ G1, M ≡ M1 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We start with the first identity in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4), such that, after defining the one-body stability operator B ∶= 1 − MS[⋅]M we find B[G − M] = −MW G + MS[G − M](G − M) and consequently, by inversion, multiplication by A = ˚ A (in the sense of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), see also (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7)) and taking a trace ⟨(G − M)A⟩ = −⟨W GX [A]M⟩ + ⟨S[G − M](G − M)X [A]M⟩, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) where we introduced the linear operator X [B] ∶= ((B∗)−1[B∗]) ∗ = (1 − S[M ⋅ M]) −1[B] for B ∈ C2N×2N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, it is important to note that the condition 1+ δ⟨ImMA⟩ = 0 (the first of the two imposed via (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' recall the definition of the cutoff function 1+ δ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5)and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5)), is stable under the linear operation A ↦ X [A]M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For a generic B ∈ C2N×2N , we find ⟨X [B]MImM⟩ = ⟨BB−1[MImM]⟩ = i 2 ⟨BImM⟩ ⟨ImM⟩ + O(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), we compute B−1[MImM] = B−1[M 2 − MM ∗] 2i = i 2 ImM η + ⟨ImM⟩ + 1 2i 1 − ⟨MM ∗⟩ 1 − ⟨M 2⟩ M 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, by means of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5, we find that ∣1 − ⟨MM ∗⟩∣ = O(η) and ∣1 − ⟨M 2⟩∣ ≳ 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Recall from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) that S[G − M] = ⟨G − M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Therefore, by means of the usual averaged local law, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6,whichinparticularshowsthat∣⟨W GB⟩∣ ≺ 1 Nη forarbitrary∥B∥ ≲ 1(see also AppendixB and [36]), we can write (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) as ⟨(G − M)A⟩ = − ⟨W G(X [A]M)○⟩ + ⟨G − M⟩⟨(G − M)(X [A]M)○⟩ − 1− δc−(X [A]M)⟨W GE−⟩ + O≺(N −1) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) where in the underlined term, we used that the E+ component of the regularisation of X [A]M is negligible thanks to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 and the regularity of A, and we introduced the short hand notation c−(X [A]M) ∶= ⟨MX [A]MME−⟩ ⟨ME−ME−⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, with the aid of W G = I − ˆΛG + wG and using ⟨GE−⟩ = 0 from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5), we undo the underline in the second to last term, such that we infer ⟨W GE−⟩ = −⟨GE− ˆΛ⟩ = −⟨(G − M)E− ˆΛ⟩ = −⟨(G − M)(E− ˆΛ)○⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 30 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES In the second equality, we used that ⟨ME− ˆΛ⟩ = 0, which follows by a simple computation using the explicit form of M given in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the last equality, we note that (E− ˆΛ)○ = E−ˆΛ − 1+ δ ⟨ImME− ˆΛ⟩ ⟨ImM⟩ E+ − 1− δ ⟨ME− ˆΛME−⟩ ⟨ME−ME−⟩ E− = E−ˆΛ, which again follows after a simple computation using the fact that ˆΛ is off-diagonal together with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We can now write (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) for A = ˚ A = (E− ˆΛ)○ = E−ˆΛ and solve the resulting equation for ⟨(G − M)E− ˆΛ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Plugging this back into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) yields ⟨(G − M)A⟩ = − ⟨W G(X [A]M)○⟩ + ⟨G − M⟩⟨(G − M)(X [A]M)○⟩ + O≺(N −1) + 1− δ c−(X [A]M) 1 − 1− δ c−(X [E− ˆΛ]M) [ − ⟨W G(X [E−Z]M)○⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) + ⟨G − M⟩⟨(G − M)(X [E−Z]M)○⟩ + O≺(N −1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since ∥X [˚ A]∥ ≲ 1 (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), the only thing left to check is, that the denominator in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) is bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For small enough δ > 0, we have that ∣1 − 1− δ(w,w)c−(X [E− ˆΛ]M)∣ ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The statement is trivial for 1− δ(w,w) = 0 and we hence focus on the complementary extreme scenario 1− δ(w,w) = 1, the intermediate ones being immediate consequences of the extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Indeed, for 1− δ(w,w) = 1, we first note that X [E−ˆΛ] = E−ˆΛ, which follows from the explicit form of M given in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 using the fact that ˆΛ is purely off-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, we use the MDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19), the chiral symmetry E−M(w) = −E−M(−w) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 to infer 1 − c−(X [E− ˆΛ]M) = 1 − ⟨ME− ˆΛMME−⟩ ⟨ME−ME−⟩ = 1 2 [1 − w + m m ⟨M 2⟩] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, specializing to w = iη with sufficiently small η, we find that, to leading order, ∣1 − η + Imm Imm ⟨M 2⟩∣ ∼ ∣1 − ⟨M 2⟩∣ ≳ 1 by means of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This principal lower bound of order one persists after a small perturbation of w allowing for a non-zero real part, but as long as 1− δ(w,w) = 1 for some δ > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ From the expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) it is apparent, that the main terms for understanding the size of ⟨(G − M)A⟩ are the underlined ones, the rest carrying additional ⟨G − M⟩-factors, hence they will become negligible errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, summarizing our investigations, we have shown that ⟨(G − M)˚ A⟩ = −⟨W G˚ A′⟩ + O≺(E av 1 ) , where we used the shorthand notation ˚ A′ ∶= (X [˚ A]M)○ + 1− δc−(X [˚ A]M) 1 − 1− δ c−(X [E−ˆΛ]M) (X [E− ˆΛ]M)○ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) in the underlined term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using the usual averaged local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23), we collected all the error terms from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) in E av 1 , defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of the second master inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let wj ∈ D(ǫ0,κ0) ℓ+1 for j ∈ [2] be spectral pa- rameters and A1 a regular matrix w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the pair of spectral parameters (w1,w2) (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By conjugation with E−, we will assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' that Imw1 > 0 and Imw2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we use the notations ej ≡ Rewj, ηj ∶= ∣Imwj∣ for j ∈ [2] and define 1 ≥ η ∶= minj ∣Imwj∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We also assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Representation as full underlined) For ∥x∥,∥y∥ ≤ 1 and any (w1,w2)-regular matrix A1 = ˚ A1, we have that (G1 ˚ A1G2 − M(w1, ˚ A1,w2))xy = −(G1 ˚ A′ 1W G2)xy + O≺(E iso 1 ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) for some (w1,w2)-regular matrix A′ 1 = ˚ A′ 1, which linearly depends on A1 = ˚ A1 (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='51)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the error term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20), we used the shorthand notation E iso 1 ∶= 1 √ Nη2 (1 + ψav 1 (Nη)1/2 + ψiso 1 Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) Note that unlike in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, now in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) the second resolvent G2 was expanded instead of G1 rendering the W factor in the middle of the underlined term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This prevents the emergence of resolvent chains in the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24b), which are ‘too long’ to be handled within our hierarchical framework of master inequalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', a chain involving four resolvents would appear in ̃Ξiso 1 defined below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Having this approximate representation of (G1 ˚ A1G2 − M(w1, ˚ A1,w2))xy as a full underlined term at hand, we turn to the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24b) via a (minimalistic) cumulant expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, starting from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) and using the same notations as in the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24a), we obtain E∣(G1 ˚ A1G2 − M(w1, ˚ A1,w2))xy∣ 2p (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) ≲E ̃Ξiso 1 ∣(G1 ˚ A1G2 − M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='))xy∣ 2p−2 + ∑ ∣l∣+∑(J∪J∗)≥2 EΞiso 1 (l,J,J∗)∣(G1 ˚ A1G2 − M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='))xy∣ 2p−1−∣J∪J∗∣ + O≺((E iso 1 )2p) , where ̃Ξiso 1 ∶= ∑σ [∣(G1 ˚ A′ 1EσG1 ˚ A1G2)xy(G1EσG2)xy∣ + ∣(G1 ˚ A′ 1EσG2)xy(G1 ˚ A1G2EσG2)xy∣] N + ∑σ [∣(G1 ˚ A′ 1EσG∗ 2(˚ A1)∗G∗ 1)xx(G∗ 2EσG2)yy∣ + ∣(G1 ˚ A′ 1EσG∗ 1)xx(G∗ 2(˚ A1)∗G∗ 1EσG2)yy∣] N and Ξiso 1 (l,J,J∗) is defined via Ξiso 1 ∶= N −(∣l∣+∑(J∪J∗)+1)/2 ∑ ab Rab∣∂l[(G1 ˚ A′ 1)xa(G2)by]∣ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) × ∏ j∈J ∣∂j(G1 ˚ A1G2)xy∣ ∏ j∈J∗ ∣∂j(G∗ 2(˚ A1)∗G∗ 2)yx∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the remainder of the proof, we need to analyze the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of the inequality derived in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Following the general strategy outlined in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, we begin with the second line and study the terms involving Ξiso 1 from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Gaussian contribution: third line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In order to do so, following Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, we need to ana- lyze in total eight terms, each of which carries one of the summands in the definition of ̃Ξiso 1 as a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since their treatment is very similar, we focus on the two exemplary terms (i) (G1 ˚ A′ 1E−G1 ˚ A1G2)xy(G1E−G2)xy , (ii) (G1 ˚ A′ 1E−G∗ 1)xx(G∗ 2(˚ A1)∗G1E−G2)yy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24) In the analysis of the Gaussian term in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 we discussed analogs of the above terms with the choice σ = +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 32 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Term (i) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the first term, we decompose, similarly to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, (˚ A′ 1)1,2E− = ((˚ A′ 1)1,2E−) 1,1 + O(∣e1 + e2∣ + ∣η1 − η2∣)E+ + O(∣e1 + e2∣ + ∣η1 − η2∣)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25) Inserting this into the first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24) and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we find ∣(G1 ˚ A′ 1E−G1 ˚ A1G2)xy∣ ≺ 1 η (1 + ψiso 2 √Nη ) + (∣e1 + e2∣ + ∣η1 − η2∣) ∑ σ ∣(G1EσG1 ˚ A1G2)xy∣ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26) In the last term, we focus on σ = −, while σ = + can be dealt with by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) and a resolvent identity, we obtain ∣(G1E−G1 ˚ A1G2)xy∣ = ∣ 1 w1 ([G(−w1) − G(w1)]˚ Aw1,w2 1 G(w2))(E−x)y∣ ≺ 1 η2 (1 + ψiso 1 √Nη ) , where in the last step we used Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 and the trivial approximation ˚ A−w1,w2 1 = ˚ Aw1,w2 1 + O(1)E+ + O(1)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the second factor in the first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24), we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) and employ the integral representation from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 with τ = +, J = Bℓκ0 , and ˜η = ℓ ℓ + 1η , for which we recall that wj ∈ D(ǫ0,κ0) ℓ+1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' in particular η ≥ (ℓ + 1)N −1+ǫ0 and hence ˜η ≥ ℓN −1+ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' After splitting the contour integral and estimating the contribution as described around (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), we find, with the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 and absorbing logarithmic corrections into ‘≺’, that ∣(G1E−G2)xy∣ ≺ 1 + ∫Bℓκ0 ∣(G(x + i˜η))x(E−y)∣ ∣(x − e1 − i(η1 − ˜η)) (x + e2 − i(η2 − ˜η))∣dx ≺ 1 + 1 ∣e1 + e2∣ + η1 + η2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='27) where in the last step we used the usual single resolvent local law from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Notice the key cancellation of the ∣e1 + e2∣ factor in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Collecting all the estimates, we have shown that ∣(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24) (i)∣ ≺ 1 η2 (1 + ψiso 1 √Nη + ψiso 2 √Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='28) Term (ii) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the first factor in the second term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24), we again employ the decomposition(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25) to find ∣(G1 ˚ A′ 1E−G∗ 1)xx∣ ≺ 1 η1/2 (1 + ψiso 1 √Nη ) + ∣e1 + e2∣ + ∣η1 − η2∣ η (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='29) with the aid of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 as well as a resolvent identity and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 for the E+ and E− in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the second factor, similarly to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='27) above, we use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 together with the decomposition (˚ Aw1,w2 1 )∗ = ˚ (A∗ 1) ¯ w2, ¯ w1 = ˚ (A∗ 1) ¯ w2,w1 = ˚ (A∗ 1) ¯ w2,x+i˜η + ∑ σ Oσ(∣x − e1∣ + ∣η1 − ˜η∣)Eσ from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 for arbitrary x to find ∣(G∗ 2(˚ A1)∗G1E−G2)yy∣ ≺ 1 η1/2 (1 + ψiso 1 √Nη ) + ∫Bℓκ0 ∣(G( ¯w2) ˚ (A∗ 1) ¯ w2,x+i˜ηG(x + i˜η))y(E−y)∣ ∣(x − e1 − i(η1 − ˜η)) (x + e2 − i(η2 − ˜η))∣dx + ∫Bℓκ0 ∑σ ∣(G( ¯w2)EσG(x + i˜η))y(E−y)∣ ∣x + e2 − i(η2 − ˜η)∣ dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='30) EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 33 ≺ 1 η1/2 (1 + ψiso 1 √Nη ) (1 + 1 ∣e1 + e2∣ + η1 + η2 ) + 1 η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='29) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='30), we obtain ∣(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24) (ii)∣ ≺ 1 η2 (1 + ψiso 1 √Nη ) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='31) This finishes the estimate for the Gaussian contribution from the third line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22), for which we have shown that ̃Ξiso 1 ≺ 1 Nη2 (1 + (ψiso 1 )2 Nη + ψiso 2 √Nη ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='32) as easily follows by combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='28) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='31) and using a Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We are now left with the terms from the last line (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) resulting from higher order cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Higher order cumulants and conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The estimate stemming from higher order cumulants is given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, plugging (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='32) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68b) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22), we find, similarly to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, that Ψiso 1 ≺ 1 + ψiso 1 Nη + ψiso 1 + ψav 1 (Nη)1/2 + (ψiso 2 )1/2 (Nη)1/4 + (ψiso 2 )1/4 (Nη)1/8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The bound given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 is an immediate consequence after a trivial Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ It remains to give a proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This is much more involved than for the previous underlined Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 crucially used that the orthogonality ⟨ImMA⟩ = 0 is (almost) preserved under the operation A ↦ X [A]M (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This is simply not available here, since we deal with two spectral parameters w1,w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We denote A1 ≡ ˚ A1, except we wish to emphasise A1 being regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Just as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, we start with G2 = M2 − M2W G2 + M2S[G2 − M2]G2 , such that we get G1 ˜A1G2 = G1 ˜A1M2 − G1 ˜A1M2W G2 + G1 ˜A1M2S[G2 − M2]G2 for ˜A1 = X12[A1] and A1 = ˚ A1 (note that ∥X12[˚ A1]∥ ≲ 1 by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), where we introduced the linear operator X12[B] ∶= (1 − S[M1 ⋅ M2]) −1[B] for B ∈ C2N×2N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='33) Extending the underline to the whole product, we obtain G1 ˜A1G2 =M1 ˜A1M2 + (G1 − M1) ˜A1M2 − G1 ˜A1M2W G2 + G1 ˜A1M2S[G2 − M2]G2 + G1S[G1 ˜A1M2]G2 , from which we conclude that G1( ˜A1 − S[M1 ˜A1M2])G2 = M1 ˜A1M2 + (G1 − M1) ˜A1M2 − G1 ˜A1M2W G2 + G1 ˜A1M2S[G2 − M2]G2 + G1S[(G1 − M1) ˜A1M2]G2 and thus G1A1G2 =M1X12[A1]M2 + (G1 − M1)X12[A1]M2 − G1X12[A1]M2W G2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='34) + G1X12[A1]M2S[G2 − M2]G2 + G1S[(G1 − M1)X12[A1]M2]G2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We note that ∥X12[˚ A1]∥ ≲ 1 by means of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, we need to further decompose X12[A1]M2 in the last three terms in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='34) as X12[A1]M2 = (X12[A1]M2) + ∑ σ 1σ δ cσ(X12[A1]M2)Eσ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='35) 34 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES where we suppressed the spectral parameters (and the relative sign of their imaginary parts, which has been fixed by Imw1 > 0 and Imw2 < 0) in the notation for the linear functionals cσ(⋅) on C2N×2N defined as c+(B) ∶= ⟨M1BM2⟩ ⟨M1M2⟩ and c−(B) ∶= ⟨M1BM ∗ 2 E−⟩ ⟨M1E−M ∗ 2 E−⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='36) Plugging (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='35) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='34) we find G1A1G2 to equal M1X12[A1]M2 + (G1 − M1)X12[A1]M2 − G1(X12[A1]M2) W G2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='37) + G1(X12[A1]M2) S[G2 − M2]G2 + G1S[(G1 − M1)(X12[A1]M2) ]G2 + ∑ σ 1σ δ cσ(X12[A1]M2)[−G1EσW G2 + G1EσS[G2 − M2]G2 + G1S[(G1 − M1)Eσ]G2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Recall that the regular component is defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the pair of spectral parameters (w1,w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In partic- ular, (X12[A1]M2) = (X12[A1]M2) 1,2 in the last term in the second line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='37) is not regular as defined via the conditions with one resolvent (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the last line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='37) we now undo the underline and find the bracket [⋯] to equal (the negative of) G1EσW G2 + G1EσS[M2]G2 + G1S[M1Eσ]G2 =G1Eσ + G1(Eσ(w2 − ˆΛ + S[M2]) + S[M1Eσ])G2 =G1Eσ − G1(EσM −1 2 − S[M1Eσ])G2 =∶ G1Eσ − G1ΦσG2 , where we used W G2 = E+ + w2G2 − ˆΛG2 in the first step and the MDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we introduced the shorthand notation Φσ ∶= Eσ 1 M2 − S[M1Eσ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='38) From the expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='37) it is apparent (and it can also be checked by hand using the explicit form of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='38)) that M1Eσ = M1(EσM −1 2 )M2 = M1X12[Φσ]M2 = M(w1,Φσ,w2), where in the last step we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This finally yields that G1A1G2 equals M(w1,A1,w2) + (G1 − M1)X12[A1]M2 − G1(X12[A1]M2) W G2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='39) + G1(X12[A1]M2) S[G2 − M2]G2 + G1S[(G1 − M1)(X12[A1]M2) ]G2 + ∑ σ 1σ δ cσ(X12[A1]M2)[−(G1 − M1)Eσ + (G1ΦσG2 − M(w1,Φσ,w2))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The last term in the last line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='39) requires further decomposition of Φσ from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='38) (completely analogous to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='35) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='36)) as Φσ = ˚Φσ + ∑ τ 1τ δ cτ(Φσ)Eτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using the explicit form of Φσ, we further observe that cτ(Φσ) ∼ δσ,τ and cτ(X12[Φσ]M2) ∼ δσ,τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='40) Therefore, by means of the first relation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='40), the expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='39) can be carried out further as M(w1,A1,w2) + (G1 − M1)X12[A1]M2 − G1(X12[A1]M2) W G2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='41) + G1(X12[A1]M2) S[G2 − M2]G2 + G1S[(G1 − M1)(X12[A1]M2) ]G2 + ∑ σ 1σ δ cσ(X12[A1]M2) [ − (G1 − M1)Eσ + (G1˚ΦσG2 − M(w1,˚Φσ,w2)) + cσ(Φσ)(G1EσG2 − M(w1,Eσ,w2))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 35 Next, we write (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='41) for both, A1 = ˚ A1 = ˚Φ+ and A1 = ˚ A1 = ˚Φ−, and solve the two resulting linear equations for G1˚Φ±G2 − M(w1,˚Φ±,w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Observe that by means of the second relation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='40) the original system of linear equations boils down to two separate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thus, plugging the solutions for G1˚Φ±G2 − M(w1,˚Φ±,w2) back into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='41) we arrive at G1A1G2 =M(w1,A1,w2) + (G1 − M1)X12[A1]M2 − G1(X12[A1]M2) W G2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='42) + G1(X12[A1]M2) S[G2 − M2]G2 + G1S[(G1 − M1)(X12[A1]M2) ]G2 + ∑ σ 1σ δ cσ(X12[A1]M2) 1 − 1σ δ cσ(X12[˚Φσ]M2) [(G1 − M1)X12[˚Φσ]M2 − G1(X12[˚Φσ]M2) W G2 + G1(X12[˚Φσ]M2) S[G2 − M2]G2 + G1S[(G1 − M1)(X12[˚Φσ]M2) ]G2 − (G1 − M1)Eσ + cσ(Φσ)(G1EσG2 − M(w1,Eσ,w2))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now need to check that the denominators in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='42) are bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For small enough δ > 0, we have that ∣1 − 1σ δ (w1,w2)cσ(X12[˚Φσ]M2)∣ ≳ 1 for σ = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' First, the statements are trivial for 1σ δ (w1,w2) = 0 and we hence focus on the complemen- tary extreme scenario 1σ δ (w1,w2) = 1, the intermediate ones being immediate consequences of the extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Indeed, for 1σ δ (w1,w2) = 1 we compute 1 − c+(X12[˚Φ+]M2) = ⟨M1⟩⟨M1M2M2⟩ ⟨M1M2⟩2 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='43) 1 − c−(X12[˚Φ−]M2) = ⟨M1E−M ∗ 2 M −1 2 E−⟩ + ⟨M1⟩⟨M1E−M ∗ 2 E−⟩ 1 + ⟨M1E−M2E−⟩ ⟨M1E−M2M ∗ 2 E−⟩ ⟨M1E−M ∗ 2 E−⟩2 for arbitrary spectral parameters w1,w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Recall that we assumed the two spectral parameters to be on different halfplanes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' s1 = −sgn(Imw1Imw2) = +, hence we shall specialise (i) the first expression in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='43) to w2 = ¯w1 and (ii) the second expression in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='43) to w2 = −w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the former, using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 and ImM1Imw1 > 0, we obtain ∣1 − c+(X12[˚Φ+]M2)∣ = ∣⟨M1⟩⟨ImM1M1⟩ ⟨ImM1⟩2 (⟨ImM1⟩ + Imw1)∣ ≥ ⟨ImM1⟩2 ≳ 1 in the bulk of the spectrum, while in the latter we find, similarly to above, again in the bulk, ∣1 − c−(X12[˚Φ−]M2)∣ ≥ ⟨ImM1⟩2 2 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' These principal lower bounds of order one persist after a small perturbationof w2around the special cases, but as long as 1σ δ (w1,w2) = 1 (for some δ > 0 small enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Next, we take the scalarproductof (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='42) withtwo deterministicvectorsx,y satisfying∥x∥,∥y∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the resulting expression,there are two particular terms, namely the ones of the form (G1S[(G1 − M1)˚ A1,2 1 ]G2)xy and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='44) cσ(X12[˚ A1,2 1 ]M2)cσ(Φσ)(G1EσG2 − M(w1,Eσ,w2))xy , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='45) whose direct (naive) estimatesare 1/(Nη2) and 1/η, respectively, and thus do not match the target size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, they have to be discussed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In our notation, we emphasised that the regularisation is defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the spectral parameters (w1,w2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', in particular, A○ 1 = A 1,2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Estimating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the term (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='44), we expand (G1S[(G1 − M1)˚ A1,2 1 ]G2)xy = ∑ σ σ⟨(G1 − M1)˚ A1,2 1 Eσ⟩(G1EσG2)xy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='46) 36 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES and observe that, by definition of ⋅○ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), we have, similarly to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 (see also (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25)), ˚ A1,2 1 Eσ = ( ˚ A1,2 1 Eσ) 1,1 + O(∣e1 − σe2∣ + ∣η1 − η2∣)E+ + O(∣e1 − σe2∣ + ∣η1 − η2∣)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='47) Now, in the second term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='46) for σ = + and Eσ = E+, we use a resolvent identity and the usual isotropic local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) to estimate it as ∣(G1G2)xy∣ ≺ 1 + 1 ∣e1 − e2∣ + η1 + η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='48) Furthermore, in the second term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) for σ = − and Eσ = E−, we employ the integral represen- tation from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 in combination with the usual isotropic local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) (see also (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='27)) to infer ∣(G1E−G2)xy∣ ≺ 1 + 1 ∣e1 + e2∣ + η1 + η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='49) Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='48) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='49) with the decomposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='47) and the usual averaged local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15), we find that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='46) can be bounded by ∑ σ (∣⟨(G1 − M1)(˚ A1,2 1 Eσ) 1,1⟩∣ + ∣e1 − σe2∣ + ∣η1 − η2∣ Nη1 ) (1 + 1 ∣e1 − σe2∣ + η1 + η2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using the definition of Ψav 1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and the apriori bound Ψav 1 ≺ ψav 1 , this immediately implies the estimate ∣(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='44)∣ ≺ 1 Nη + 1 √ Nη ψav 1 (Nη)1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='50) Estimating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Forthe term (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='45), we firstnote thatthe two prefactorscσ(X12[A 1,2 1 ]M2)andcσ(Φσ) are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, in each of the two cases σ = ±, the bound on one of the prefactors needs to be improved: In the first case, σ = +, we use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) and compute c+(Φ+) = ⟨M1⟩(1 − ⟨M1M2⟩) ⟨M1M2⟩ = O(∣e1 − e2∣ + η1 + η2) from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='36) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining this with the bound ∣(G1G2 − M(w1,E+,w2))xy∣ ≺ ( 1 √Nη1 + 1 √Nη2 ) ⋅ 1 ∣e1 − e2∣ + η1 + η2 which is obtained completely analogous to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='48), we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='45) for σ = + can be estimated by 1/√Nη (recall η ∶= min{η1,η2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Similarly, in the second case, σ = −, we perform a computation similar to the one leading to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) and use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) in order to obtain that c−(X12[˚ A1,2 1 ]M2) equals i 2 ⟨M1 ˚ A1,2 1 M ∗ 2 E−⟩ ⟨M1E−M ∗ 2 E−⟩ + 1 2i ⟨M1 ˚ A1,2 1 M2E−⟩ ⟨M1E−M ∗ 2 E−⟩ 1 + ⟨M1E−M ∗ 2 E−⟩ 1 + ⟨M1E−M2E−⟩ = O(∣e1 + e2∣ + η1 + η2) Combining this with the bound ∣(G1E−G2 − M(w1,E−,w2))xy∣ ≺ 1 √Nη ⋅ 1 ∣e1 + e2∣ + η1 + η2 which is obtained completely analogous to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='49), we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='45) can be estimated by 1/√Nη – now in both cases σ = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Summarizing our investigations, we have shown that (G1 ˚ A1G2 − M(w1, ˚ A1,w2))xy = −(G1 ˚ A′ 1W G2)xy + O≺(E iso 1 ) , where we used the shorthand notation ˚ A′ 1 ∶= (X12[˚ A1]M2) + ∑ σ 1σ δ cσ(X12[˚ A1]M2) 1 − 1σ δ cσ(X12[˚Φσ]M2) (X12[˚Φσ]M2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='51) in the underlined term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='50) and the bound on (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='45) established above with the usual single resolvent local laws (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) and the bounds on deterministic approximations in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we collected all the error terms from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='42) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of the third master inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let wj ∈ D(ǫ0,κ0) ℓ+1 for j ∈ [2] be spectral parame- ters and A1 a regular matrix w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (w1,w2) and A2 a regular matrix w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (w2,w1) (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By conjugation with E−, we again assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' that Imw1 > 0 and Imw2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Just as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we use the notations ej ≡ Rewj, ηj ∶= ∣Imwj∣ for j ∈ [2] and define 1 ≥ η ∶= minj ∣Im wj∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We also assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Representation as full underlined) For any (w1,w2)-regular matrix A1 = ˚ A1 and (w2,w1)-regular matrix A2 = ˚ A2, we have that ⟨(G1 ˚ A1G2 − M(w1, ˚ A1,w2)) ˚ A2⟩ = −⟨W G1 ˚ A1G2 ˚ A′ 2⟩ + O≺(E av 2 ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='52) for some (w2,w1)-regular matrix A′ 2 = ˚ A′ 2, which linearly depends on A2 = ˚ A2 (analogously to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='51), see (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) for an explicit formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the error term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='52), we used the shorthand notation E av 2 ∶= 1 Nη (1 + (ψav 1 )2 Nη + ψav 2 Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='53) Note that similarly to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 but contrary to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, we again expanded the first resolvent G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Otherwise, the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8, given in Appendix E, is very similar to the one of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We only mention that the quadratic error (ψav 1 )2 stems from terms of the form ⟨S[G1 ˚ A1,2 1 G2](G2 − M2)˚ A2,1 2 ⟩, appearing in the analogue of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='42) (see (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) in Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Having the approximate representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='52), we turn to the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24c) via cumulant expansion of the full underlined term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Starting from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), we obtain, as in the proofs of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24a) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24b), E∣⟨(G1 ˚ A1G2 − M(w1, ˚ A1,w2))˚ A2⟩∣ 2p (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='54) ≲E ̃Ξav 2 ∣⟨(G1 ˚ A1G2 − M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='))˚ A2⟩∣ 2p−2 + ∑ ∣l∣+∑(J∪J∗)≥2 EΞav 2 (l,J,J∗)∣⟨(G1 ˚ A1G2 − M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='))˚ A2⟩∣ 2p−1−∣J∪J∗∣ + O≺((E av 2 )2p) , where ̃Ξav 2 ∶= 1 N 2 ∑ σ ∣⟨G1 ˚ A1G2 ˚ A2G1EσG1 ˚ A1G2 ˚ A′ 2Eσ⟩∣ + ⋯ with the other terms being analogous, just 1 and 2 in the first half G1 ˚ A1G2 ˚ A2G1 of the chain inter- changed or the entire half taken as adjoint, and Ξav 2 (l,J,J∗) is defined as Ξav 2 ∶= N −(∣l∣+∑(J∪J∗)+3)/2 ∑ ab Rab∣∂l(G1 ˚ A1G2 ˚ A′ 2)ba∣ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='55) × ∏ j∈J ∣∂j⟨G1 ˚ A1G2 ˚ A2⟩∣ ∏ j∈J∗ ∣∂j⟨G∗ 2 ˚ A∗ 2G∗ 1 ˚ A∗ 1⟩∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, in the remainder of the proof, we need to analyze the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We begin with the second line and study the terms involving Ξav 2 from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='55) afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Gaussian contribution: second line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Along the principal strategy outlined in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, we need to analyze in total eight terms, each of which carries one of the summands in the definition of ̃Ξav 2 as a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since their treatment is very similar, we focus on the exemplary term ⟨G1 ˚ Aw1,w2 1 G2 ˚ Aw2,w1 2 G1G1 ˚ Aw1,w2 1 G2(˚ A′ 2)w2,w1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='56) Now, we represent G1G1 via the integral representation from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 with τ = +, J = Bℓκ0 , and ˜η = ℓ ℓ + 1η , for which we recall that w ∈ D(ǫ0,κ0) ℓ+1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' in particular η ≥ (ℓ + 1)N −1+ǫ0 and hence ˜η ≥ ℓN −1+ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' After splitting the contour integral and bounding the individual contributions as described in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), we 38 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES obtain, with the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, ∣(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='56)∣ ≺ 1 η2 (1 + ψav 4 Nη ) + ∫Bℓκ0 ∣⟨G1 ˚ Aw1,w2 1 G2 ˚ Aw2,w1 2 G(x + i˜η)˚ Aw1,w2 1 G2(˚ A′ 2)w2,w1⟩∣ (x − e1)2 + η2 1 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, we decompose ˚ Aw2,w1 2 and ˚ Aw1,w2 1 in the integrand as ˚ Aw2,x+i˜η 2 = ˚ Aw2,w1 2 + ∑ σ Oσ(∣x − e1∣ + ∣η1 − ˜η∣)Eσ ˚ Ax+i˜η,w2 1 = ˚ Aw1,w2 1 + ∑ σ Oσ(∣x − e1∣ + ∣η1 − ˜η∣)Eσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='57) While the properly regularised term contributes an η−2(1+ψav 4 /(Nη))-error, a typical cross term shall be estimated as ∫Bℓκ0 ∣⟨G1 ˚ Aw1,w2 1 G2 ˚ Aw2,x+i˜η 2 [G(x + i˜η) − G2](˚ A′ 2)w2,w1⟩∣ (∣x − e1∣ + η1) (∣x − e2∣ + η2) ≺ 1 η2 (1 + ψiso 2 √Nη ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='58) where in the second step we wrote out the averaged trace and estimated each summand in isotropic form with the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, using ψiso 2 instead of ψav 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, for ‘error × error’-type terms are bounded by η−2, simplyby using a trivial Schwarz inequal- ity in combination with a Ward identity and the usual local law from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 to infer ∣⟨G1B1G2B2∣ ≤ √ ⟨G1B1B∗ 1G∗ 1⟩⟨G2B2B∗ 2G∗ 2⟩ ≤ 1 η √ ⟨ImG1B1B∗ 1⟩⟨ImG2B2B∗ 2⟩ ≺ 1 η , which is valid for arbitrary bounded matrices ∥B1∥,∥B2∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This finishes the estimate for the Gaussian contribution from the second line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='54), for which, collecting the above estimates, we have shown that ̃Ξav 2 ≺ 1 N 2η2 (1 + ψiso 2 √Nη + ψav 4 Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='59) We are now left with the terms from the last line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='54) resulting from higher order cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Higher order cumulants and conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The estimate stemming from higher order cumulants is given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68c) in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, plugging (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='59) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68c) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='54), we find, similarly to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, that Ψav 2 ≺ 1 + (ψav 1 )2 + (ψiso 1 )2 + ψav 2 Nη + ψiso 2 + (ψav 4 )1/2 (Nη)1/2 + (ψiso 2 )1/2 (Nη)1/4 + (ψiso 3 )3/8 + (ψiso 4 )3/8 (Nη)3/16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The bound given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 is an immediate consequence after a trivial Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of the fourth master inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let wj ∈ D(ǫ0,κ0) ℓ+1 for j ∈ [3] be spectral parame- ters and A1 a regular matrix w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (w1,w2) and A2 a regular matrix w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (w2,w3) (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By conjugation with E−, we will assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' that Imw1 > 0, Imw2 < 0, and Imw3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As before, we use the notations ej ≡ Rewj, ηj ∶= ∣Imwj∣ for j ∈ [3] and define 1 ≥ η ∶= minj ∣Im wj∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We also assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Representation as full underlined) For ∥x∥,∥y∥ ≤ 1 and any (w1,w2)-regular matrix A1 = ˚ A1 and (w2,w3)-regular matrix A2 = ˚ A2, we have that (G1 ˚ A1G2 ˚ A2G3 − M(w1, ˚ A1,w2, ˚ A2,w3))xy = −(G1 ˚ A′ 1W G2 ˚ A2G3)xy + O≺(E iso 2 ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='60) for some other (w1,w2)-regular matrix A′ 1 = ˚ A′ 1, which linearly depends on A1 = ˚ A1 (analogously to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='51), see (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='33) for an explicit formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the error term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='60), we used the shorthand notation E iso 2 ∶= 1 √ Nη3 (1 + ψiso 1 + ψav 1 ψiso 1 Nη + ψiso 2 Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='61) EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 39 Note that similarly to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20), we again expanded the second resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9, given in Appendix E, is very similar to the one of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We only mention that the errors carrying ψiso 1 ψav 1 and ψiso 1 stem from terms of the form (G1S[(G1 − M1)A 1,2 1 ]G2 ˚ A2G3)xy and cσ(X12[˚ A1]M2)cσ(Φσ)(G1EσG2 ˚ A2G3 − M(w1,Eσ,w2, ˚ A2,w3))xy , respectively, appearing in the analogue of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='42) (see (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26) in Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Having the repre- sentation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='60) we turn to the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24d) via cumulant expansion of the underlined term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, starting from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='60), we obtain E∣(G1 ˚ A1G2 ˚ A2G3 − M(w1, ˚ A1,w2, ˚ A2,w3))xy∣ 2p (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='62) ≲E ̃Ξiso 2 ∣(G1 ˚ A1G2 ˚ A2G3 − M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='))xy∣ 2p−2 + O≺((E iso 1 )2p) + ∑ ∣l∣+∑(J∪J∗)≥2 EΞiso 2 (l,J,J∗)∣(G1 ˚ A1G2 ˚ A2G3 − M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='))xy∣ 2p−1−∣J∪J∗∣ , where ̃Ξiso 2 ∶= ∑σ ∑3 j=1 ∣(G1 ˚ A′ 1EσGj ˚ Aj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' G3)xy(G1 ˚ A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ˚ Aj−1GjEσG2 ˚ A2G3)xy∣ N + ∑σ ∑3 j=1 ∣(G1 ˚ A′ 1EσG∗ j ˚ A∗ j−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ˚ A∗ 1G∗ 1)xx(G∗ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ˚ A∗ jG∗ j EσG2 ˚ A2G3)yy∣ N and Ξiso 2 (l,J,J∗) is defined as Ξiso 2 ∶= N −(∣l∣+∑(J∪J∗)+1)/2 ∑ ab Rab∣∂l[(G1 ˚ A′ 1)xa(G2 ˚ A2G3)by]∣ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='63) × ∏ j∈J ∣∂j(G1 ˚ A1G2 ˚ A2G3)xy∣ ∏ j∈J∗ ∣∂j(G∗ 3 ˚ A∗ 2G∗ 2 ˚ A∗ 1G∗ 2)yx∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We need to analyze the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of the inequality derived in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We begin with the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Gaussian contribution: second line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='FollowingRemark5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, we needto analyze intotaltwelve terms, each of which carries one of the summands in the definition of ̃Ξiso 2 as a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Again, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 for the A’s, we pick two exemplary terms (G1 ˚ Aw1,w2 1 G2 ˚ Aw2,w3 2 G3E−G2 ˚ Aw2,w3 2 G3)xy(G1 ˚ (A′ 1) w1,w2E−G3)xy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='64) (G1(˚ A′ 1)w1,w2G∗ 2( ˚ A∗ 1) ¯ w2, ¯ w1G∗ 1)xx(G∗ 3 ˚ (A∗ 2) ¯ w3, ¯ w2G∗ 2G2 ˚ Aw2,w3 2 G3)yy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='65) which shall be treated in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The other terms are analogous and hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The term (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the first factor, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 with parameters τ = +, J = B(ℓ+ 1 2 )κ0 , and ˜η = 2ℓ 2ℓ + 1η , (in order to have some flexibility before approaching the boundary of the domain D(ǫ0,κ0) ℓ ) to bound it as ∣(G1 ˚ Aw1,w2 1 G2 ˚ Aw2,w3 2 G3E−G2 ˚ Aw2,w3 2 G3)xy∣ ≺ 1 η3/2 (1 + ψiso 3 √Nη ) + ∫B(ℓ+ 1 2 )κ0 ∣(G1 ˚ Aw1,w2 1 G2 ˚ Aw2,w3 2 G(x + i˜η) ˚ (E−A2) −w2,w3G3)xy∣ (∣x − e3∣ + η3) (∣x + e2∣ + η2) dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 40 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Next, we decompose ˚ Aw2,w3 2 and ˚ (E−A2) −w2,w3 according to the integration variable with the aid of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 (iii), analogously to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This leaves us with four terms, which shall be estimated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' While the fully regularised term gives 1 η3/2 (1 + ψiso 3 √Nη )(1 + 1 ∣e2 + e3∣ + η2 + η3 ) , the cross terms can be estimated as 1 η2 (1 + ψiso 2 √Nη ) , analogously to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As an exemplary error term, we consider ∫B(ℓ+ 1 2 )κ0 ∣(G1 ˚ Aw1,w2 1 G2E+G(x + i˜η)E−G3)xy∣dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='66) and use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 with new parameters τ = −, J = Bℓκ0 , ˜η = ℓ ℓ + 1η , to find, dropping the integration domains for ease of notation, ∣(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='66)∣ ≺ 1 η1/2 (1 + ψiso 1 √Nη ) + ∫ dx ∫ dy ∣(G1 ˚ Aw1,w2 1 G(y − i˜η))x(E−y)∣ (∣y − e2∣ + η2) (∣y + x∣ + η) (∣y + e3∣ + η3) ≺ 1 η3/2 (1 + ψiso 1 √Nη ) (1 + 1 ∣e2 + e3∣ + η2 + η3 ) , where in the last step we used Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 for decomposing ˚ Aw1,w2 1 accordingly, and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This finishes the bound on the first factor in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The second factor can easily be estimated as ∣(G1 ˚ (A′ 1) w1,w2E−G3)xy∣ ≺ 1 η1/2 (1 + ψiso 1 √Nη ) + ∣e2 + e3∣ + η2 + η3 η using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Notice the cancellation of ∣e2 + e3∣ between the two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The term (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the first factor in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='65), we realise that (˚ A′ 1)w1,w2 = (˚ A′ 1)w1, ¯ w2, which without approximation immediately yields that ∣(G1(˚ A′ 1)w1,w2G∗ 2( ˚ A∗ 1) ¯ w2, ¯ w1G∗ 1)xx∣ ≺ 1 η (1 + ψiso 2 √Nη ) with the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the second factor, we apply a Ward identity to G∗ 2G2 and again use that the regularisation is insensitive to complex conjugation in the second spectral parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this way, and decomposing ˚ Aw2,w3 2 = ˚ A ¯ w2,w3 2 + O(∣e2 − e3∣ + ∣η2 − η3∣)E+ + O(∣e2 + e3∣ + ∣η2 − η3∣)E− by means of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 (ii), we find that the second factor is stochastically dominated by 1 η2 (1 + ψiso 1 + ψiso 2 √Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This finishes the estimate for the Gaussian contribution from the second line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='62), for which, collecting the above estimates, we have shown that ̃Ξiso 2 ≺ 1 Nη3 ⎡⎢⎢⎢⎢⎣ (1 + ψiso 3 √Nη )(1 + ψiso 1 √Nη ) + (1 + ψiso 1 + ψiso 2 √Nη ) 2⎤⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='67) We are now left with the terms from the last line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='62) resulting from higher order cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Higher order cumulants and conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The estimate stemming from higher order cumulants is EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 41 given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68d) in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, plugging (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='67) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68d) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='62), we find, similarly to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, that Ψiso 2 ≺ 1 + ψiso 1 + ψav 1 ψiso 1 + (ψiso 1 )2 + ψiso 2 Nη + ψiso 2 + (ψiso 1 ψiso 3 )1/2 (Nη)1/2 + (ψiso 3 )3/8 + (ψiso 4 )3/8 (Nη)3/16 The bound given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 is an immediate consequence after a trivial Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Contributions from higher order cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The goal of the present section is to estimate the terms originating from higher order cumulants in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='54), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In order to do so, we assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For any J,J∗ ⊂ Z2 ≥0 ∖ {(0,0)}, l ∈ Z2 ≥0 with ∣l∣ + ∑(J ∪ J∗) ≥ 2 it holds that (Ξav 1 ) 1/(1+∑(J∪J∗)) ≺ 1 Nη1/2 (1 + ψiso 1 (Nη)1/2 + (ψiso 2 )1/4 (Nη)1/8 ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68a) (Ξiso 1 ) 1/(1+∑(J∪J∗)) ≺ 1 √ Nη2 (1 + ψiso 1 (Nη)1/2 + (ψiso 2 )1/4 (Nη)1/8 ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68b) (Ξav 2 ) 1/(1+∑(J∪J∗)) ≺ 1 Nη (1 + (ψiso 1 )2 Nη + ψiso 2 (Nη)1/2 + (ψiso 3 )3/8 + (ψiso 4 )3/8 (Nη)3/16 ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68c) (Ξiso 2 ) 1/(1+∑(J∪J∗)) ≺ 1 √ Nη3 (1 + (ψiso 1 )2 Nη + ψiso 2 (Nη)1/2 + (ψiso 3 )3/8 + (ψiso 4 )3/8 (Nη)3/16 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68d) For k = 1,2, l ∈ Z2 ≥0 and a multiset J ⊂ Z2 ≥0 ∖ {(0,0) } we now define slightly (notationally) simplified versions of Ξav/iso k , namely Ξav k (l,J) ∶= N −(∣l∣+∑ J+3)/2 ∑ ab ∣∂l((GA)k−1GA′)ba∣ ∏ j∈J ∣∂j ⟨(GA)k⟩∣ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='69) Ξiso k (l,J) ∶= N −(∣l∣+∑ J+1)/2 ∑ ab ∣∂l[(GA)xa(G(AG)k−1)by]∣ ∏ j∈J ∣∂j((GA)kG)xy∣, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='70) where ∑ J ∶= ∑j∈J∣j∣, ∣(j1,j2)∣ ∶= j1 + j2 and ∂(j1,j2) ∶= ∂j1 ab∂j2 ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Here, for notational simplicity, we do not carry the dependence on the spectral parameters of the resolvents but assume that implicitly each resolvent has its own spectral parameter and that each A is correctly regularised with respect to its neighboring resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular compared to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='55), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='63), it is not necessary to distinguish the sets J,J∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Throughout the proof, we denote φk ∶= ψiso k /√Nη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The naive estimate for the derivatives simply is ∣∂l((GA)k−1GA′)ba∣ ≺ η−(k−1)/2(1 + φk−1) , ∣∂j ⟨GA⟩∣ ≺ 1 Nηk/2 ∑ k1+k2+⋯=k ∏ i (1 + φki) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='71) due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) and recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='71) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='69) we obtain ∣Ξav 1 ∣ ≺ (Nη1/2)−1−∣J∣N (2−∣l∣−∑ J)√ Nη (1 + φ1) ∣J∣ , ∣Ξav 2 ∣ ≺ (Nη)−1−∣J∣N (2−∣l∣−∑ J)√ Nη (1 + φ1)(1 + φ2 + φ2 1) ∣J∣ , ∣Ξiso 1 ∣ ≺ ( √ Nη)−1−∣J∣η1+∣J∣/2N (4−∣l∣+∣J∣−∑ J)/2(1 + φ1) ∣J∣ , ∣Ξiso 2 ∣ ≺ ( √ Nη3/2)−1−∣J∣η1+∣J∣/2N (4−∣l∣+∣J∣−∑ J)/2(1 + φ1)(1 + φ2 + φ2 1) ∣J∣ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='72) 42 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES and therefore have proved (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68a) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68c) in all cases except ∣l∣ + ∑ J = 2 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68b) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68d) in all cases except ∣l∣ + ∑ J − ∣J∣ < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the remaining cases we need a more refined estimate using the following Ward lemma: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let x be any deterministic vector of bounded norm, let w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ,wk ∈ D(ǫ0,κ0) ℓ+1 be spectral parameters and A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ,Ak deterministic matrices of bounded norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then for Gi = G(wi) it holds that 1 N ∑ a ∣(G1 ˚ Aw1,w2 1 ⋯˚ Awk−1,wk k−1 GkAk)xa∣ ≺ 1 √Nη 1 η(k−1)/2 (1 + φ1 + ⋯ + φ2k) 1/2 , which improves upon the term-wise bound by a factor of (Nη)−1/2 at the expense of replacing 1 + φk by 1 + √φ1 + ⋯ + φ2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The proof of the above Ward lemma is largely based on yet another more general estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let x,y be normalised vectors, let w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ,wk+1 ∈ D(ǫ0,κ0) ℓ+1 be spectral parameters and A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ,Ak be deterministic matrices of bounded norm such that a of them are regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ˚ Awi,wi+1 i = Ai for all i ∈ I for some I ⊂ [k] of cardinality a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then with Gi = G(wi) it holds that ∣(G1A1G2⋯AkGk+1)xy∣ ≺ 1 ηk−a/2 (1 + φ1 + ⋯ + φa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='73) We defer the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12 to the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By Cauchy-Schwarz and the norm bound on the middle Ak we have ( 1 N ∑ a ∣(G1 ˚ Aw1,w2 1 ⋯˚ Awk−1,wk k−1 GkAk)xa∣) 2 ≲ 1 N (G1 ˚ Aw1,w2 1 ⋯˚ Awk−1,wk k−1 GkG∗ k ˚ A ¯ wk, ¯ wk−1 k−1 ⋯˚ A ¯ w2, ¯ w1) 1 G∗ 1) xx ≺ 1 Nηk (1 + φ1 + ⋯ + φ2k) due to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12 for 2k resolvents and a = 2k − 2 regularised A-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ The rest of the proof is split into several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Treatment of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68a) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68c) for ∣l∣ + ∑J = 2: For the case ∣l∣ + ∑ J = 2 we either have ∣l∣ ∈ {0,2 } or ∑J = 1 = ∣J∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the former case an off-diagonal resolvent is guaranteed to be present in the first factor of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='69) (by parity) and in the latter case the second factor consists of a single off-diagonal resolvent chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In either case we may use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11 to gain a factor of 1/√Nη compared to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='71) and obtain ∣Ξav 1 ∣ ≺ (Nη1/2)−1−∣J∣(1 + φ1)(∣J∣−1)+(1 + φ1 + 1(∣J∣ ≥ 1)φ1/2 2 ) , ∣Ξav 2 ∣ ≺ (Nη)−1−∣J∣(1 + φ2 1 + φ2)(∣J∣−1)+(1 + φ3 1 + φ3/2 2 + 1(∣J∣ ≥ 1)(φ3 + φ4)3/4), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='74) where we used the fact that for ∣J∣ = 0 only a single factor of (1 + φ1) needs to be replaced by a factor of (1 + (φ1 + φ2)1/2) for Ξav 2 and no factor needs to be replaced for Ξav 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we used φ1(φ3 + φ4)1/2 + φ2 1φ1/2 2 ≲ φ3 1 + φ3/2 2 + (φ3 + φ4)3/4 by a simple Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='74) implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68a) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68c) by another simple Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Treatment of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68b) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68d) for ∣l∣+∑J −∣J∣ ∈ {2,3 }: In this case we can simply use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11 for the two resolvent chains in the first factor of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='70) involving x,y to gain a factor of (Nη)−1 compared to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='71) at the expense of replacing 1 + φ1 by 1 + φ1/2 1 + φ1/2 2 in case of Ξiso 2 which proves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68b) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68d) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Treatment of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68b) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68d) for ∣l∣+∑J −∣J∣ = 0: In this case we necessarily have ∣l∣ = 0 and ∣J∣ ≥ 2 and ∣j∣ = 1 for all j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular all factors of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='70) consist of two resolvent chains evaluated in EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 43 (x,a),(y,b) or (x,b),(y,a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This allows to use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11 four times (twice for the a- and twice for the b-summation) to gain a factor of (Nη)−2 comparedto (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='71) at the expense of replacing one factor of (1 + φ1) by (1 + (φ1 + φ2)1/2) in case of Ξiso 1 and one factor of (1 + φ1)(1 + φ2 1 + φ2) by (1 + (φ1 + φ2)1/2)(1 + φ1 + φ2 + (φ3 + φ4)1/2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='75) in case of Ξiso 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This concludes the proof in case of Ξiso 1 and together with (1 + (φ1 + φ2)1/2)(1 + φ1 + φ2 + (φ3 + φ4)1/2) ≲ 1 + (φ1 + φ2)3/2 + (φ3 + φ4)3/4 also in case of Ξiso 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Treatment of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68b) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='68d) for ∣l∣ + ∑J − ∣J∣ = 1: In this case we necessarily have ∣J∣ ≥ 1 and either ∣l∣ = 0 or ∣j∣ = 1 for all j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In either case we can use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11 twice for the first factor and once for some other factor in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='70) to gain a factor of (Nη)−3/2 compared to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='71) at the expense of replacing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='75) in case of Ξiso 1 and one factor of (1+φ1)(1+φ2 1+φ2) by (1+(φ1+φ2)1/2)((1+φ1)(1+φ1+φ2)1/2+(φ3+φ4)1/2) in case of Ξiso 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Together with (1 + (φ1 + φ2)1/2)((1 + φ1)(1 + φ1 + φ2)1/2 + (φ3 + φ4)1/2) ≲ 1 + (φ3 + φ4)3/4 + φ3/2 2 + φ2 1 this concludes the proof also in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ It remains to give the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The proof is via induction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' we assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='73) has been established for re- solvent chains of up to k resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For k + 1 resolvents and a = k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' in case when all deterministic matrices are regular, the claim follow by definition of ψiso k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Therefore we may assume that some Aj is not regular which we decompose into its regular component ˚ A wj,wj+1 j and a linear combination of E±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By linearity it thus suffices to check (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='73) for the cases Aj = E±, and moreover, by chiral symmetry GjE−Gj+1 = −E−G(−wj)E+Gj+1 and ˚ Awj−1,wjE− = ˚ Awj−1,−wj (recall Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) the estimate for E− follows from the estimate for E+ upon replacing wj by −wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Therefore it suffices to check (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='73) in case Aj = E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' If sj = −sgn(Im wjImwj+1) = +, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the adjacent spectral parameters lie in opposite half-planes, then we use the resolvent identity to write Aj−1GjE+Gj+1Aj+1Gj+2 = Aj−1 Gj − Gj+1 wj − wj+1 Aj+1Gj+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We discuss each of the two resulting summands separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the summand involving Gj+1, if Aj−1 was not counted as regularised, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' j−1 /∈ I, then the claim followsby induction and the trivial estimate ∣wj − wj+1∣ ≥ η since k has been reduced by one, while a has been preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' On the other hand, if Aj−1 was correctly regularised, then we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 to write ˚ A wj−1,wj j−1 = ˚ A wj−1, ¯ wj j−1 = ˚ A wj−1,wj+1 j−1 + O(∣ ¯wj − wj+1∣)E+ + O(∣ ¯wj − wj+1∣)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='76) Inserting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='76) into Aj−1Gj+1Aj+1Gj+2/(wj −wj+1) the claimed bound followsfrom induction since for the ˚ A wj−1,wj+1 j−1 term a has been preserved and k has been reduced by one compensating for ∣wj − wj+1∣ ≥ η, while for E± both k,a have been reduced by one and ∣ ¯wj − wj+1∣/∣wj − wj+1∣ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, for the summand involving Gj, the argument is completely analogous, apart from the two error terms in ˚ A wj,wj+1 j+1 = ˚ A wj,wj+2 j+1 + O(∣wj − ¯wj+1∣ + ∣wj − sj+1wj+2∣)Esj+1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='77) + O(∣wj − ¯wj+1∣ + ∣wj + sj+1 ¯wj+2∣)E−sj+1 , appearing for an Aj+1 = ˚ A wj+1,wj+2 j+1 , which has been correctly regularised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Here, we applied Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and denoted, as usual, sj+1 = −sgn(Im wj+1Imwj+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, for the error terms, we assume that the 44 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES second summand in each O(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=') is non-zero (otherwise we are back to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='76)) and argue by induction: Indeed, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) and applying a resolvent identity, we find ∣wj − ¯wj+1∣ + ∣wj − sj+1wj+2∣ wj − wj+1 GjEsj+1Gj+2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='78) = ∣wj − ¯wj+1∣ + ∣wj − sj+1wj+2∣ (wj − wj+1)(wj − sj+1wj+2)sj+1(G(wj) − G(sj+1wj+2))Esj+1 , such that, in the resulting chain we have reduced k by two and a by one, and the prefactor in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='78) is bounded by 1/η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The argument for the second error in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='77) is completely analogous, after realizing that (∣wj − ¯wj+1∣ + ∣wj + sj+1 ¯wj+2∣)/(∣wj − wj+1∣∣wj + sj+1wj+2∣) ≤ 1/η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' On the contrary, if sj = −sgn(ImwjImwj+1) = −, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the adjacent spectral parameters lie the same half-plane (without loss of generality the upper one), then we use the integral representation from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 to write Aj−1GjE+Gj+1Aj+1 = 1 2πi ∫Γ Aj−1G(z)Aj+1 (z − wj)(z − wj+1) dz , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='79) where Γ is an appropriately chosen contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' If j − 1,j + 1 /∈ I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' both Aj−1,Aj+1 were not counted as regularised, then the claim follows by induction and estimating the integral by η−1 (up to log factors) since k has been reduced by one, and a has been preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' On the other hand, if both Aj−1,Aj+1 were counted as regularised, then we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 to write them as ˚ A wj−1,wj j−1 = ˚ A wj−1,z j−1 + O(∣wj − z∣)E+ + O(∣wj − z∣)E− , ˚ A wj+1,wj+2 j+1 = ˚ A z,wj+2 j+1 + O(∣wj+1 − z∣)E+ + +O(∣wj+1 − z∣)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='80) The resulting term with ˚ A wj−1,z j−1 , ˚ A z,wj+2 j can be estimated by induction since k has been reduced by one, a has been preserved and the integral may be estimated by η−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The other terms with either one or two E± can also be estimated by induction since the integral is at most logarithmically divergent, k has been reduced by one and a by at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, if in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='79) one of Aj−1,Aj+1 were counted as regularised, then we use the relevant expansion from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='80), so that for the resulting term with ˚ A, k has been reduced by one, and a has been preserved, so that the η−1 estimate on the integral is affordable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The other term with E± can also be estimated by induction with both a,k reduced by one, and the integral being at most logarithmically divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of the reduction inequalities, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9 During the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9, we will heavily rely on the following integral representation for the absolute value ∣G∣ of a resolvent (see also [28, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Integral representation for the absolute value of a resolvent) Let w = e + iη ∈ C ∖ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then the absolute value of the resolvent G(w) can be represented as ∣G(e + iη)∣ = 2 π ∫ ∞ 0 ImG(e + i √ η2 + s2) ds √ η2 + s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This immediately follows from the functional calculus for H and the identity 1 ∣x − iη∣ = 1 iπ ∫ ∞ 0 ( 1 x − i(η2 + s2)1/2 − 1 x + i(η2 + s2)1/2 ) ds √ η2 + s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To keep the notation simpler within this proof we may often denote Ai = ˚ Ai = ˚ Awi,wi+1 i , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' sometimes we drop the spectral parameters wi = ei + iηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We start with the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25), for which, similarly to [28, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6], we get Ψav 4 ≲ Nη + N 2η2 (⟨∣G1∣A1∣G2∣A∗ 1⟩⟨∣G2∣A2∣G3∣A∗ 2⟩⟨∣G3∣A3∣G4∣A∗ 3⟩⟨∣G4∣A4∣G1∣A∗ 4⟩) 1/2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 45 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, spectral decomposition, and a Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, we use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) to write ⟨∣G1∣A1∣G2∣A∗ 1⟩ = 4 π2 ∬ ∞ 0 ⟨ImG(w1,s)˚ Aw1,w2 1 ImG(w2,t)(˚ Aw1,w2 1 )∗⟩ dsdt √ η2 1 + s2√ η2 2 + t2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) where we defined wi,s ∶= ei + i √ η2 i + s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The very large s,t–regimes in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) can be easily shown to be negligible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' see [28, Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' even if not stated explicitly we assume that the upper integration limit can be replaced by N 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Additionally, we can restrict to the case when η ∶= minj ∣Imwj∣ ≤ 1, when this is not the case we use the local law in the regime η > 1 from Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 (see [28, Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1] for a detailed argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We remark that this argument is not circular since in the proof of the local law for η > 1 sketched below Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5 one does not use the reduction inequalities in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In order to estimate the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) we write ImG = 1 2i(G − G∗) for both ImG to obtain four terms with two resolvents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' to keep the presentation concise we only present the estimate for one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' From now on we only consider only the term ⟨∣G1∣A1∣G2∣A∗ 1⟩, the bound for all the other terms in the last line of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) is completely analogous and so omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the following we will often use the approximations from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 (omitting the trivial ∧1 in the errors for notational simplicity): ˚ Aw1,w2 = ˚ Aw1,s,w2,t + O(∣ √ η2 1 + s2 − η1∣ + ∣ √ η2 2 + t2 − η2∣)E+ + O(∣ √ η2 1 + s2 − η1∣ + ∣ √ η2 2 + t2 − η2∣)E− , (˚ Aw1,w2)∗ = (˚ A∗)w2,t,w1,s + O(∣e1 − e2∣ + √ η2 1 + s2 + √ η2 2 + t2)E+ + O(∣e1 + e2∣ + √ η2 1 + s2 + √ η2 2 + t2)E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) We point out that when taking the adjoint of the first formula to arrive at the second we used that for any w1,w2 it holds (˚ Aw1,w2)∗ = (˚ A∗)w2,w1, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Recall that within this proof we always assume that η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' From now on for the error terms we will always use the bounds ∣ √ η2 1 + s2 − η1∣ ≲ s , √ η2 1 + s2 ≤ η1 + s , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) and a similar bound with η1,s replaced with η2,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The first bound is not optimal for small η1, but good enough for our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) we write ∬ ∞ 0 ⟨G(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s)˚ Aw1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w2 1 G(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t)(˚ Aw1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w2 1 )∗⟩ dsdt √ η2 1 + s2√ η2 2 + t2 = ∬ ∞ 0 ⟨G(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s)˚ A w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t 1 G(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t)(˚ A∗ 1)w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s⟩ dsdt √ η2 1 + s2√ η2 2 + t2 + ∑ σ∈{+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='−} ∬ ∞ 0 ⟨G(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s)EσG(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t)(˚ A∗ 1)w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s⟩O(η1 + η2 + s + t) dsdt √ η2 1 + s2√ η2 2 + t2 + ∑ σ∈{+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='−} ∬ ∞ 0 ⟨G(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s)˚ A w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t 1 G(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t)Eσ⟩O(η1 + η2 + s + t) dsdt √ η2 1 + s2√ η2 2 + t2 + ∑ σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='τ∈{+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='−}∬ ∞ 0 ⟨G(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s)EσG(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t)Eτ⟩O(η2 1 + η2 2 + s2 + t2) dsdt √ η2 1 + s2√ η2 2 + t2 + ∬ ∞ 0 ⟨G(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s)[ ∑ σ Oσ(∣e1 − σe2∣)Eσ]G(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t)(˚ A∗ 1)w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s⟩ dsdt √ η2 1 + s2√ η2 2 + t2 + ∬ ∞ 0 ⟨G(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s)˚ A w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t 1 G(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t)[O(∣e1 − e2∣)E+ + O(∣e1 + e2∣)E−]⟩ dsdt √ η2 1 + s2√ η2 2 + t2 + ∬ ∞ 0 ⟨G(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s)[ ∑ σ Oσ(∣e1 − σe2∣)Eσ]G(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t)[∑ τ Oτ(∣e1 − τe2∣)Eτ]⟩ dsdt √ η2 1 + s2√ η2 2 + t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) 46 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES We now estimate the terms in the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the following estimates we will always omit log N-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We start with ����������� ∬ ∞ 0 ⟨G(w1,s)˚ A w1,s,w2,s 1 G(w2,t)(˚ A∗ 1)w2,t,w1,s⟩ dsdt √ η2 1 + s2√ η2 2 + t2 ����������� ≺ 1 + ψav 2 Nη , which readily follows by the definition of Ψav 2 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and from the assumption Ψav 2 ≺ ψav 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the third to the fifth line in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) we use the bound ����������� ∬ ∞ 0 ⟨G(w1,s)EσG(w2,t)B⟩O(η1 + η2 + s + t) dsdt √ η2 1 + s2√ η2 2 + t2 ����������� ≺ ∬ ∞ 0 ⎛ ⎝ 1 √ η2 1 + s2 ∧ 1 √ η2 2 + t2 ⎞ ⎠ [η1 + η2 + s + t] dsdt √ η2 1 + s2√ η2 2 + t2 ≲ 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) for any deterministic norm bounded matrices B and for σ ∈ {+,−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the fifth line of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) we used the bound (s2 + t2) ∧ 1 ≤ (s + t) ∧ 1 (recall that ∧1 is omitted in the error terms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) for notational simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that here we used: ∣⟨G(w1,s)EσG(w2,t)B⟩∣ ≺ 1 √ η2 1 + s2 ∧ 1 √ η2 2 + t2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) which holds uniformly in matrices with ∥B∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We point out that to obtain the bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) we used spectral decomposition of the resolvents and that ⟨wi,Eσwj⟩ = δi,σj to bound ∣⟨G(w1,s)EσG(w2,t)B⟩∣ = ∣ 1 2N ∑ i ⟨wi,Bwσi⟩ (λi − w1,s)(λi − σw2,t)∣ ≲ 1 N ∑ i 1 ∣λi − w1,s∣∣λi − σw2,t∣ ≺ 1 ∣Imw1,s∣ ∨ ∣Imw2,t∣ , where in the last inequality we used the single resolvent local law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, for the last three lines in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) we use that for any norm bounded matrix B, by resolvent identity, we have (recall that E+ = I) ∣⟨G(w1,s)BG(w2,t)⟩∣ ≺ 1 ∣w1,s − w2,t∣ , ∣⟨G(w1,s)BG(w2,t)E−⟩∣ ≺ 1 ∣w1,s + w2,t∣ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) which after the integration in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) gives a bound of order one, as a consequence of ∣e1 ± e2∣ ∣w1,s ± w2,t∣ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that here it is important that the error terms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) involving ∣e1 −e2∣ are always multiplied with the matrix E+, while errors of order ∣e1 + e2∣ are in the direction of E−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining the computations in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) we conclude that ∣⟨∣G1∣A1∣G2∣A∗ 1⟩∣ ≺ 1 + ψav 2 Nη , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) which, after plugging it in the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2), clearly implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26) for Ψiso 3 , we find Ψiso 3 ≲ √ Nη + Nη2((G1A1∣G2∣A∗ 1G∗ 1)xx(G∗ 4A∗ 3∣G3∣A3G4)yy⟨∣G2∣A2∣G3∣A∗ 2⟩) 1/2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) again by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, spectral decomposition, and a Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, using again the integral representation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), we find that (G1A1∣G2∣A∗ 1G∗ 1)xx = 2 π ∫ ∞ 0 (G1A1ImG(w2,s)A∗ 1G∗ 1)xx ds √ η2 2 + s2 , EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 47 recalling the notation w2,s = e2 + i √ η2 2 + s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The estimate for this term is fairly similar to the one in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3), hence we present only the main differences and skip the details;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' actually the current case is easier since we now have only one ∣G∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' After splitting ImG = 1 2i(G − G∗) and handling both terms separately, we can write, similarly to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) and using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5), the following approximation: ∫ ∞ 0 (G1A1G(w2,s)A∗ 1G∗ 1)xx ds √ η2 2 + s2 = ∫ ∞ 0 (G1 ˚ A w1,w2,s 1 G(w2,s)(˚ A∗ 1)w2,s,w1G∗ 1)xx ds √ η2 2 + s2 + E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) Here E is an error coming from all the errors in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the first term in the second line of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) we use the bound ����������� ∫ ∞ 0 (G1 ˚ A w1,w2,s 1 G(w2,s)(˚ A∗ 1)w2,s,w1G∗ 1)xx ds √ η2 2 + s2 ����������� ≺ 1 η (1 + ψiso 2 √Nη ) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) which follows by the definition of Ψiso 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the error term we do not write the details, since once we replace (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) with (here B,B1,B2 are deterministic norm bounded matrices) ∣(G1B1G(w2,s)B2G∗ 1)xx∣ ≤ (G1B1B∗ 1G∗ 1) 1/2 xx (G1B∗ 2G(w2,s)G(w2,s)∗B2G∗ 1) 1/2 xx ≺ 1 η √ η2 2 + s2 ∣(G1EσG(w2,s)BG∗ 1)xx∣ ≺ 1 η∣w1 − w2,s∣ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) respectively, the estimate ∣E∣ ≺ 1 η (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) follows completely analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The estimates (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) follow by repeated applications of the resolvent identity (after commuting Eσ with G in case of the second formula), the trivial bound ∥G∥ ≤ 1/η and the single resolvent local law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) we conclude ∣(G1A1∣G2∣A∗ 1G∗ 1)xx∣ ≺ 1 η (1 + ψiso 2 √Nη ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) The bound in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), together with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) to estimate the averaged term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), concludes the proof (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26) for Ψiso 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Analogously to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), for Ψiso 4 we find that Ψiso 4 ≲ √ Nη + Nη5/2((G1A1∣G2∣A∗ 1G∗ 1)xx(G∗ 5A∗ 4∣G4∣A4G5)yy⟨∣G2∣A2G3A3∣G4∣A∗ 3G∗ 3A∗ 2⟩) 1/2 ≲ √ Nη + N 3/2η5/2((G1A1∣G2∣A∗ 1G∗ 1)xx(G∗ 5A∗ 4∣G4∣A4G5)yy) 1/2 × (⟨∣G2∣A2∣G3∣A∗ 2⟩⟨∣G3∣A3∣G4∣A∗ 3⟩⟨∣G4∣A∗ 3∣G3∣A3⟩⟨∣G3∣A∗ 2∣G2∣A2⟩) 1/4 where in the last inequality we used spectral decomposition and a bound as in [28, Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6] to bound the trace with four G’s and four A’s in terms of a product of traces containing only two G’s and two A’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, using the bounds (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), we conclude the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26) for Ψiso 4 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Properties of the MDE and the stability operator: Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 In the first part of this appendix, we derive several elementary properties of the MDE − 1 M = w − ˆΛ + S[M], w ∈ C ∖ R, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) 48 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19)) and its unique solution M (under the usual constraint Im M⋅Imw > 0) where the operator S was given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) and ˆΛ ∈ C2N×2N is from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Afterwards, in the second part, we turn to the associated two-body stability operator B ≡ B(w1,w2) ∶= 1 − M(w1)S[⋅]M(w2) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and its adjoint B∗, understood with respect to the standard (normalised) inner product ⟨S,T ⟩ ∶= ⟨S∗T ⟩ for S,T ∈ C2N×2N, which is given by B∗ ≡ B∗(w1,w2) ∶= 1 − S[(M(w1))∗ ⋅ (M(w2))∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) Moreover, we also explain the relation between the regularisation from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 and the stability operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, after proving and combining LemmasA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 on M and B, respectively, we will complete the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The Matrix Dyson Equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) and its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Existence and uniqueness of the solution to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) under the constraint ImM ⋅Imw > 0 has already been shown in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By [2, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1], this solution can also be represented as the Stieltjes transform of a compactly supported semi-definite matrix-valued probability measure on R, which has the immediate consequence that ∥M(w)∥ ≤ ∣Imw∣−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let M be the unique solution to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) and write its 2 × 2-block representation as M = (M11 M12 M21 M22) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) Then we have the following: (a) The average trace ⟨M⟩ coincides with the solution m of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4), ⟨M(w)⟩ = m(w), and the blocks in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We have M ∗(w) = M( ¯w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (b) The solution has a continuous extension to the real line from the upper half plane, denoted by M(e) ∶= limη↓0 M(e + iη);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the limit from the lower half plane is M ∗(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The self-consistent density of states of the MDE, defined as ρ(e) = 1 π ⟨ImM(e)⟩, is identical to the free convolution of µˆΛ ⊞ µsc from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Both ρ and its Stieltjes transform m are Hölder continuous with a small universal exponent c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ∣ρ(e1) − ρ(e2)∣ ≤ C∣e1 − e2∣c, e1,e2 ∈ R, and ∣m(w1) − m(w2)∣ ≤ C′∣w1 − w2∣c, w1,w2 ∈ C+, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) where C,C′ depend only on ∥Λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (c) We have the chiral symmetry M(w)E− = −E−M(−w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) In particular, for purely imaginary spectral parameter, w = iImw, it holds that m = iImm as well as M11 = iImM11 and M22 = iImM22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, the off-diagonal blocks of ImM are vanishing on the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (d) Fix κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For any spectral parameter in the κ-bulk, w ∈ C ∖ R with Rew ∈ Bκ, we have ∥M(w)∥ ≤ C(κ,∥Λ∥) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) for some constant depending only on κ and an upper bound on the norm ∥Λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, ρ(e) is real analytic on Bκ with derivatives controlled uniformly max{∣∂kρ(e)∣ ∶ e ∈ Bκ} ≤ C(k,κ,∥Λ∥) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) with a constant C(k,κ,∥Λ∥) for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For part (a), a direct computation shows that M from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) with the blocks given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) indeed solves (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) if m is replaced with ⟨M⟩ in these formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The calculation uses the simple ob- servation that ⟨M11⟩ = ⟨M22⟩ from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18), hence S[M] = ⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Furthermore, the MDE also implies that ⟨M⟩ solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4), but this equation has a unique solution by the theory of free convolutions with a EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 49 semicircular density, hence m = ⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally M ∗(w) = M( ¯w) follows from ¯m(w) = m( ¯w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This proves (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For part (b), since S[M] = ⟨M⟩, we observe that M solves − 1 M = w − ˆΛ + ⟨M⟩, which is exactly the MDE for a deformed Wigner matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14 The point is that the Hermitised H from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) does not satisfy the uniform lower bound in the flatness condition on the self-energy operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' S[T ] ≥ c⟨T ⟩ does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Nevertheless, for the purpose of computing M we can replace H with the deformed Wigner model W +ˆΛ with self-energy given S[T ] = ⟨T ⟩ and which is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thus we can use several results from the analysis of the MDE with flatness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The Hölder- continuity of the scDos was proven in [2, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2], which easily extends to the Hölder-continuity of its Stieltjes transform m, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' [1, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular ⟨M(w)⟩ extends continuously to the real line and thus the scDos ρ(e) ∶= 1 π ⟨ImM(e)⟩ = 1 π Imm(e) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since it has the same Stieltjes transform as the free convolution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) by part (a), we proved that the scDos defined via MDE is the same as the free convolution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The continuous extension of M (and not only its trace) requires an additional argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For any open interval I ∈ R define ∥M∥I ∶= sup{∥M(e + iη)∥ ∶ e ∈ I,η > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Suppose for some open I ∈ R we have ∥M∥I < ∞, then we have the Lipschitz continuity ∥M(w1) − M(w2)∥ ≤ ∥M∥2 I∣w1 − w2∣, Rew1,Re w2 ∈ I following from the resolvent identity applied to M(w) = (ˆΛ − w − m)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thus M(w) continuously extends to any e ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' So the key question for the extension (and for many other results on the MDE) is the boundedness ∥M∥I < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the bulk spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for any e ∈ R with ρ(e) > 0, we can use the bound ∥M(w)∥ ≤ ∣Imm(w) + Im w∣−1 that is obtained by taking the imaginary part of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), yielding ImM = (Imw + ⟨ImM⟩)MM ∗ , andusing∥MM ∗∥ = ∥M∥2 and∥Im M∥ ≤ ∥M∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Bythe Höldercontinuity(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) insmallneighborhood I of e (whose size depend on the lower bound on ρ(e)) we obtain ∥M∥I ≲ ρ(e)−2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thus M continuously extends to I with the same bound and it is locally Lipschitz continuous with a Lipschitz constant of order ρ(e)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the entire κ-bulk this extension is controlled by a constant depending only on κ and ∥Λ∥ (via (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This proves (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Near the spectral edges we have only an N-independent upper bound for ∥M∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using the spectral decomposition of ˆΛ with eigenvalues νi and normalised eigenvectors yi, i ∈ ±[N], we have M(w) = ∑ i ∣yi⟩⟨yi∣ νi − w − m(w), thus ∥M(w)∥ ≤ 2N mini ∣νi − w − m(w)∣ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) On the other hand the imaginary part of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) implies Imm = 1 2N ∑ i Imm + Imw ∣νi − w − m∣2 thus 1 2N ∑ i 1 ∣νi − w − m∣2 = Imm Im m + Imw ≤ 1 so ∣νi − w − m∣ ≥ 1/ √ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) this gives the uniform bound ∥M(w)∥ ≤ (2N)3/2, w ∈ C ∖ R, 14That is, a matrix H = W + ˆΛ, where W is a Hermitian matrix with normalised i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (up to the symmetry) entries of variance 1/(2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 50 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES which guarantees the continuous extension of M to the real line with a uniform Lipschitz constant (2N)3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As we have seen, in the bulk this regularity can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 15 For part (c), the symmetry ρ(e) = ρ(−e) immediately implies the symmetry m(w) = −m(−w) for its Stieltjes transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) is an immediate consequences of the formulas (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, for part (d), the bound (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) was already proven above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The real analyticity of ρ and m in the bulk with the bounds on the derivative (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) follows from taking derivatives in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) and using again the lower bound on Imm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Finally, we prove some regularity property of the κ-bulk, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let 0 < κ′ < κ be two small constants, then dist(∂Bκ′,Bκ) ≥ c(κ − κ′) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) with some N-independent constant c = c(∥Λ∥) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, Bκ is a finite union of disjoint compact intervals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the number of these components depends only on κ and ∥Λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As in the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, we interpret Bκ as the κ-bulk of the deformed Wigner matrix W + ˆΛ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' a model with the flatness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The statement on the number of components directly follows from the real analyticity of ρ and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The same argument would also imply(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) with a constant c that depends on κ and an upper bound on ∥Λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To remove the κ-dependence, we need to use the detailed shape analysis for ρ from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, the flatness condition and ∥M∥I < C(κ) for any interval I ⊂ Bκ (equivalent to [4, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16)]) implies that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5 in [4] holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Therefore Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 in [4] applies to our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This theorem says that in the regime where ρ is small, it is approximately given by explicit 1/3-Hölder continuous functions, moreover ρ itself is 1/3-Hölder continuous with Hölder constant depending only on the so- called model parameters of the problem, which in our case is just an upper bound on Λ (note that [4] was written for much more complicated self-energy operators to include the MDE analysis for random matrices with correlated entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Noticing the κ1/3 power in the definition of Bκ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), this means that the boundaries of Bκ are Lipschitz continuous functions of κ when κ is small with a Lipschitz constant depending only on an upper bound on ∥Λ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that the proof of the independence of c = c(∥Λ∥) of κ required a much more sophisticated analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, for our main proof, c = c(κ,∥Λ∥) > 0 in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) is sufficient, note that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) is only used in choosing δ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' More precisely, for fixed L = L(ǫ) and κ0 > 0, given the family (ℓκ0)ℓ∈[L] of parameters for the domains D(ǫ0,κ0) ℓ , we would have that dist(∂B(ℓ−1)κ0,Bℓκ0) ≥ c(ℓκ0,∥Λ∥)κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, the cutoff parameter δ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) is chosen much smaller than c(ℓκ0,∥Λ∥)κ0 for every ℓ ≤ L(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The stability operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and its spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Throughout the entire paper, the two- body stability operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and its adjoint (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' These operators depend on two (a priori) different spectral parameters w1,w2 via the solutions M1 = M(w1) and M2 = M(w2) of the MDE (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For these solutions, we have the following basic lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let w1,w2 ∈ C ∖ R be two spectral parameters and M1 = M(w1),M2 = M(w2) the corresponding solutions to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (a) Then we have the M-Ward identity, M1 − M2 = [(w1 − w2) + (⟨M1⟩ − ⟨M2⟩)]M2M1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) In particular, M1 and M2 commute and it holds that (1 − ⟨MM ∗⟩) ⟨ImM⟩ = Imw ⟨MM ∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) 15We remark that under some extra condition on Λ further improvements away from the bulk are possible for m but not for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For example, if the singular values νi of Λ are 1/2-Hölder continuous in the sense that ∣νi − νj∣ ≤ C0[∣i − j∣/N]1/2, then m is also uniformly bounded and 1/3-Hölder continuous with a constant depending on C0, see Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 51 (b) Fix κ > 0 and let Rew1,Re w2 ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for Imw1Imw2 > 0, we have the perturbative estimate ∥M(w1) − M(w2)∥ = O(∣w1 − w2∣ ∧ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Part (a) is an immediate consequence of the MDE (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) using the fact that M = (ˆΛ − (w + m)) −1 is a resolvent of ˆΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The special case (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) with w1 = w and w2 = ¯w, and taking a trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For part (b), we focus on the case of small imaginary parts for the spectral parameters (the comple- mentary regime being trivial) and use that M is analytic away from the real axis and differentiate (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' w, such that we find ∂wM = 1 1 − ⟨M 2⟩M 2 by means of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12), the denominator is lower bounded as ∣1 − ⟨M 2⟩∣ = ∣(1 − ⟨MM ∗⟩) − 2i⟨MIm M⟩∣ ≥ 2∣⟨(ImM)2⟩∣ ≥ 2⟨ImM⟩2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) which shows that ∥∂wM∥ ≲ 1 in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now the claim follows from the fundamental theorem of calculus together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Armed with this information, we can now turn to the following lemma, collecting several basic spectral properties stability operator B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Its proof will be given at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let w1,w2 ∈ C ∖ R and M1,M2 be the respective solutions of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (a) The associated two-body stability operator B = 1 − M1S[⋅]M2 has two non-trivial eigenvalues β± (the other (2N)2 − 2 are equal to one), given by β± = 1 ∓ ⟨M1E±M2E±⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) The corresponding right- and left-eigenvectors B[R±] = β±R± , B∗[L∗ ±] = ¯β±L∗ ± , take the explicit form R± = M1E±M2 , L± = E± , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) up to a normalisation ensuring that ⟨L±,R±⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (b) The eigenvalues (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) can be lower bounded as ∣β±∣ ≳ (∣Rew1 ∓ Rew2∣ + ∣Imw1∣ + ∣Imw2∣) ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) In particular, the inverse stability operator B−1 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (c) Fix κ > 0 and denote s ∶= −sgn(Imw1 Imw2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for Rew1,Re w2 ∈ Bκ, we have that ∣β−s∣ ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By the last item, given s ∶= −sgn(Imw1 Imw2), we will always refer to (β ∶= 1 − s⟨M1EsM2Es⟩, R ∶= M1EsM2 , L ∶= Es) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) as the critical eigentriple (and accordingly β as the critical eigenvalue etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ), consisting of the eigenvalue and the corresponding right- and left-eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, the estimate (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) shows that, if we have (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5)) 1± δ(w1,w2) ∶= φδ(Rew1 ∓ Rew2) φδ(Imw1) φδ(Im w2) = 0 for some δ > 0, then the inverse stability operator B−1 is bounded and none of the eigenvalues β± is really critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the complementary regime, 1± δ(w1,w2) = 1, and Rew1,Re w2 ∈ Bκ, we shall now explain the interplay between the critical eigentriple (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) and the regularisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 52 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let w1,w2 ∈ C ∖ R with Rew1,Rew2 ∈ Bκ for some fixed κ > 0 and denote the relative sign of imaginary parts by s ∶= −sgn(Imw1 Im w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, let M1 = M(w1),M2 = M(w2) be the respective solutions of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) and A ∈ C2N×2N a bounded deterministic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (a) If 1s δ(w1,w2) = 1 for some δ > 0 small enough, the critical left- and right-eigenvectors (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) are normalised as ⟨L,R⟩ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, if 1± δ(w1,w2) = 1, the respective denominator in the regularisation ˚ Aw1,w2 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6)) is bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (b) The operator X12, acting as X12[B] ∶= ((B∗ 12)−1[B∗]) ∗ = (1 − S[M1 ⋅ M2]) −1[B], B ∈ C2N×2N , where B12 ∶= 1−M1S[⋅]M2, is well defined and bounded on the s-regular component ˚ As (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the pair of spectral parameters (w1,w2)) of any bounded A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This means, for ˚ As ∶= A − 1s δ(w1,w2) ⟨M1AM2Es⟩ ⟨M1EsM2Es⟩Es (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) it holds that ∥X12[˚ As]∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, combining Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 (b) with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 (a), Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 (c), and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 (a), we conclude the perturbative statements from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For part (a), similarly to the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5 (c) given below, we focus on the extreme case w2 = s ¯w1, where the critical eigentriple is given by (β = 1 − s⟨M(w1)EsM(s ¯w1)Es⟩, R = M(w1)EsM(s ¯w1), L = Es) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) Now by means of the chiral symmetry M(w1)E− = −E−M(−w1), we readily obtain ⟨L,R⟩ = s⟨M1M ∗ 1 ⟩ = s ⟨ImM1⟩ Imw1 + ⟨ImM1⟩ ∼ 1, where we used(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12)inthe secondstep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thisprincipalnormalisationof orderpersistsaftersmallpertur- bation of w2 around the extreme case, but as long as 1s δ(w1,w2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Our claim for the denominators in the regularisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) follows immediately from the representation in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For part (b), we first note that, by means of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5, the statement is trivial for constellations of spectral parameters w1,w2 satisfying 1s δ(w1,w2) = 0 and we can hence focus on the complementary extreme case 1s δ(w1,w2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then it follows from the explicit form X12[B] = B + ∑ σ σ ⟨M1BM2Eσ⟩ 1 − σ⟨M1EσM2Eσ⟩Eσ and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5 that X12[B] = s 1 β ⟨M1BM2Es⟩Es + O(1)[B], (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) where O(1) is a shorthand notation for a linear operator E ∶ C2N×2N → C2N×2N satisfying ∥E[B]∥ ≲ ∥B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, plugging ˚ As from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) yields the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ It remains to give the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For (a), we first observe that, due to the simple structure of S[⋅], indeed (2N)2 −2 of the (2N)2 eigenvalues of B are equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The expressions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) can be verified by direct computation, invoking Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 in combination with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For (b) with w1 ≠ ±w2, we first find that 1 β± = 1 1 ∓ ⟨M1E±M2E±⟩ = 1 + ⟨M1⟩ ∓ ⟨M2⟩ w1 ∓ w2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) as a consequence of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 (a) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, using that ∣⟨M⟩∣ ≤ ⟨MM ∗⟩1/2 < 1, which follows from MM ∗ = ImM/(Im w + ⟨ImM⟩) (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 (a)), we conclude that ∣β±∣ ≳ ∣Rew1 ∓ Rew2∣ ∧ 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 53 by application of a triangle inequality in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, we estimate min {∣β+∣,∣β−∣} ≥ ∣1 − ⟨M1M ∗ 1 ⟩1/2⟨M2M ∗ 2 ⟩1/2∣ ≳ (∣Imw1∣ + ∣Imw2∣) ∧ 1, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) where in the first step we used ⟨MM ∗⟩ < 1 together with a Schwarz inequality, and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, for (c), we consider the case of small imaginary parts for the spectral parameters (the com- plementary regime being trivial) and focus on the extreme case w1 = −sw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13), we obtain ∣β−s∣ = ∣1 − ⟨M 2 1 ⟩∣ ≥ 2⟨Im M1⟩2 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24) This principal lower bound persists after small perturbations of w2, and the complementary regime can be dealt with by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 In this appendix, we give a short proof of the usual single resolvent local law in the bulk given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the literature, bulk local laws are established under the usual flatness assumption (see [36, Assumption E]) on the self-energy operator S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' c⟨R⟩ ≤ S[R] ≤ C⟨R⟩ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) for some constants c,C > 0 and any positive semi-definite matrix R ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, for our model, the stability operator S[R] = ∑σ σ⟨REσ⟩Eσ violates the lower bound in the flatness condition (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), which is why we need to provide a separate argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The main idea is that lacking of the lower bound in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) is compensated by the orthogonality relation ⟨GE−⟩ = ⟨ME−⟩ = 0 as a consequence of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The following argument heavily relies on [36, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1], where a general high-moment bound on the underlined term in ⟨(G − M)B⟩ = −⟨W GX [B]M⟩ + ⟨G − M⟩⟨(G − M)X [B]M⟩ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and its isotropic counterpart (see (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) below)has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We stress that this estimate from [36] does not require the lower bound in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) for the self-energy operator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As usual, we suppressed the spectral parameter w ∈ C ∖ R satisfying Rew ∈ Bκ for some fixed κ > 0 from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The expansion (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) for an arbitrary deterministic matrix B ∈ C2N×2N has already been established in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15), where we introduced the linear operator X [B] ∶= (1 − S[M ⋅ M]) −1[B] acting on matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For given B, we now decompose it into its (−)-regular and (−)-singular component (see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18), the cutoff function being irrelevant here), B = ˚ B− + ⟨MBME−⟩ ⟨ME−ME−⟩E− , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the second summand, we note that ⟨GE−⟩ = ⟨ME−⟩ = 0, and we can hence focus on the regular component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' assume that B = ˚ B− is (−)-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this case, for a bounded deterministic ∥B∥ ≲ 1 we thus have ∥X [B]∥ ≲ 1 from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' With the high-moment bound on the underlined term from [36, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, part (b)] one can conclude the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 in the averaged case, ∣⟨(G−M)B⟩∣ ≺ (Nη)−1, by a standard bootstrap argument (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', [36, Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the isotropic case, we evaluate (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) for B = 2N ∣y⟩ ⟨x∣, where x,y ∈ C2N are deterministic vectors in with ∥x∥,∥y∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' More precisely, we subtract its (−)-singular component (which can be dealt with separately as explained above) and insert B = ˚ B− = 2N ∣y⟩ ⟨x∣ − ⟨x,ME−My⟩ ⟨ME−ME−⟩ E− in the expansion (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2), which leaves us with (G − M)xy = − (W G)x(My) + ⟨G − M⟩(G − M)x(My) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) + [⟨x,ME−My⟩ ⟨ME−ME−⟩ + ⟨x,M 2y⟩ 1 − ⟨M 2⟩ ][⟨W GE−M⟩ − ⟨G − M⟩⟨(G − M)E−M⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 54 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES Afterrealizingthatthe denominatorsin(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3)are boundedawayfrom zero (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5andLemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 in the isotropic case, ∣(G − M)xy∣ ≺ (Nη)−1/2, can be concluded again by a standard bootstrap argument, now using the high-moment bound from [36, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, part (a)] and the already proven averaged law ∣⟨(G − M)B⟩∣ ≺ (Nη)−1 with ∥B∥ ≲ 1 as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Bounds on the deterministic approximations: Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 The goal of this appendix is to define the deterministic approximation M(w1,B1,w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bk−1,wk) to a resolvent chain G(w1)B1G(w2)⋯Bk−1G(wk) and prove the bounds from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' While the definition of M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) is done for any num- ber k of spectral parameters w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk, the bounds in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 are proven for at most five and the deterministic matrices B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bk−1 being regular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' to the surrounding spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Fix k ∈ N and let w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk ∈ C∖R be spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As usual, the corresponding solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) (see also Appendix A) are denoted by M(wj), j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for deterministic matrices B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bk−1 we recursively define M(w1,B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='Bk−1,wk) = (B1k) −1[M(w1)B1M(w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) + ∑ σ=± k−1 ∑ l=2 σM(w1)⟨M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wl)Eσ⟩EσM(wl,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk)] , where we introduced the shorthand notation Bmn ≡ B(wm,wn) = 1 − M(wm)S[⋅]M(wn) for the stability operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that the recursion (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) is well defined, since on the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), there are only M(wm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wn) appearing for which the number of spectral parametersis strictly smaller than on the lhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' n− m + 1 < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As a preparation for the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we shall now show that M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) satisfies multiple recursive relations,called recursive Dyson equations, by using a so-called meta argument, that relies on the fact that M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) actually approximates a chain of products of resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, we only picked one of the recursive relations (namely (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) with j = 1) for actually defining M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) in Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Although the second recursion relation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) will not be used in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, it is obtained completely analogous to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and we hence give it for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' A similar meta argument has been done several times, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For convenience of the reader we repeat it in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (Recursive Dyson equations for M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk), see [28, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1]) Fix k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk ∈ C ∖ R be spectral parameters and B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bk−1 ∈ C2N×2N deterministic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then for any 1 ≤ j ≤ k we have the relations M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) = M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wj−1,Bj−1M(wj)Bj,wj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) + ∑ σ=± j−1 ∑ l=1 σM(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bl−1,wl,Eσ,wj,Bj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk)⟨M(wl,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wj−1)Bj−1M(wj)Eσ⟩ + ∑ σ=± k ∑ l=j+1 σM(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bj−1M(wj)Eσ,wl,Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk)⟨M(wj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wl)Eσ⟩ and M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) = M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wj−1,Bj−1M(wj)Bj,wj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 55 + ∑ σ=± j−1 ∑ l=1 σM(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bl−1,wl,EσM(wj)Bj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk)⟨M(wl,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wj)Eσ⟩ + ∑ σ=± k ∑ l=j+1 σM(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bj−1,wj,Eσ,wl,Bl,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk)⟨M(wj)BjM(wj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wl)Eσ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' If j = 1 or j = k, we define B0 = E+ resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Bk = E+ in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The formulas (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) shall be derived by expanding the jth resolvent Gj in the resolvent chain G1B1 ⋯GjBj ⋯ Bk−1Gk corresponding to M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) in an underlined term, once to the right (for (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2), see (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9)) and once to the left (for (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3), see (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Altogether, this yields 2k different recursions for M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk), which are listed in the above lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, it would be possible to prove directly that all these different recursions define the same M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This strategy has been used in a much simpler setup [26] dealing with Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Here, we find it simpler to use the alternative meta argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The principal idea is to derive the respective relations (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) on the level of resolvent chains G1B1⋯Bk−1Gk, which, after taking the expectation and using that Gi ≈ Mi from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, yields the same relation on the level of the deterministic approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the purpose of proving identities about M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk), we may use the most convenient distribution for X, namely Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the sake of this proof, we thus assume the single entry distribution χ of X to be a standard complex Gaussian χ = NC(0,1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' X in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 is a complex Ginibre matrix, in which case it holds that (recall the discussion below (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3)) E f(W )W g(W ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) Let w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk ∈ C ∖ R be arbitrary (but fixed!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=') spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now conduct the meta argu- ment, consisting of three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We consider the resolvent chain G1B1 ⋯ Bk−1Gk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) Expanding G1 via the identity G1 = M1 − M1W G1 + M1S[G1 − M1]G1 and using S[G1 − M1] = ⟨G1 − M1⟩ from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5), we find that G1B1 ⋯ Bk−1Gk =M1B1 ⋯ Bk−1Gk − M1W G1B1 ⋯ Bk−1Gk + ⟨G1 − M1⟩M1G1B1 ⋯ Bk−1Gk =M1B1 ⋯ Bk−1Gk + ∑ σ=± k−1 ∑ l=2 σM1⟨G1B1 ⋯ Bl−1GlEσ⟩EσGlBl ⋯ Bk−1Gk (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) − M1W G1B1 ⋯ Bk−1Gk + ⟨G1 − M1⟩M1G1B1 ⋯ Bk−1Gk + M1S[G1B1 ⋯ Bk−1Gk]Mk , where in the last step we distributed the derivatives coming from the definition of the underline in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) according to the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) can be rewritten as G1B1 ⋯ Bk−1Gk =(B1k)−1[M1B1 ⋯ Bk−1Gk + ∑ σ=± k−1 ∑ l=2 σM1⟨G1B1 ⋯ Bl−1GlEσ⟩EσGlBl ⋯ Bk−1Gk − M1W G1B1 ⋯ Bk−1Gk + ⟨G1 − M1⟩M1G1B1 ⋯ Bk−1Gk] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) Apart from the last two terms in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7), this is the exact same relation on the level of resolvents as in Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 for M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='Letthe originalmatrixsize N be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Foranyd ∈ N, we considerthe dN×dN Ginibre random matrix X(d) with entries having variance 1/(dN), and the deformation Λ(d) ∶= Λ ⊗ Id ∈ CdN×dN, where Id ∈ Cd×d is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Analogously to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15), we also define the Hermitisations 56 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES ˆΛ (d) and W (d), as well as the resolvents G(d) i = G(d)(wi) ∶= (W (d) + ˆΛ (d) − wi)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' It is crucial to observe that the correspondingly modified MDE − 1 M (d) = w − ˆΛ (d) + S(d)[M (d)] under the usual Imw ImM (d) > 0 constraint with S(d)[R] ∶= ̃Ẽ W (d)R̃ W (d) = ∑ σ σ⟨R E(d) σ ⟩E(d) σ , where E(d) σ ∶= Eσ ⊗ Id , has the unique solution M (d) = M ⊗Id, where M is the unique solution of the MDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) on C2N×2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In particular, if we define B(d) i ∶= Bi ⊗ Id for all i ∈ [k], then it holds that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) defined with M (d) i and B(d) i as inputs, also satisfies M (d)(w1,B(d) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',B(d) k−1,wk) = M(w1,B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',Bk−1,wk) ⊗ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We now multiply the analogue of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) in boldface matrices by some B(d) k = Bk ⊗ Id with Bk ∈ C2N×2N and take the averaged trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, by means of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4), taking the expectation of the resulting expression removes the underlined term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, using the one-to-one correspondence between the terms in the second line of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) and the terms on the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), mentioned below (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7), it follows by telescopic replacement and a simple induction on the length k of the chain, that lim d→∞E ⟨G(d) 1 B(d) 1 ⋯ G(d) k B(d) k ⟩ = ⟨M(w1,B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk)Bk⟩ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) by means of the usual global law [36, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1] for the last term on the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, due to the tensorisation, we have that ∣⟨G(d) 1 − M (d) 1 ⟩∣ ≺ 1/(Nd) since ∣Imw1∣ ≳ 1, where the implicit constant potentially depends on N but not on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We emphasise that the tensorisation by Id is indeed a necessary step, since the matrices Mi and Bi are N-dependent and hence one cannot take the limit N → ∞ in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Having (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) at hand, the recursive relations in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) can be proven as follows: For (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2), let 1 ≤ j ≤ k and expand Gj in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) according to Gj = Mj − MjW Gj + MjS[Gj − Mj]Gj , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) which yields, analogously to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), G1 ⋯ Bj−1GjBj ⋯ Gk = G1 ⋯ Bj−1MjBj ⋯ Gk (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) + ∑ σ=± j−1 ∑ l=1 σG1 ⋯ Bl−1Gl⟨Gl ⋯ Gj−1Bj−1MjEσ⟩EσGjBj ⋯ Gk + ∑ σ=± k ∑ l=j+1 σG1 ⋯ Bj−1Mj⟨GjBj ⋯ Bl−1GlEσ⟩EσGlBl ⋯ Gk − G1 ⋯ Bj−1MjW GjBj ⋯ Gk + ⟨Gj − Mj⟩G1 ⋯ Bj−1MjGjBj ⋯ Gk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, after taking the trace against some arbitrary Bk ∈ C2N×2N , by performing the tensorisation from Step 2, taking an expectation, and using (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8), we obtain (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2), but in a trace against Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, since Bk was arbitrary, we conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the second recursion (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3), the argument is identical except from the fact that we expand Gj in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) according to Gj = Mj − GjWMj + GjS[Gj − Mj]Mj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) □ The recursive relations from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 can be used to show the bounds from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 on the deterministic counterparts in the definition of Ψav/iso k in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) for k ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Recall that all deterministic matrices Ai appearing in the respective averaged or isotropic chain are regular in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the following, we will distinguish the two regimes η ≤ 1 and η > 1 and argue for each of them separately, iteratively using Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Before going into the iteration, recall that ∥M(w1)∥ ≲ min(1, 1 ∣Im w1∣) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, which immediately yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 57 Regime η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) for k = j = 2, we find that M(w1,A1,w2) = M(w1)X12[A1]M(w2) = B−1 12 [M(w1)A1M(w2)], (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) where X12[B] ∶= (1−S[M(w1) ⋅ M(w2)])[B] for B ∈ C2N×2N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since A1 is regular, we conclude (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k = 1 (by means of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 (b)), which immediately translates to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) and k = 2, we again use (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) with j = 2, such that we obtain M(w1,A1,w2,A2,w3) =M(w1,X12[A1]M(w2)A2,w3) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) + ∑ σ σM(w1,X12[A1]M(w2)Eσ,w3)⟨M(w2,A2,w3)Eσ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for k = 2 in combination with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) and the lower bound (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) on the eigenvalues of the stability operator B, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k = 2 readily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) and k = 3 we need a different representation of M(w1,A1,w2,A2,w3) as B−1 13[M(w1)A1M(w2,A2,w3) + ∑ σ σM(w1)EσM(w2,A2,w3)⟨M(w1,A1,w2)Eσ⟩], which follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) with j = 1 (or simply by Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This implies ⟨B−1 13[⋯]A3⟩ = ⟨[⋯]X31[A3]⟩ and thus, since ∥[⋯]∥ ≲ 1 from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) with k = 1 and ∥X31[A3]∥ ≲ 1 (recall Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 (b)), we have proven (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In order to see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k = 3, we first need to show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k = 2 remains valid, if only one of the two involved matrices A1,A2 is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Henceforth, we will assume that A1 = ˚ A1 and A2 is arbitrary, the other case being similar and hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We start with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and use the lower bound (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) on the eigenvalues of B in the first term in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13), such that the remaining terms to be investigated are in the last line of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13), where we study each factor separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thereby, we focus on the case Imw1 > 0 and s1 = s2 = + (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7)), other constellations being completely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, in the second factor in the last line of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) we use ∣⟨M(w2,A2,w3)E−⟩∣ = ∣⟨M(w2)A2M(w3)X32[E−]⟩∣ ≲ 1 for σ = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For σ = +, we find, using cyclicity of the trace, that ∣⟨M(w2,A2,w3)E+⟩∣ equals ∣⟨A2M(w3,E+,w2)⟩∣ = 1 ∣w3 − w2∣∣⟨A2(M(w3) − M(w2))⟩∣ ≲ 1 + 1 ∣w3 − w2∣ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the first factor in the last line of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13), we use the usual bound (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) for σ = − and conclude the desired estimate together with the bound on the second factor for σ = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, for σ = +, the argument is slightly more involved: Using the usual notations ej = Rewj and ηj = ∣Imwj∣, recall from the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6 (see the estimate of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='45)) that ⟨M1X12[A 1,2 1 ]M2M ∗ 2 E−⟩ = O(∣e1 + e2∣ + η1 + η2) , which readily implies that ⟨M1X12[A 1,2 1 ]M2M3E−⟩ = O(∣e2 − e3∣ + ∣e1 + e2∣ + η1 + η2 + η3) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) by means of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Employing the associated decomposition in the first factor in the last line of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) (and using the analogous cτ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=')-notation as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='36)), we find it being equal to M(w1,(X12[A1]M(w2)) 1,3,w3) + ∑ τ cτ(X12[A 1,2 1 ]M2)M(w1,Eτ,w3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The first summand is easily bounded by one, as follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12), the term with τ = + is also bounded by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The remaining term with τ = − can be estimated with the aid of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) as ∣e2 − e3∣ + ∣e1 + e2∣ + η1 + η2 + η3 ∣w1 + w3∣ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Collecting all the estimates from above, we find that ∥M(w1, ˚ A1,w2,A2,w3)∥ is bounded by 1 η + (1 + ∣e1 + e3∣ + ∣e2 − e3∣ + η1 + η2 + η3 ∣e1 + e3∣ + η1 + η3 )(1 + 1 ∣e3 − e2∣ + η2 + η3 ) ≲ 1 η , 58 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES which shows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) remains valid if only one of the two involved matrices A1, A2 is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Having this at hand, we can now turn to the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) for k = 4, we find M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='.,w4) =M(w1,X12[A1]M(w2),A2,w3,A3,w4) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) + ∑ σ σM(w1,X12[A1]M(w2)Eσ,w3,A3,w4)⟨M(w2,A2,w3)Eσ⟩ + ∑ σ σM(w1,X12[A1]M(w2)Eσ,w4)⟨M(w2,A2,w3,A3,w4)Eσ⟩, where the first and second line of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) are bounded by 1 η and we can thus focus on the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Struc- turally, this term is the analog of the last line in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and also proving it being bounded by 1 η is com- pletely analogous to the arguments above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This concludes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k = 3, from which (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for k = 4 immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, we turn to the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) for j = 1 (or simply by Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) we find the different representation M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',w5) =B−1 15[M(w1)A1M(w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',w5) + ∑ σ σM(w1)EσM(w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',w5)⟨M(w1,A1,w2)Eσ⟩ + ∑ σ σM(w1)EσM(w3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',w5)⟨M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',w3)Eσ⟩ + ∑ σ σM(w1)EσM(w4,A4,w5)⟨M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',w4)Eσ⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining ∥[⋯]∥ ≲ η−1, as follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for k ∈ [3] and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) for k ∈ [4], with the usual bound (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), we conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This finishes the proof in the first regime where η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Regime η > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this second regime, we note that all inverses of stability operators are bounded (see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, it easily follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) that every summand in the definition of M(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=',wk) carries at least k factors of (different) M(wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, as mentioned in the beginning of the proof, we have ∥M(wi)∥ ≲ 1/η, which implies the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Motivating derivation of the regularisation In this appendix, we shall motivate and derive the regularisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) introduced in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 by considering two basic examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' First, in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, we compute E ∣⟨W G(iη)A⟩∣ 2, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) which is the leading contribution to ⟨(G − M)B⟩ with A = X [B]M, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We will show that, in order to be able to reduce its naive size 1/(Nη)2 to the target 1/(N 2η), we need that ⟨A,V±⟩ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' we need A ∈ C2N×2N to be orthogonal to two certain directions V± in C2N×2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that, we chose the spectral parameter w = iη to be on the imaginary axis, assuming that 0 ∈ Bκ for some κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this case, both cutoff functions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) in the actual definition of the regularisation satisfy 1± δ(iη,iη) = 0 for η > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, at least a posteriori, we really catch both directions V± and not only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This calculation is rather foundational and unambiguously reveals two directions V±, for which we need that ⟨A,V±⟩ = 0, in order to reduce the naive size of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Second, in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we consider the averaged chain ⟨GΛ1(w1)A1GΛ2(w2)A2⟩, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) where the resolvents are even allowed to have generally different deformations, Λ1 ≠ Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Let M1 ∶= M Λ1(w1) and M2 ∶= M Λ2(w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For simplicity, we will assume that the stability operators Bm(∗)n(∗) ∶= 1 − M (∗) m S[⋅]M (∗) n , m,n ∈ [2], (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) for all constellations of adjoints, have at most one critical eigenvalue βm(∗)n(∗) which is not of order one (with associated right and left eigenvectors Rm(∗)n(∗) and Lm(∗)n(∗), respectively, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 59 shown in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5(c), this is the case, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', if Λ ≡ Λ1 = Λ2 and Rew1,Re w2 ∈ BΛ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (This remains true for other more general random matrix models with a flat (see (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1)) self-energy operator [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=') Again, the main question is what special property A1,A2 must have so that (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) be smaller than its naive size of order 1/η obtained from a simple Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Instead of directly computing the second moment of the corresponding underline term (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1)), we will make a pragmatic ansatz on the regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We then start a proof for a bound on (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and find that certain deterministic terms are too big for general A1,A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We shall see that there exist two matrices ˜V± ∈ C2N×2N (which turn out to be certain right eigenvectors Rm(∗)n(∗) of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3), see (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) later), such that, if ⟨Ai, ˜V±⟩ = 0, these critical terms are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We observe that, for the situation Λ1 = Λ2 and w1 = w2 = iη, the expressions for ˜V± in fact coincide with those for V± obtained in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1, showing that the foundational and the pragmatic approaches lead to the same regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3, motivated by the previous tandem of foundational and pragmatic computa- tions in Sections D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, respectively, we list generally valid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for arbitrary w1,w2 also away from the imaginary axis) explicit formulas for the directions V± in case that Λ1 = Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' These explicit formulas are identical to those used in the regularisation introduced in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Variance calculation of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the following, we simply write G = G(iη) for ease of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, using a cumulant expansion and neglecting cumulants of order at least three (or assuming that X is Ginibre), one gets E∣⟨W GA⟩∣ 2 = 1 N ∑ ab RabE⟨∆abGA⟩∂ba⟨A∗G∗W⟩ = 1 N ∑ ab RabE⟨∆abGA⟩⟨GA∗G∗∆ba⟩ (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) + 1 N 2 ∑ abcd RabRcdE⟨∆abG∆dcGA⟩⟨A∗G∗∆baG∗∆cd⟩ = 1 N 2 ∑ σ σE⟨EσGAEσA∗G∗⟩ + 1 N 2 ∑ στ στE⟨EσG∗EτGA⟩⟨EσGEτ(GA)∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The rescaled cumulant Rab ∶= Nκ(ab,ba) has been introduced below (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) and ∆ab ∈ C2N×2N con- tains only one non-zero entry at position (a,b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (∆ab)cd = δacδbd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As we will show, the cumulant expansion (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) yields that (up to a constant) E ∣⟨W GA⟩∣ 2 ≈ E∣⟨ImGA⟩∣ 2 (Nη)2 + E∣⟨ImGAE−⟩∣ 2 (Nη)2 + O ( 1 N 2η ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) Indeed, the first summand in the last line of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) is estimated by 1/(N 2η), the target size, with the aid of a trivial Schwarz inequality and a Ward identity using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By writing out the summation in the last summand, we get in total four terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Since their treatment is very similar, we focus on two exemplary terms with σ = τ = + (analogous to σ = τ = −) and σ = −τ = − (analogous to σ = −τ = +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the former, we apply a Ward identity and find it to be given by E ∣⟨ImGA⟩∣ 2 (Nη)2 , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) which, without any further information on A, using that ⟨GA⟩ ∼ 1 from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, is too big, compared to the targeted 1/(N 2η)-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, this drastically improves if ⟨ImM,A⟩ = 0 (recall that ImM is self adjoint): Since ⟨(G − M)A⟩ and ⟨W GA⟩ are roughly of the same size (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2)), the contribution (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) basically becomes a lower-order correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We have thus identified the first of the two directions V±, to which A has to be orthogonal to in order to reduce the naive size of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), namely V+ = α+ ImM for some non-zero α+ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) The latter case, σ = −τ = −, is slightly more involved due to the asymmetry of the two factors in the last summand in the last line of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4): For the first factor, again a Ward identity is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the 60 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES second factor, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) together with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 (with Im G(w) instead of G(w) in the integral) in the approximate form G∗G∗ ∼ ImG/η, as follows by replacing the Cauchy kernel in the integral ∫ ImG(x + iη) x2 + η2 dx ∼ ImG(iη) η by a δ-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Overall, this leaves us (roughly) with E∣⟨ImGAE−⟩∣ 2 (Nη)2 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) for the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, arguing for (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) completely analogous as done for (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), we find the second direction V−, to which A has to be orthogonal to, in order to reduce the naive size of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), namely V− = α− ImME− for some non-zero α− ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) We point out that the first term in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) would have worked in the exact same way also for spectral parameters w = e + iη with e ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, the second direction V− would not have been visible in this scenario, since the second term in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) would have been replaced by (at least for an upper bound) E∣⟨ImG(e + iη)AE−⟩∣ 2 N 2η (∣e∣ + η) + E∣⟨ImG(e + iη)AE−⟩⟨ImG(−e + iη)E−A∗⟩∣ N 2η (∣e∣ + η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' General structural regularisation in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We begin with the general rather structural regular- izing decomposition of a matrix A (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2)), which shall be conducted as (dropping the tilde, which has been temporarily introduced below (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3)) A○ ≡ ˚ A ∶= A − ⟨V+,A⟩U+ − ⟨V−,A⟩U− (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) for some Uσ,Vσ ∈ C2N×2N to be determined but subject to the conditions ⟨Vσ,Uτ⟩ = δσ,τ and ⟨Uσ,Uσ⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We point out, that the following calculations are largely insensitive to the form of the self-energy operator S[⋅] (but see Footnote 16) and hence the conclusions for Uσ and Vσ derived in this section are valid beyond our concrete model (up to the fact that, due to the chiral symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), the regularisation involves a two-dimensional projection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The goal of the present subsection is to show that V± must be chosen as certain right eigenvectors Rm(∗)n(∗) of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This follows by expanding (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and identifying several terms, whose size is too big for general deterministic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, these terms can be neutralised, if ⟨Ai,Rm(∗)n(∗)⟩ = 0 for certain right eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, as already mentioned in Section 3, for the directions U± there are a priori no further constraints or conditions (apart from orthogonality and normalisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, as it turns out to be convenient for our proofs, we will choose the matrices Uσ in such a way, that a resolvent identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the transformation of a product into a difference, GΛ1(w1)UσGΛ2(w2) ≈ (GΛ1(w1) − GΛ2(σw2))Uσ , can be applied (here, the symbol ‘≈’ neglects lower order terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, the condition ⟨Vσ,Uτ⟩ = δσ,τ will guarantee that the regularisation is idempotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (˚ A)○ = ˚ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that our general ansatz (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) is restricted to the non-degenerate situation, where Uσ and Vσ are non-orthogonal, ⟨Vσ,Uσ⟩ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This is guaranteed for our concrete model with deformations Λ1 = Λ2 (see Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) but requires some non-trivial arguments in more general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Although the regularisation is inherently two-dimensional (at least for our model), we also define ˚ Aσ = A○σ ∶= A − ⟨Vσ,A⟩Uσ , σ ∈ {+,−}, and refer to A○σ as the σ-regular component (or σ-regularisation) of A and to ⟨Vσ,A⟩Uσ as its σ-singular component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that (A○+)○− = (A○−)○+ = ˚ A, since ⟨Vσ,Uτ⟩ = δσ,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As usual, we use the common notation ηi ∶= ∣Imwi∣ for i ∈ [2] and abbreviate (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7)) si ∶= −sgn(ImwiImwi+1), i ∈ [2], (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) where the indices are understood cyclically modulo 2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This means that, in particular, s1 = s2 due to the short length of the chain (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the following, we will drop the arguments by EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 61 writing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', M1 = M Λ1(w1) and G2 = GΛ2(w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, we take A1 = ˚ A1 and A2 = ˚ A2 to be regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' orthogonal to some yet to be specified V±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, by means of G1 = M1 − M1W G1 + M1S[G1 − M1]G1 , we immediately find G1A1G2 = M1A1G2 − M1W G1A1G2 + M1S[G1 − M1]G1A1G2 , from which we conclude that B12[G1A1G2] = M1A1M2 + M1A1(G2 − M2) − M1W G1A1G2 + M1S[G1 − M1]G1A1G2 + M1S[G1A1G2](G2 − M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This implies ⟨(G1A1G2 − M A1 12 )A2⟩ = ⟨M1A1(G2 − M2)X21[A2]⟩ − ⟨M1W G1A1G2X21[A2]⟩ + ⟨M1S[G1 − M1]G1A1G2X21[A2]⟩ + ⟨M1S[G1A1G2](G2 − M2)X21[A2]⟩ where we defined M A1 12 ∶= B−1 12 [M1A1M2] = M1X12[A1]M2 = M(w1,A1,w2) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) (recall (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) and see Appendix C) and used the shorthand notation Xmn[B] = ((B∗ nm)−1[B∗]) ∗ = (B−1 m∗n∗)∗[B], B ∈ C2N×2N , where the adjoint of Bnm is understood like in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' So far, the regularisation of A1 and A2 has been rather structural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To make it more concrete, we must allow Vσ and Uσ to be potentially different depending on which of the Ai is regularised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In order to do so, we also temporarily introduce the additional index i, referring to the considered Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' That is, we will write Vσ,i instead of Vσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The matrices Vsi,i (recall (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) for the definition of si) shall be determined by requiring that ∥M A1 12 ∥ = ∥M1X 12[A1]M2∥ ≲ ∥A1∥ for i = 1 and ∥X21[A2]∥ ≲ ∥A2∥ for i = 2, meaningthatthe (adjointof the)stabilityoperatorhasa boundedinverse onregularobservables(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='sub- tracting the si-singular component amounts to removing the ‘bad direction’ of the stability operators X12 and X12, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' From this condition, we find the characterisation of Vs1,1 and Vs2,2, namely Vs1,1 = R1∗2∗ = (R21)∗ and Vs2,2 = R2∗1∗ = (R12)∗ , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) up to a normalisation constant, which can be specified only after determining Uσ (recall that ⟨Vσ,Uτ⟩ = δσ,τ and ⟨Uσ,Uσ⟩ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Recall from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3), that we denote by Rm(∗)n(∗) and Lm(∗)n(∗) the (normalised) right and left eigenvectors of Bm(∗)n(∗) corresponding to the (potentially) critical eigenvalue βm(∗)n(∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Indeed, in order to verify that (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) is the right choice for Vsi,i, we use the decomposition Xmn = (B−1 m∗n∗)∗ = 1 ¯βm∗n∗ ∣Lm∗n∗⟩ ⟨Rm∗n∗∣ + O(1), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) where O(1) is a shorthand notation for a linear operator E ∶ C2N×2N → C2N×2N satisfying ∥E[B]∥ ≲ ∥B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This linear operator is represented by a contour integration of the form 1 2πi ∮ dz z − B∗ m∗n∗ where the contour encircles all non-critical eigenvalues of B∗ m∗n∗ and remains at an order one distance from the entire spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Note that for general non-Hermitian operators the resolvent (z −B∗ m∗n∗)−1 wouldnotnecessarilybe bounded(independentlyofN)justbecause z iswellawayfrom the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 62 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES However, the explicit form of S (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20)) implies 16that B∗ m∗n∗ = 1+T where T is a rank-two operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For such operators elementary linear algebra shows that ∥ 1 z − B∗ m∗n∗ ∥ ≲ [dist(z,Spec(B∗ m∗n∗))] −2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the non-Hermitian instability only affects a two-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) we find X12[˚ As1 1 ] = 1 ¯β1∗2∗ (⟨R1∗2∗,A1⟩ − ⟨Vs1,1,A1⟩⟨R1∗2∗,Us1,1⟩)L1∗2∗ + O(1)[A1] for the decomposition of A1 and X21[˚ As2 2 ] = 1 ¯β2∗1∗ (⟨R2∗1∗,A2⟩ − ⟨Vs2,2,A2⟩⟨R2∗1∗,Us2,2⟩)L2∗1∗ + O(1)[A2], for the decomposition of A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This implies that for (⋯) to be vanishing for every ˚ Asi i , the matrix Vsi,i has to be chosen according to (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) (recall ⟨Vσ,i,Uτ,i⟩ = δσ,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17 Overall, subtracting the si-singular component already accounts for removing the ‘bad direction’ of a involved stability operator and thus – in particular – reduces the naive size of the deterministic approximation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, removing the si-singular component is not sufficient: Although ⟨Vsi,i,U−si,i⟩ = 0 and thus U−si,i is si-regular, we observe that ⟨G1U−s1,1G2U−s2,2⟩ (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) still (potentially) has large fluctuations: In our concrete i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' model, take z ≡ z1 = z2 (to be suppressed from the notation) and w ≡ w1 = −w2 with e = Rew1 and η = Im w1 > 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', which implies that s1 = s2 = + and Uσ = Eσ for σ = ± (see the discussion below (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this situation, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) and thus (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) takes the form ⟨G(e + iη)E−G(−e − iη)E−⟩ = −⟨G(e + iη)G(e + iη)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By construction of Vsi,i, the corresponding deterministic approximation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) is bounded by one, but this is dominated by the fluctuation of order 1/(Nη2) in the relevant small regime η ∼ N −1+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This example shows again, what we have already established in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1: For our concrete model, at least close to the imaginary axis, the regularisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) is necessarily a two-dimensional operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For determining the other directions V−si,i, we note that the regularisation should be designed in such a way, that it covers also the cases where one (or both) of the resolvents G1,G2 are taken as an adjoint (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, requiring that the same arguments leading to (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) should also be followed for (i) ⟨G1A1G∗ 2A2⟩ and (ii) ⟨G∗ 1A1G2A2⟩ (considering ⟨G∗ 1A1G∗ 2A2⟩ would again lead to a conclusion for Vsi,i as the relative sign of imaginary parts is preserved), we find that V−s1,1 = (R2∗1)∗ and V−s2,2 = (R12∗)∗ in case (i), and V−s1,1 = (R21∗)∗ and V−s2,2 = (R1∗2)∗ in case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In general, the right eigenvectors for these two cases are not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, as pointed out in Footnote 17, there is a certain tolerance in choosing the V±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Therefore, within this tolerance and in order to have a consistent and conceptually simple choice, we take V−s1,1 from case (i) and V−s2,2 from case (ii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' V−s1,1 = R1∗2 = (R2∗1)∗ and V−s2,2 = R2∗1 = (R1∗2)∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) Here, in both situations the spectral parameter being the right neighbor of Ai receives a complex con- jugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In comparison, if we took V−s1,1 from case (ii) and V−s2,2 from case (i), we would have ended up with the alternative regularisation from Footnote 10, where the left neighbor of Ai received a complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Again, the relations in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) are understood up to a normalizing constant, which can be specified only after determining Uσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 16This is the only place in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 where the special form of S is currently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For more general S operator an appropriate generalisation of the symmetrised (saturated) self-energy operator [2, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5] to two different spectral parameters is needed, see [45, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='30)] in the commutative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 17In case that Λ1 = Λ2, by the lower bound (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), the choices in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) not necessarily have to be made exact, but tolerate an error of the order given in the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Having such a tolerance might be important if one treats the Λ1 ≠ Λ2 case (contrary to Λ1 = Λ2 as done in this paper) and still has to satisfy the constraints ⟨Vσ, Uτ ⟩ = δσ,τ and ⟨Uσ, Uσ⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 63 Now, it is very important to observe that,for our concrete model with Λ1 = Λ2 and w1 = w2 = iη (in particular, s1 = s2 = −), our choices for V± in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) agree with those in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7)and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9)obtained from a variance calculation with only a single resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This follows from the explicit formulas for the critical right eigenvector in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17), Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4 (a), and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Explicit formulas for our concrete model and Λ1 = Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In this subsection, we will give ex- plicitformulasforV± andU± forourconcrete modelwithone fixeddeformationΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Infact, forΛ1 = Λ2, the so far unspecified matrices Uσ can be characterised by requiring that, jointly with the symmetry re- lationE−Gz(−w)E− = −Gz(w), a resolventidentity can be appliedto G2UσG1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This yields, together with the normalisation ⟨Uσ,Uσ⟩ = 1, that18 U+ = E+ and U− = E− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The singular (or critical) eigenvectors of the stability operators characterizing Vsi,i can also be ex- plicitlycalculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Using(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), we infer, bymeansof (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17)andthe normalisation/orthogonality condition ⟨Vσ,i,Uτ,i⟩ = δσ,τ, that Vs1,1 = M2Es1M1 ⟨M2Es1M1Es1⟩ , V−s1,1 = M ∗ 2 E−s1M1 ⟨M ∗ 2 E−s1M1E−s1⟩ , Vs2,2 = M1Es2M2 ⟨M1Es2M2Es2⟩ , V−s2,2 = M ∗ 1 E−s2M2 ⟨M ∗ 1 E−s2M2E−s2⟩ , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) matching the definition of the regularisation given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The normalisation is obvious and the orthogonality readily follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 in combination with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Finally, we remark that in order to define the regularisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) and work with (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16), it is not necessary to have the explicit forms for Vσ,i at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Instead, the single instance of relevant explicit formulas is the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7, more precisely, the bound in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4, where one needs that for ∣Imw1∣ ∼ N −1+ǫ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', (R1∗1)∗ is close to ImM1 (up to a normalisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' But this is true beyond our model, as easily follows after taking the imaginary part of the general matrix Dyson equation (see [37]) − 1 M = w − A + S[M], Imw ⋅ Im M > 0 with self-adjoint matrix of expectations A = A∗ and (flat, see (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1)) self-energy operator S[⋅].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In fact, this yields (1 − MS[⋅]M ∗)(Im M) = (Imw)MM ∗ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' for ∣Imw∣ ≪ 1 very small, ImM is an approximate right eigenvector of the stability operator 1 − MS[⋅]M ∗ corresponding to the critical eigenvalue (recall the discussion below (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9 In this appendix, we carry out the proofs of the two Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Similarly to the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, we get from Appendix D and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) that ⟨(G1A1G2 − M1X12[A1]M2)A2⟩ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) = ⟨M1A1(G2 − M2)X21[A2]⟩ − ⟨M1W G1A1G2X21[A2]⟩ + ⟨M1S[G1 − M1]G1A1G2X21[A2]⟩ + ⟨M1S[G1A1G2](G2 − M2)X21[A2]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We note that ∥X12[˚ A1]∥ ≲ 1 and ∥X21[˚ A2]∥ ≲ 1 by means of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, analogously to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='35), we need to further decompose X21[A2]M1 in the last three terms in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='34) as X21[˚ A2]M1 = (X21[˚ A2]M1)○ + ∑ σ 1σ δ cσ(X21[˚ A2]M1)Eσ , 18Note that the assignment of ± is a priori not determined, but we chose it in that way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This is also reflected in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 64 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES where we again suppressedthe spectral parameters(and the relative sign of their imaginary parts, which has been fixed by Imw1 > 0 and Imw2 < 0) in the notation for the linear functionals cσ(⋅) on C2N×2N defined as c+(B) ∶= ⟨M2BM1⟩ ⟨M2M1⟩ and c−(B) ∶= ⟨M2BM ∗ 1 E−⟩ ⟨M2E−M ∗ 1 E−⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2) Continuing the expansion of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1), we arrive at ⟨M1 ˚ A1(G2 − M2)X21[˚ A2]⟩ − ⟨W G1 ˚ A1G2(X21[˚ A2]M1)○⟩ + ⟨S[G1 − M1]G1 ˚ A1G2(X21[˚ A2]M1)○⟩ + ⟨S[G1 ˚ A1G2](G2 − M2)(X21[˚ A2]M1)○⟩ + ∑ σ 1σ δ cσ(X21[˚ A2]M1)[ − ⟨W G1 ˚ A1G2Uσ⟩ + ⟨S[G1 − M1]G1 ˚ A1G2Eσ⟩ + ⟨S[G1 ˚ A1G2](G2 − M2)Eσ⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We emphasise that, in case of ˚ A2 and its linear dependents, the regular component is defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' the pair of spectral parameters (w2,w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, analogously to the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, we undo the underline in [⋯], such that our expansion of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1) becomes ⟨(G1 ˚ A1G2 − M1X12[˚ A1]M2)˚ A2⟩ = ⟨M1 ˚ A1(G2 − M2)X21[˚ A2]⟩ − ⟨W G1 ˚ A1G2(X21[˚ A2]M1)○⟩ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) + ⟨S[G1 − M1]G1 ˚ A1G2(X21[˚ A2]M1)○⟩ + ⟨S[G1 ˚ A1G2](G2 − M2)(X21[˚ A2]M1)○⟩ + ∑ σ 1σ δ cσ(X21[˚ A2]M1)[ − ⟨˚ A1G2Eσ⟩ + ⟨G1 ˚ A1G2˚Φσ⟩ + cσ(Φσ)⟨G1 ˚ A1G2Eσ⟩], where Φσ ∶= Eσ 1 M1 − S[M2Eσ] (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4) was further decomposed with the aid of cσ(Φτ) ∼ δσ,τ and we used the notation (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We can now write (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) for both, ˚ A2 = ˚Φ+ and ˚ A2 = ˚Φ−, and solve the two resulting equation for ⟨G1 ˚ A1G2˚Φσ⟩ and ⟨G1 ˚ A1G2˚Φ−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Observe that by means of cτ(X21[˚Φσ]M1) ∼ δσ,τ , the original system of linear equations boils down to two separate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thus, plugging the solutions for ⟨G1 ˚ A1G2˚Φ±⟩ back into (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3) we arrive at ⟨(G1 ˚ A1G2 − M1X12[˚ A1]M2)˚ A2⟩ = − ⟨W G1 ˚ A1G2(X21[˚ A2]M1)○⟩ + ⟨G1 − M1⟩⟨G1 ˚ A1G2(X21[˚ A2]M1)○⟩ + ⟨M1 ˚ A1(G2 − M2)X21[˚ A2]⟩ + ⟨S[G1 ˚ A1G2](G2 − M2)(X21[˚ A2]M1)○⟩ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) + ∑ σ 1σ δ cσ(X21[˚ A2]M1) 1 − 1σ δ cσ(X21[˚Φσ]M1) [ − ⟨W G1 ˚ A1G2(X21[˚Φσ]M1)○⟩ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) + ⟨G1 − M1⟩⟨G1 ˚ A1G2(X21[˚Φσ]M1)○⟩ + ⟨M1 ˚ A1(G2 − M2)X21[˚Φσ]⟩ + ⟨S[G1 ˚ A1G2](G2 − M2)(X21[˚Φσ]M1)○⟩ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7) − ⟨˚ A1(G2 − M2)Eσ⟩ + cσ(Φσ)⟨(G1 ˚ A1G2 − M ˚ A1 12 )Eσ⟩] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) We now need to check that the denominators in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) are bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For small enough δ > 0, we have that ∣1 − 1σ δ (w2,w1)cσ(X21[˚Φσ]M1)∣ ≳ 1 for σ = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Completely analogous to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 65 Next, there are two particular terms, namely the ones of the form ⟨S[G1 ˚ A1,2 1 G2](G2 − M2)˚ A2,1 2 ⟩, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) appearing in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7), and cσ(X21[˚ A2,1 2 ]M1)cσ(Φσ)⟨(G1 ˚ A1,2 1 G2 − M1X12[˚ A1,2 1 ]M2)Eσ⟩, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) appearing in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8), whose naive size 1/(Nη2) does not match the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, they have to be dis- cussed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10), we emphasised the pair of spectral parameters with respect to which the regularisation has been conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Moreover, for the following estimates, we recall the a priori bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Estimating (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We begin by expanding ⟨S[G1 ˚ A1,2 1 G2](G2 − M2)˚ A2,1 2 ⟩ = ∑ σ σ ⟨G1 ˚ A1,2 1 G2Eσ⟩⟨(G2 − M2)˚ A2,1 2 Eσ⟩ (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11) and note that, analogously to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='47), ˚ Ai,j i Eσ = (˚ Ai,j i Eσ) i,i + O(∣ei − σej∣ + ∣ηi − ηj∣)E+ + O(∣ei − σej∣ + ∣ηi − ηj∣)E− (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) as well as ˚ Ai,j i Eσ = (˚ Ai,j i Eσ) j,j + O(∣ei − σej∣ + ∣ηi − ηj∣)E+ + O(∣ei − σej∣ + ∣ηi − ηj∣)E− (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) for i ≠ j ∈ [2] and σ = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the first term in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), for σ = + and Eσ = E+, we use a resolvent identity and the usual averaged local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) in combination with (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6), in order to bound it as ∣⟨G1 ˚ A1,2 1 G2⟩∣ ≺ 1 + 1 ∣e1 − e2∣ + η1 + η2 max i∈[2] ∣⟨(Gi − Mi)(˚ A1,2 1 )○i,i⟩∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) For σ = − and Eσ = E−, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) and employ the integral representation from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 with τ = +, J = Bℓκ0 , and ˜η = ℓ ℓ + 1η , for which we recall that wj ∈ D(ǫ0,κ0) ℓ+1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' in particular η ≥ (ℓ + 1)N −1+ǫ0 and hence ˜η ≥ ℓN −1+ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' After splitting the contour integral and bounding the individual contributions as described in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11), we obtain, with the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, ∣⟨G1A 1,2 1 G2E−⟩∣ ≺ 1 + ∫Bℓκ0 ∣⟨G(x + i˜η)A 1,2 1 E−⟩∣ ∣(x − e1 − i(η1 − ˜η)) (x + e2 − i(η2 − ˜η))∣dx ≺1 + ∫Bℓκ0 ∣⟨(G(x + i˜η) − M(x + i˜η))(A 1,2 1 E−) x+i˜ η,x+i˜ η⟩∣ ∣(x − e1 − i(η1 − ˜η)) (x + e2 − i(η2 − ˜η))∣ dx , where in the second step, we freely added and subtracted M(x − i˜η) by residue calculus, used (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='13), and absorbed logarithmic corrections from the integral into ‘≺’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' This finally yields that ∣⟨G1A 1,2 1 G2E−⟩∣ ≺ 1 + 1 ∣e1 + e2∣ + η1 + η2 ⋅ ψav 1 Nη1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) Combining (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) with the estimate ∣⟨(G2 − M2)A 2,1 2 Eσ⟩∣ ≺ ∣e1 − σe2∣ + ∣η1 − η2∣ Nη + ψav 1 Nη1/2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) for the second term in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='11),whichreadily followsfrom (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15), we find that (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) can be bounded as ∣⟨S[G1A 1,2 1 G2](G2 − M2)A 2,1 2 ⟩∣ ≺ 1 Nη + (ψav 1 )2 (Nη)2 , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) where we used the trivial estimate ψav 1 ≺ η−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Estimating (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the term (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10), we first note that the two prefactors cσ(X21[A 2,1 2 ]M1) and cσ(Φσ) are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' However, completely analogous to the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6, in each of the two 66 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES cases σ = ±, the bound on one of the prefactors can be improved: In the first case, σ = +, we use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) and compute c+(Φ+) = ⟨M1⟩(1 − ⟨M1M2⟩) ⟨M1M2⟩ = O(∣e1 − e2∣ + η1 + η2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ∣⟨G1 ˚ A1G2 − M(w1, ˚ A1,w2)⟩∣ ≺ 1 Nη + 1 ∣e1 − e2∣ + η1 + η2 max i∈[2] ∣⟨(Gi − Mi)(A 1,2 1 )○i,i⟩∣ which is obtained completely analogous to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='14), we conclude that (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) for σ = + can be estimated by 1/(Nη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Similarly, in the second case, σ = −, we perform a computation similar to the one leading to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) and use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='12) in order to obtain that c−(X12[A 1,2 1 ]M2) equals i 2 ⟨M1A 1,2 1 M ∗ 2 E−⟩ ⟨M1E−M ∗ 2 E−⟩ + 1 2i ⟨M1A 1,2 1 M2E−⟩ ⟨M1E−M ∗ 2 E−⟩ 1 + ⟨M1E−M ∗ 2 E−⟩ 1 + ⟨M1E−M2E−⟩ = O(∣e1 + e2∣ + η1 + η2) Combining this with the bound ∣⟨(G1A 1,2 1 G2 − M(w1,A 1,2 1 ,w2))E−⟩∣ ≺ 1 Nη + 1 ∣e1 + e2∣ + η1 + η2 ⋅ ψav 1 Nη1/2 which is obtained completely analogous to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15), we conclude that (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) can be estimated by 1/(Nη) – now in both cases σ = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Summarizing our investigations, we have shown that ⟨(G1 ˚ A1G2 − M(w1, ˚ A1,w2))˚ A2⟩ = −⟨W G1 ˚ A1G2 ˚ A′ 2⟩ + O≺(E av 2 ) , where we used the shorthand notation ˚ A′ 2 ∶= (X21[˚ A2]M1) + ∑ σ 1σ δ cσ(X21[˚ A2]M1) 1 − 1σ δ cσ(X21[˚Φσ]M1) (X21[˚Φσ]M1) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='18) in the underlined term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='17) and the bound on (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='10) establishedabove with the usual single resolvent local laws (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) and the bounds on deterministic approximations in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we collected all the error terms from the expansion around (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='5)–(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='8) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' We denote Ai ≡ ˚ Ai, except we wish to emphasise Ai being regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' As usual, we use the customary shorthand notations and start with G2 = M2 − M2W G2 + M2S[G2 − M2]G2 , such that we get G1 ˜A1G2 ˚ A2G3 = G1 ˜A1M2 ˚ A2G3 − G1 ˜A1M2W G2 ˚ A2G3 + G1 ˜A1M2S[G2 − M2]G2 ˚ A2G3 for ˜A1 = X12[A1] with A1 = ˚ A1 (note that ∥X12[˚ A1]∥ ≲ 1 by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='6) and the linear operator X12 has been introduced in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' The definition of X23 is completely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Extending the underline to the whole product, we obtain G1( ˜A1−S[M1 ˜A1M2])G2 ˚ A2G3 =G1 ˜A1M2 ˚ A2G3 − G1 ˜A1M2W G2 ˚ A2G3 + G1 ˜A1M2S[G2 ˚ A2G3]G3 + G1 ˜A1M2S[G2 − M2]G2 ˚ A2G3 + G1S[(G1 − M1) ˜A1M2]G2 ˚ A2G3 , which leaves us with G1 ˚ A1G2 ˚ A2G3 − M(w1,A1,w2,A2,w3) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) = (G1 [X12[˚ A1]M2(˚ A2 + S[M2X23[˚ A2]M3])]G3 − M(w1,[⋯],w3)) − G1X12[˚ A1]M2W G2 ˚ A2G3 + G1X12[˚ A1]M2S[G2 − M2]G2 ˚ A2G3 + G1S[(G1 − M1)X12[˚ A1]M2]G2 ˚ A2G3 + G1X12[˚ A1]M2S[G2 ˚ A2G3 − M2X23[˚ A2]M3]G3 , where we used Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2 for assembling the purely deterministic terms on the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' To continue, we first note that ∥X12[˚ A1]∥ ≲ 1 and ∥X23[˚ A2]∥ ≲ 1 (again, the matrices being regular removes the potentially ‘bad direction’ of the stability operators X12 and X23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 67 Then, we need to further decompose X12[A1]M2 in the last four terms in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) as X12[A1]M2 = (X12[A1]M2) + ∑ σ 1σ δ cσ(X12[A1]M2)Eσ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) where, similarly as for ⋅○, we suppressed the spectral parameters w1,w2 in the notation for the linear functionals cσ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='), which have been defined in see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, plugging (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20) into (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='19) we find G1 ˚ A1G2 ˚ A2G3 − M(w1, ˚ A1,w2, ˚ A2,w3) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) = (G1 [X12[˚ A1]M2(˚ A2 + S[M2X23[˚ A2]M3])]G3 − M(w1,[⋯],w3)) − G1(X12[˚ A1]M2) W G2 ˚ A2G3 + G1(X12[˚ A1]M2) S[G2 − M2]G2 ˚ A2G3 + G1S[(G1 − M1)(X12[˚ A1]M2) ]G2 ˚ A2G3 + G1(X12[˚ A1]M2) S[G2 ˚ A2G3 − M2X23[˚ A2]M3]G3 + ∑ σ 1σ δ cσ(X12[˚ A1]M2)[ − G1EσW G2 ˚ A2G3 + G1EσS[G2 − M2]G2 ˚ A2G3 + G1S[(G1 − M1)Eσ]G2 ˚ A2G3 + G1EσS[G2 ˚ A2G3 − M2X23[˚ A2]M3]G3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, as in the earlier sections (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=', the display above (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='4)), in the last line of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) we now undo the underline and find the bracket [⋯] to equal (the negative of) G1Eσ( ˚ A2 + S[M(w2, ˚ A2,w3)])G3 − G1ΦσG2 ˚ A2G3 , where we denoted Φσ ∶= Eσ 1 M2 − S[M1Eσ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' It is apparent from the expansion (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='21) (and it can also be checked by hand) that M(w1,Eσ ˚ A2 + EσS[M(w2, ˚ A2,w3)],w3) = M(w1,Φσ,w2, ˚ A2,w3), which finally yields G1 ˚ A1G2 ˚ A2G3 − M(w1, ˚ A1,w2, ˚ A2,w3) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) = (G1 [X12[˚ A1]M2(˚ A2 + S[M2X23[˚ A2]M3])]G3 − M(w1,[⋯],w3)) − G1(X12[˚ A1]M2) W G2 ˚ A2G3 + G1(X12[˚ A1]M2) S[G2 − M2]G2 ˚ A2G3 + G1S[(G1 − M1)(X12[˚ A1]M2) ]G2 ˚ A2G3 + G1(X12[˚ A1]M2) S[G2 ˚ A2G3 − M2X23[˚ A2]M3]G3 + ∑ σ 1σ δ cσ(X12[˚ A1]M2)[ − (G1Eσ(˚ A2 + S[M(w2, ˚ A2,w3)])G3 − M(w1,[⋯]w3)) + (G1˚ΦσG2 ˚ A2G3 − M(w1,˚Φσ,w2, ˚ A2,w3)) + ∑ σ cσ(Φσ)(G1EσG2 ˚ A2G3 − M(w1,Eσ,w2, ˚ A2,w3))], where we further decomposed Φσ in the last line of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) (while using the first relation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='40)) just as X12[A1]M2 in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, we write (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22) for both, A1 = ˚ A1 = ˚Φ+ and A1 = ˚ A1 = ˚Φ−, and solve the two resulting linear equations for G1˚Φ±G2 − M(w1,˚Φ±,w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Observe that by means of the second relation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='40) the original system of linear equations boils down to two separate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Thus, plugging the solutions for G1˚Φ±G2 ˚ A2G3 − M(w1,˚Φ±,w2, ˚ A2,w3) back into (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='22), we arrive at G1 ˚ A1G2 ˚ A2G3 − M(w1, ˚ A1,w2, ˚ A2,w3) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) = (G1 [X12[˚ A1]M2(˚ A2 + S[M2X23[˚ A2]M3])]G3 − M(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='[⋯],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w3)) − G1(X12[˚ A1]M2) W G2 ˚ A2G3 + G1(X12[˚ A1]M2) S[G2 − M2]G2 ˚ A2G3 + G1S[(G1 − M1)(X12[˚ A1]M2) ]G2 ˚ A2G3 + G1(X12[˚ A1]M2) S[G2 ˚ A2G3 − M2X23[˚ A2]M3]G3 + ∑ σ 1σ δ cσ(X12[˚ A1]M2) 1 − 1σ δ cσ(X12[˚Φσ]M2) [ − (G1[Eσ(˚ A2 + S[M(w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ˚ A2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w3)])]G3 − M(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='[⋯]w3)) 68 EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES + (G1 [X12[˚Φσ]M2(˚ A2 + S[M2X23[˚ A2]M3])]G3 − M(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='[⋯],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w3)) − G1(X12[˚Φσ]M2) W G2 ˚ A2G3 + G1(X12[˚Φσ]M2) S[G2 − M2]G2 ˚ A2G3 + G1S[(G1 − M1)(X12[˚Φσ]M2) ]G2 ˚ A2G3 + G1(X12[˚Φσ]M2) S[G2 ˚ A2G3 − M2X23[˚ A2]M3]G3 + cσ(Φσ)(G1EσG2 ˚ A2G3 − M(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='Eσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' ˚ A2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='w3))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' It has been shown in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='7 that the denominators are bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Next, we take the scalar product of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) with two deterministic vectors x,y satisfying ∥x∥,∥y∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' In the resulting expression, in case that 1σ δ (w1,w2) = 1 (as we assumed in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' )), there are three particular terms, namely the ones of the form (G1S[(G1 − M1)A 1,2 1 ]G2 ˚ A2G3)xy , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24) as appearing twice, in the fourth and second to last line, (G1A 1,2 1 S[G2 ˚ A2G3 − M(w2, ˚ A2,w3)]G3)xy , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25) as appearing, again twice, in the fourth and second to last line, cσ(X12[˚ A1]M2)cσ(Φσ)(G1EσG2 ˚ A2G3 − M(w1,Eσ,w2, ˚ A2,w3))xy , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26) as appearing in the last line, whose naive sizes 1/(Nη3), 1/(Nη3), and 1/ √ Nη4 do not match the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Hence, they have to be discussed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Estimating (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the terms of the first type, we begin by expanding (G1S[(G1 − M1)A 1,2 1 ]G2 ˚ A2G3)xy = ∑ σ σ⟨(G1 − M1)A 1,2 1 Eσ⟩(G1EσG2 ˚ A2G3)xy and recall from (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='16) that first factor can be estimated by ∣⟨(G1 − M1)A 1,2 1 Eσ⟩∣ ≺ ∣e1 − σe2∣ + ∣η1 − η2∣ Nη + ψav 1 Nη1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='27) In the second factor, we distinguish the two cases σ = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For σ = +, we find G1G2A 2,3 2 G3 = G1A 2,3 2 G3 − G2A 2,3 2 G3 (e1 − e2) + i(η1 + η2) by a simple resolvent identity, which together with ˚ Aw2,w3 2 = ˚ Aw1,w3 2 + O(∣e1 − e2∣ + ∣η1 − η2∣ + ∣e1 − e3∣ + ∣η1 − η3∣)E+ + O(∣e1 − e2∣ + ∣η1 − η2∣ + ∣e1 + e3∣ + ∣η1 − η3∣)E− from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='3 (note the difference between the E+-error and the E−-error!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=') and the usual isotropic law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) yields the estimate ∣(G1G2A 2,3 2 G3)xy∣ ≺ 1 η + 1 ∣e1 − e2∣ + η1 + η2 ⎛ ⎝1 + ψiso 1 √ Nη2 ⎞ ⎠ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='28) where we again used the a priori bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For σ = − we employ the integral representation from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='1 and argue similarly as for (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) such that we finally obtain ∣(G1E−G2A 2,3 2 G3)xy∣ ≺ 1 η + 1 ∣e1 + e2∣ + η1 + η2 ⎛ ⎝1 + ψiso 1 √ Nη2 ⎞ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='29) Now, combining (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='27) with (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='28) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='29), we find ∣(G1S[(G1 − M1)A 1,2 1 ]G2 ˚ A2G3)xy∣ ≺ 1 √ Nη3 (1 + ψav 1 ψiso 1 Nη ) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='30) where we used that ψav 1 ≺ η−1/2 trivially by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' EIGENVECTOR OVERLAPS FOR NON-HERMITIAN RANDOM MATRICES 69 Estimating (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For terms of the second type, we again start by expanding (G1A 1,2 1 S[G2 ˚ A2G3 − M(w2, ˚ A2,w3)]G3)xy = ∑ σ σ⟨(G2 ˚ A2G3 − M(w2, ˚ A2,w3))Eσ⟩(G1A 1,2 1 EσG3)xy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Then, for the first factor, we recall from the estimate of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='9) that ∣⟨(G2A 2,3 2 G3 − M(w2,A 2,3 2 ,w3))Eσ⟩∣ ≺ 1 Nη + 1 ∣e2 − σe3∣ + η2 + η3 ⋅ ψav 1 Nη1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Treating the second factor analogously to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='28) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='29) above, we find ∣(G1A 1,2 1 EσG3)xy∣ ≺ ∣e2 − σe3∣ + ∣η2 − η3∣ η + ⎛ ⎝1 + ψiso 1 √ Nη2 ⎞ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining the two estimates, we have shown that ∣(G1A 1,2 1 S[G2 ˚ A2G3 − M(w2, ˚ A2,w3)]G3)xy∣ ≺ 1 √ Nη3 (1 + ψiso 1 Nη + ψav 1 ψiso 1 Nη ) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='31) where we again used that ψav 1 ≺ η−1/2 trivially by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Estimating (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' For the third term, we recall the (improved) estimates c+(Φ+) = O(∣e1 − e2∣ + η1 + η2) c−(X12[˚ A1]M2) = O(∣e1 + e2∣ + η1 + η2) on the anyway bounded prefactors, which have been shown in the course of estimating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' By arguing analogously to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='28) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='29), we also find ∣(G1EσG2 ˚ A2G3 − M(w1,Eσ,w2, ˚ A2,w3))xy∣ ≺ 1 √ Nη3 + 1 ∣e1 − σe2∣ + η2 + η3 ψiso 1 √ Nη2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Now, combining these estimates, we conclude ∣(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='26)∣ ≺ 1 √ Nη3 (1 + ψiso 1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='32) Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Summarizing our investigations, we have shown that (G1 ˚ A1G2 ˚ A2G3 − M(w1, ˚ A1,w2, ˚ A2,w3))xy = −(G1 ˚ A′ 1W G2 ˚ A2G3)xy + O≺(E iso 2 ) , where we used the shorthand notation ˚ A′ 1 = (X12[A1]M2) + ∑ σ 1σ δ cσ(X12[A1]M2) 1 − 1σ δ cσ(X12[˚Φσ]M2) (X12[˚Φσ]M2) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='33) in the underlined term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Combining (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='30), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='31), and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='32) with the usual single resolvent local laws (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='15) and the bounds on deterministic approximations in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='2, we collected all the error terms from (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='23) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' □ References [1] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Ajanki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Erdős, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Krüger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' Quadratic vector equations on complex upper half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' American MathematicalSociety Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' 1261 (2019) [2] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE1T4oBgHgl3EQf8wWT/content/2301.03549v1.pdf'} +page_content=' H.' metadata={'source': 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b/_tE2T4oBgHgl3EQfmwdA/content/tmp_files/2301.04001v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..72068dafaef99da1bf9b1dc9c46af27548ceac27 --- /dev/null +++ b/_tE2T4oBgHgl3EQfmwdA/content/tmp_files/2301.04001v1.pdf.txt @@ -0,0 +1,1740 @@ +Big Ideas in Sports Analytics and Statistical Tools +for their Investigation +A Preprint +Benjamin S. Baumer +Statistical & Data Sciences +Smith College +Northampton, MA 01063 +bbaumer@smith.edu +Gregory J. Matthews +Mathematics and Statistics +Loyola University Chicago +Chicago, IL 60660 +gmatthews1@luc.edu +Quang Nguyen +Statistics & Data Science +Carnegie Mellon University +Pittsburgh, PA 15213 +nmquang@cmu.edu +January 11, 2023 +Abstract +Sports analytics—broadly defined as the pursuit of improvement in athletic performance +through the analysis of data—has expanded its footprint both in the professional sports +industry and in academia over the past 30 years. In this paper, we connect four big ideas +that are common across multiple sports: the expected value of a game state, win probability, +measures of team strength, and the use of sports betting market data. For each, we explore +both the shared similarities and individual idiosyncracies of analytical approaches in each +sport. While our focus is on the concepts underlying each type of analysis, any implementation +necessarily involves statistical methodologies, computational tools, and data sources. Where +appropriate, we outline how data, models, tools, and knowledge of the sport combine to +generate actionable insights. We also describe opportunities to share analytical work, but +omit an in-depth discussion of individual player evaluation as beyond our scope. This paper +should serve as a useful overview for anyone becoming interested in the study of sports +analytics. +Keywords sports analytics · R packages · sports data · pairwise comparisons · datasets +1 +Introduction +Insights derived from the analysis of data have transformed the world of sports over the last few decades. +While baseball—a naturally discrete sport with more than a century’s worth of professional data—may be the +sport with the longest relationship with sports analytics, one would be hard-pressed to identify a professional +sport today in which sports analytics is not having an impact. In basketball, analytics has driven a shift in +the conventional wisdom about shot selection. Most teams are shooting more three-pointers, settling for fewer +long two-point shots, deploying more versatile defenders, and relying less on the strategy of pounding the ball +into the paint in an attempt to get a high-percentage shot (Schuhmann, 2021). In American football, teams +are going for it on fourth down far more often than in the past, a direct result of statistical analysis showing +that most teams were previously overly conservative (Lopez, 2020). And, of course, in baseball, teams are +using defensive shifts to maximize the probability of recording an out, encouraging hitters to improve their +launch angles, and optimizing pitcher repertoires to minimize contact (Healey, 2017). +These are just the most obvious examples of strategic changes that are fueled by insights extracted from data +by practitioners of sports analytics. Similar insights are now being made in less obvious settings, including +esports (Clark et al., 2020; Maymin, 2021). These insights come both from academia, where researchers +typically use public data to produce high-caliber, peer-reviewed scientific work, as well as from industry, where +highly-trained analysts work with with players, coaches, and team officials to put new ideas into immediate +arXiv:2301.04001v1 [stat.AP] 10 Jan 2023 + +A preprint - January 11, 2023 +effect thanks to high-resolution, often proprietary data. A growing pool of people move seamlessly between +these two worlds, leading to the formation of partnerships and the cross-pollination of ideas. +Every sport is different, with its own set of rules, strategies, methods of data collection, number of players, +and the magnitude of the role of chance. At the same time, many sports are similar, either because one +evolved from the other, or the structure of the games share certain attributes. Sports that are closely related +historically may or may not share common applications of analytical methods. For example, despite belonging +to the same bat-and-ball family, baseball and cricket differ in strategies such as batting order or sacrifice +plays. Conversely, with just a few small tweaks, analytical metrics might work just as well across sports that +are unrelated and quite different. For instance, an Elo rating could be equally valid for chess players and ice +hockey teams. +In this paper, we explore four key ideas that have widespread applicability across many sports: the expected +value of a game state (Section 2), win probability (Section 3), measures of team strength (Section 4), and the +use of sports betting market data (Section 5). In each case, we define the concept mathematically, explain +how it originated, and give examples of its applications in multiple sports. Our goal is to unify the conceptual +threads, while doing some justice to the customizations necessary to make a metric meaningful in a particular +sport. We include copious references to original works of scholarship. +Doing the work of sports analytics requires computing with data. While the sources of sports data are too +numerous to list, in Section 6 we highlight a few computational tools (including a table of R packages) that +make this kind of work possible. Section 7 lists several opportunities for disseminating work publicly. We +conclude in Section 8 with a short discussion of some ideas that are not explored in this paper. Notably, we +omit a treatment of individual player ratings for team sports, since this concept has been covered ably in +these pages by Albert (2015), and its inclusion would double the length of this manuscript. We do, however, +discuss individual player ratings in the context of one-person teams (e.g., chess, tennis) in Section 4. +We encourage readers to explore Cochran et al. (2017) and Albert et al. (2016) for collections of articles in +sports analytics that provide broad coverage of the field. +2 +The expected value of a game state +In many sports, the first step towards an analytical understanding is the estimation of the expected value of +a game state at any given point in it. Mathematically, we define X to be a random variable indicating the +number of points (or runs) that a team will score over some determined amount of time (e.g., remainder of +game, quarter, period, or inning). Let s ∈ S be a tuple that encodes the state of a game. Then our task is to +estimate: +E[X|s] = +� +x≥0 +Pr[X = x|s] · x , +(1) +for any state s ∈ S, where Pr[X = x|s] is the probability of scoring x points given that the game is in state s +and S is the set of all possible states. +The concept of a state is easier to grasp in a sport that can be modeled as discrete (in the sense of discrete +event simulation). By discrete, we mean a sport that can be easily broken into short, distinct segments of +action which are typically summarized categorically. Each of these segments might represent a state s. For +example, each pitch in baseball is either a ball or a strike. If the ball is put in play, then there may be a +complex sequence of movements by the players, but ultimately (within a few seconds) that sequence will end +and no more action will be permitted until the next pitch. At the beginning and end of each phase of action, +we will know definitively which team is on offense and defense, which runners are on which bases, the score, +how many outs there are, etc. Tennis could similarly be viewed as a series of discrete actions defined by each +point. To say that a player is winning 6-2, 3-1, 40-15 and serving with one fault committed is to characterize +the state of the match. In American football, the game can be broken down into a discrete sequence based +on each down. Contrast this to sports like lacrosse, soccer, or any variant of hockey, which feature largely +running clocks and continuous player movement. In these sports, it is not obvious how to break up the action +into discrete chunks. +In this Section, we illustrate how the fundamental concept of the expected value of a game state leads to +compelling findings in a variety of sports. +2 + +A preprint - January 11, 2023 +Table 1: George Lindsey’s expected run matrix. Note how (when reading across the rows) the expected runs +decrease as outs increase for the same configuration of baserunners, while (when reading down the columns) +expected runs generally increase as baserunners advance. 000 means no runners on base, and 110 means +runners on second and third bases. +Out +Base +0 +1 +2 +000 +0.461 +0.243 +0.102 +001 +0.813 +0.498 +0.219 +010 +1.194 +0.671 +0.297 +011 +1.390 +0.980 +0.355 +100 +1.471 +0.939 +0.403 +101 +1.940 +1.115 +0.532 +110 +1.960 +1.560 +0.687 +111 +2.220 +1.642 +0.823 +2.1 +Discrete event analysis +First, we explore results derived from the expected value of a state in sports where discrete event analysis +is common. We draw primarily on baseball and American football, but applications in other sports (e.g., +tennis) are common (see for example, Kovalchik & Reid (2019)). +2.1.1 +In baseball, the expected run matrix +In baseball, s is typically determined by two factors: the configuration of the runners on base (there are 8 +possibilities) and the number of outs (3 possibilities). Thus, there are |S| = 24 = 8 · 3 basic states of an inning +in baseball1, and we are often interested in the number of runs that will be scored from some state until the +end of the inning. In this example using baseball, E[X|s] is the expected number of runs scored between now +and the end of the inning given that the inning is currently in state s. The collection of estimates E[X|s] for +all 24 states is called the expected run matrix 2, and it is foundational in baseball analytics. +Early work on this topic can be found in Lindsey (1963), who used play-by-play data to compute an empirical +estimate for the mean number of runs scored in the remainder of the inning for each of these 24 possible +states of an inning. This line of work led to analysis of all types of common baseball strategies. For example, +many baseball teams elect to attempt a sacrifice bunt with a runner on first and no one out in the inning, +with the goal of moving the runner to second base, at the cost of the batter being out. Figure 1 shows a +reproduction of Lindsey (1963)’s original calculations, and Table 1 shows the expected run matrix in its most +common form. +Tango et al. (2007) (and many subsequent analyses) conclude that the sacrifice bunt is rarely worth it, +because most teams would be expected to score more runs with a runner on first and no outs than they +would with a runner on second and one out. +It is worth emphasizing that the values in E[X|s] are estimates, and the precision of those estimates has +many subtleties. +First, the values within the expected run matrix change over time. For example, any estimation of the values +in the expected run matrix based on data from a high-scoring era (e.g., the early 2000s) will yield different +values than equivalent analysis in a low-scoring era. In a high run-scoring environment, where there are +many home runs, the value of a walk may be higher, since a player who walks is more likely to score on a +subsequent home run. Conversely, in a low run-scoring environment where hits are hard to come by, stolen +bases and sacrifice bunts may be comparatively more valuable. Thus, a careful estimate of E[X|s] would +include a time parameter t, indicating when the estimate is appropriate. +125, if you include the absorbing state of 3 outs that describes the end of an inning. +2There is no inherent dimensionality to E[X|s]. The matrix nomenclature stems from its values typically being +displayed in 8 × 3 grid. However, when computing with E[X|s], it is most often convenient to treat it as a 24 × 1 +vector. +3 + +A preprint - January 11, 2023 +Figure 1: Table 1 from Lindsey’s original paper. The column labeled E(T, B) gives the expected run matrix +as a vector, based on Lindsey’s analysis of Major League Baseball data from 1959 and 1960. +4 + +TABLE I +DISTRIBUTION OF +SCORES +S INREMAINDEROFI +HALF-INNING +Data +B +TN(T,B) +P(o|T,B) +P(xT,B) +P(2[T,B) +P(>2|T,B) +E(T,B) +a/VN +59/60 +0 +6561 +·747 +.136 +.068 +.049 +·461 +.012 +I +4664 +.855 +.085 +.039 +.021 +·243 +.011 +2 +3710 +·933 +.042 +.018 +.007 +.102 +.008 +59/60 +I +1728 +.604 +.166 +.127 +.103 +.813 +.031 +I +I +2063 +-734 +.124 +.092 +.050 +·498 +.022 +I +2 +2119 +.886 +.045 +.048 +.021 +.219 +.016 +59/60 +2 +294 +·381 +·344 +.129 +.146 +1.194 +.083 +2 +I +657 +.610 +.224 +.104 +.062 +.671 +.043 +2 +2 +779 +·788 +.158 +.038 +.016 ++297 +.024 +59/60 +3 +67 +.12 +.64 +.11 +.13 +I.39 +.09 +3 +I +202 +·307 +·529 +.104 +.060 +·980 +.072 +2 +327 +·738 +.208 +.030 +.024 ++355 +.040 +59/60 +12 +367 +-395 +.220 +.131 +.254 +1.471 +.087 +12 +I +700 +·571 +.163 +.119 +.147 +·939 +.051 +I2 +2 +896 +·791 +.100 +.061 +.048 ++403 +.032 +59/60 +13 +0 +119 +.13 +·41 +.18 +.28 +I·94 +.15 +13 +I +305 +·367 +·400 +.105 +.128 +1.115 +.077 +13 +2 +419 +.717 +.167 +.045 +.071 +·532 +.054 +59/60 +23 +73 +.18 +.25 +.26 +·31 +1.96 +.18 +23 +I +176 +.27 +.24 +.28 +.21 +1.56 +.10 +23 +2 +211 +.668 +.095 +.170 +.067 +.687 +.080 +59/60 +F +0 +92 +.18 +.26 +.21 +·35 +2.22 +.20 +F +I +215 +·303 +.242 +.172 +.283 +1.642 +.105 +F +2 +283 +.671 +.092 +.102 +.135 +.823 +.085 +ZN=27027 +52/60 +F +0 +173 +.17 +.27 +..17 +·39 +2.254 +.145 +F +1 +419 +·310 +.242 +.186 +·262 +I.632 +.080 +F +2 +527 +.645 +.114 +.110 +.131 +.861 +.06A preprint - January 11, 2023 +Figure 2: Table 1 from Carter and Machol’s original paper. Note the monotonic increase in expected point +values as the team gets closer to the endzone. +Second, the characterization of S as having 24 states is only the simplest possible. The inning, or the score of +the game, or even the weather, could be incorporated into S, as those conditions might reasonably affect the +estimate of E[X|s]. More definitively, the identity of the current batter, pitcher, or batter on deck, might also +affect the estimate of E[X|s]. Indeed, Tango et al. (2007) show that when a particularly weak-hitting batter +is up (i.e., the pitcher), a sacrifice bunt becomes a more effective strategy. +See Albert & Bennett (2001) for a fuller discussion of the use of the expected run matrix in baseball and +Marchi et al. (2018) for examples of how to estimate the expected run matrix using Retrosheet data and the +R statistical computing language (R Core Team, 2022). +2.1.2 +In American football, expected points +The concept of estimating the value of the state of a game is easily extended to other sports. For example, in +American football, s is determined by situational variables such as down, yardage to the next first down, +time remaining in the game, and field position. +The task of estimating expected points of possession in football goes back to Carter & Machol (1971), who +estimate the expected points for 1st and 10 plays in the NFL, given any yard line on the football field. Due +to limitations regarding the amount of data collected, the authors divide football field into 10-yard buckets, +centered at their midpoints (e.g. 5, 15, 25, 35, etc.), before averaging the value of the next scoring instance +across the field to obtain the expected points. Figure 2 shows a reproduction of Carter & Machol (1971)’s +estimates. As expected, the estimated expected points increases monotonically as the teams gets closer to +the endzone. One limitation of this approach is the linearity assumption, which results in a high negative +expected point value when the offensive team is 95 yards away from the opponent’s goal line. +Early work on expected point values in American football can also be found in Carroll et al. (1988). In +particular, the authors consider a similar approach to Carter & Machol (1971) and propose a linear model +for expected points in the NFL. They determine that every extra 25 yards is associated with 2 more points +scored on average for a football team. +5 + +TABLE I +THE +EXPECTED POINT VALUES OF POSSESSION OF THE FOOTBALL WITH FIRST +DOWN AND TEN YARDS TO GO FOR VARIOUS TEN-YARD STRIPS +Center of the ten-yard strip +(yards from the target goal line): X +Expected point value: E(X) +95 + I, 245 +85 +0.637 +75 ++o.236 +65 +0.923 +55 +1.538 +45 +2.392 +35 +3.167 +25 +3.681 +15 +4·572 +6.041A preprint - January 11, 2023 +Other attempts at modeling expected points in football are Goldner (2012) and Goldner (2017), who propose +a Markov framework. In particular, the author considers a football drive as an absorbing Markov chain, +consisting of distinct absorbing states that include touchdowns, field goals, and other possession outcomes. +An absorbing state is a link in a Markov chain from which there are no possible transitions (i.e., it is the end +of the chain). For any given play, the expected points are calculated using the absorption probabilities for +different scoring events. +A more in-depth overview of the history of expected points in sports is provided in Yurko et al. (2019) (Section +1.1). Most importantly, Yurko et al. (2019) use publicly available data provided by the nflscrapR package +(Horowitz et al., 2020) to model the expected points on a play-by-play level in football. The authors introduce +a multinomial logistic regression approach, which takes into account the current down, time remaining, yards +from endzone, yards to go, and indicators for goal down situation and whether there are less than two minutes +remaining in the half. Their model estimates the probabilities of the following possible scoring outcomes +after each play: no score, safety, field goal, and touchdown for both the offensive and defensive teams, all of +which have a point value. The expected points for a play can then be calculated accordingly, by summing up +the products of the scoring event point values and their associated probabilities (see Equation 1). +In addition, Pelechrinis et al. (2019) develop an expected points framework in the same spirit as the previous +work, but account for the strength of the opponents in their method. They state that by failing to account +for opponent strength appropriately, about 124.8 points per team each season (or about 3.8 wins per season) +are not credited correctly. This is a substantial amount in a 16-game season. +2.1.3 +In American football, 4th down strategy +The concept of expected points in American football has many applications. One of the most notable and +well-studied topics is the evaluation of 4th down strategy. There is near universal consensus in the literature +that NFL teams have been too conservative in the past when making 4th down decisions. +Romer (2006) examines 4th down decisions in the NFL using expected points by focusing only on examples +from the first quarter of a game (to avoid issues with end-of-half and end-of-game decision making). They +concluded that teams don’t go for it enough if teams are trying to maximize their probability of winning the +game. +Numerous other papers (see Lopez (2020) for details) use the analysis of the expected number of points to +improve fourth down strategy. In addition to Romer (2006), later work by Yam & Lopez (2019) uses win +probability (see Section 3), rather than expected points, and a causal inference framework to reach similar +conclusions that NFL teams are too conservative in going for it on 4th down. In addition, they estimate that +a better strategy would be worth about 0.4 wins per season on average, a substantial amount comparable to +the effect size reported by Pelechrinis et al. (2019) above. +Lopez (2020) presents an introduction to NFL tracking data, and examines 4th down behavior as an example +of the type of problem that can be more thoroughly studied with the increase in granularity of the tracking +data over traditional NFL data. In the past, when looking at down and distance data to study whether NFL +coaches are making good decisions about whether to “go for it” or punt on 4th down, the distance data is only +a rounded approximation of the true distance “to go” (i.e. 1 yard, 1 foot, and 1 inch will all be recorded as +4th and 1. In fact, anything up to 2 yard will recorded as 4th and 1 (Lopez, 2022). However, a coach on the +field during a game will be able to clearly see the difference between 1 inch and 1 yard, and this information +will factor into their decision making. With tracking data, the “to go” distance can be much more accurately +assessed and therefore evaluation of 4th down coaching decisions can now account for this “extra” information +that is available to a coach on the field of play, but not recorded in traditional NFL data. Many past analyses +of the decision to go for it or not on 4th down conclude that coaches in the NFL are too conservative in their +decision making. Lopez (2020) also concludes that coaches are too conservative on 4th down decision making, +but notes further that past estimates of the magnitude of how conservative coaches are on 4th down may be +overstated due to the way in which to go yardage was recorded only approximately in the past. +2.1.4 +Other applications of expected points in American football +Researchers have also applied the notion of expected points to investigate other aspects of the game of +football, including quarterback performance and coaching decisions. +For quarterback evaluation, White & Berry (2002) present a tiered logistic regression method that can be, in +general, applied to any regression setting with a polychotomous response. Using this technique, they estimate +6 + +A preprint - January 11, 2023 +the value of NFL plays using a simple expected points model with down, yards to go, and yards to goal as +predictors. Accordingly, the model results are utilized to obtain ratings and rankings for NFL passers. +Alamar (2010) implements an expected points framework to examine play calling in the NFL. However, rather +than assessing each play on its own, they evaluate the play in the context of the drive. Based on play-by-play +data from 2005 through 2008, they determine that teams are under-utilizing passing plays in some situations. +Another application of expected points is to evaluate kickoff decisions made by football coaches, as demon- +strated by Urschel & Zhuang (2011). Specifically, they look at surprise on-sides kicks versus regular kickoffs +and the decision to accept a touchback versus returning the kickoff. Using data from the 2009 NFL season, +they conclude, as many have, that coaches in the NFL tend to make conservative decisions. +2.2 +Continuous event analysis +Even in sports where the concept of a state is more difficult to define, the value of a possession can be +estimated with the help of tracking data. Over the past decade or so, professional sports leagues have +collected tracking data which record the locations of all players and the ball (or puck) throughout a game. +This high-resolution data allows researchers to produce advanced analyses of the captured spatiotemporal +information and better understand the game. This is a great leap forward from older resources such as +traditional box-score results and play-by-play data. +2.2.1 +In basketball, expected point value +In basketball, Cervone et al. (2014) and Cervone et al. (2016) introduce expected possession value (EPV) as +a means toward an assessment of a player’s on-court performance. This metric is a continuous-time estimate +of the expected number of points for the offensive team on a given possession using player and ball locations. +The EPV takes into account all possible outcomes (a shot attempt, a pass, etc.) for a given player with +the ball, with different weights being assigned to each decision. The computation of the EPV statistic is +done using a (technically discrete) Markov model conditioned on spatial locations. Consequently, the authors +derive a metric called EPV-Added (EPVA), measuring a player’s EPV contribution in a given situation +relative to a league-average player. +A demonstration of the EPV model presented in Cervone et al. (2016) is available at https://github. +com/dcervone/EPVDemo. Figure 3 illustrates how the provided tracking data informs the evolution of EPV +throughout the play. It displays a snapshot of a possession during the NBA regular season matchup between +the Miami Heat and the Brooklyn Nets on November 1, 2013. Miami is the team on offense in this possession, +whose outcome is a 26-foot three-point miss by Mario Chalmers. The plot consists of two elements: 1) +(bottom) the player locations on the court at a particular moment in this possession: when the ball just left +Chalmers’s hands, and 2) (top) a line graph showing how the EPV changes continuously throughout the play +until the three-point attempt. For this possession, the estimated EPV for the Miami Heat reaches its peak at +1.276 points at the moment the shot is taken. +Note that Miami starts the play with an EPV of approximately 1.0 points, which indicates their implied +average points per possession. Chalmers’ shot is worth 3 points, so the EPV of 1.276 points implies that the +model estimate of the probability of Chalmers making this shot is 42.5%. A breakthrough in this work is +that this estimate is conditional on the locations of the other 9 players on the basketball court. +Another framework for estimating expected points in basketball is proposed by Sicilia et al. (2019). The +authors offer a different point of view on expected points, where they first consider a classification model which +returns the probabilities for whether a player would commit a foul (shooting and non-shooting), turnover, or +attempt a shot. The values associated with each of those “terminal actions” are then used to compute the +expected points within a basketball play. +See also Bornn et al. (2017) for more information on how tracking data have enabled advanced statistical +analyses of basketball in recent years. The strategy of maximizing expected points in basketball has led +directly to the proliferation of three-point shooting in the NBA. +2.2.2 +In American football, yards gained +In Section 2.1.2, we discussed advances in American football analytics based on discrete game states defined by +down, yards to first down, field position, etc. The advent of player tracking data makes it possible to analyze +American football using continuous states. For example, Yurko et al. (2020) use tracking data provided by +the 2019 NFL Big Data Bowl (see Section 7) to model the expected yards gained for a ball-carrier during the +7 + +A preprint - January 11, 2023 +Figure 3: Player locations and estimated EPV for a possession during the Miami Heat (red) vs. Brooklyn +Nets (black) NBA game on November 1, 2013. The captured moment is when Miami’s Mario Chalmers +just releases a three-point shot, which ends up as a missed field goal. Figure created by Cervone, et al. +https://github.com/dcervone/EPVDemo +course of a play. As an extension to pre-existing approaches, the authors use conditional density estimation to +obtain a probability distribution for the number of yards gained during the play, rather than only producing +a single estimate for the expected yards gained. Accordingly, the probability of various types of outcomes at +the end of a play such as a touchdown or a first-down gain can be computed from the distribution of the +end-of-play yard line. +Expected point value is also the main component of a novel NFL quarterback evaluation metric introduced by +Reyers & Swartz (2021). The authors take advantage of player tracking data to account for different passing +and running options on the football field that are available to the quarterback. The expected points and +probabilities associated with the possible quarterback options are estimated using the method of ensemble +learning via stacking. +2.2.3 +In other sports +The notion of expected possession value has also been extended to association football (soccer). Fernandez et +al. (2021) implement deep learning methods to examine the instantaneous expected value of soccer possessions. +This approach considers passes, ball drives, and shots in soccer as the main set of actions used to compute the +EPV metric. Many applications can be derived from this framework, including predicting which footballer on +the pitch is most likely to receive the next pass from the current on-ball player. +Macdonald (2012) uses expected goals to evaluate ice hockey players, but does not have access to player +tracking data necessary to evaluate possessions. Kumagai et al. (2021) offer an EPV metric for ice hockey +via a Bayesian space-time framework. +2.3 +Optimal strategies that don’t maximize expected points +Earlier in Section 2, we defined the expected value of a possession based on the state s of the game in terms +of the expected number of points (runs) X that would be scored in the remainder of some period of time. +We then showed how this value could be used to analyze the relative effectiveness of certain strategies, with +8 + +EPV +1.50 +Deron Williams +1.25 +1.00 +)PaulPierce +0.75 +0.50 +)JoeJohnson +)KevinGarnett +1 +5 +)BrookLopez +5 +DMarioChalmers +)DwyaneWade +LeBronJames +)UdonisHaslem + Chris BoshA preprint - January 11, 2023 +the simple idea that strategies that yield higher expected values are preferable. Generally, the goal of any +sport is to score more points than the other team, which most often means trying to score as many points as +possible, leading to a general strategy of maximizing expected points. However, there are situations in which +maximizing the number of expected points is not the desired strategy. +For example, in the bottom of the ninth inning of a tied baseball game, the optimal strategy for winning +the game is maximizing the probability of scoring at least one run, which may differ from the strategy of +maximizing expected runs. If we let U be the set of all strategies, then we assert that it is not always the +case that the strategy u ∈ U that maximizes the expected number of points will maximize the probability of +winning: +arg max +u∈U +Pr[X > 0|s, u] ̸= arg max +u∈U +E[X|s, u] . +Consider the situation where runners are on first and third base, and the score is tied in the bottom of the +ninth inning with no one out. Information derived from Table 1 reveals that the expected number of runs +scored in the remainder of the inning is 1.94 runs, while the probability of scoring zero runs is 0.13. The +defense is in a tight spot, facing an 87% probability of losing the game. However, by walking the hitter to +load the bases, they create the opportunity to force the lead runner at home and thus reduce the chances of +scoring to 82%, even though they raise the expected number of runs scored to 2.22. In this case, the defensive +team is wise to pursue the strategy of maximizing the expected number of runs scored, because it minimizes +the probability of scoring at least one run. +Maximizing the probability of scoring is optimal in any sudden-death situation, which has (but currently +does not) included overtime in American football (Martin et al., 2018). +The situation gets even more interesting when teams modify both their offensive and defensive strategies +simultaneously. For example, in hockey teams will often pull their goalie when trailing in the final period. +This strategy severely weakens their defense but strengthens their offense. The hope is to score a quick +goal to get back in the game, but the risk is falling further behind. Beaudoin & Swartz (2010) show that +NHL coaches do not always employ the optimal strategies, usually by waiting too long to pull their goalies. +Skinner (2011) develops a general framework for these desperation strategies, which include the onside kick +in American football, pulling the infield and/or outfield in baseball, and of course, the fabled Hack-a-Shaq +strategy in basketball. +3 +Win probability +A related, but different concept to expected points is the notion of win probability. Win probability is simply +an estimate of the probability that a team will win the game, given its current state s. Extending the +mathematical framework we defined in Section 2, let Wi be a binary random variable that indicates a win for +team i. Then, +Pr[Wi|s] , +is the win probability for team i in the state s. +This win probability is closely related to the expected value of a state. Albert (2015) defines the win +probability as: +Pr[Wi|s] = +� +X≥0 +Pr[X|s] · Pr[Wi|X, s] , +where Pr[Wi|X, s] is the probability that team i will win the game given that they score X points from state +s. +Win probability is easily extended to provide a measure of the impact of sports plays and individual player +contributions, as discussed in Albert (2015). Given its popularity, recent books on sports analytics often +dedicate multiple chapters entirely to win probability. These include Albert & Bennett (2001), Schwarz +(2004), Tango et al. (2007), Albert et al. (2016), and Winston et al. (2022). +In this section, we discuss notable previous work on win probability in baseball, American football, basketball, +and several other sports. +3.1 +Baseball +The notion of win probability in baseball goes back to at least as early as Lindsey (1961), who calculates the +expected win probability after each inning based on the distribution of runs scored in each inning. Inspired by +9 + +A preprint - January 11, 2023 +Lindsey (1961)’s work, Mills & Mills (1970) utilize win probability to introduce Player Win Average (PWA), +a measure of a player’s contribution to the game outcome. In particular, PWA is computed as +PWA = +Win Points +Win Points + Loss Points, +where the win and loss points represent how much the player positively and negatively impacts their team’s +probability of winning after each play. In effect, the win points are the sum of the changes in Pr[Wi|s] from +one state to the next. +Additionally, a mathematical model for estimating win probability in baseball is presented in Tango et al. +(2007). The authors use Markov chains to look at win expectancy throughout the course of a baseball game. +This approach considers different states of the game such as base, inning, outs and score, and outputs win +probabilities accordingly. +See Albert (2015) for a more detailed historical overview of the use of win probability in baseball. +3.2 +American football +In recent years, a number of statistical methods have been used to build well-calibrated win probability +models in American football. These are flexible algorithms that have high predictability, can account for +nonlinear interactions between the explanatory variables, require few assumptions, and produce feature +importance scores. +Lock & Nettleton (2014) implement a random forest framework to provide a win probability estimate before +each play in a football game. Covariates included in this tree-based method are the current down, score +differential, time remaining, adjusted score, point spread, number of timeouts remaining for each team, total +points scored, current yard line, and yards to go for a first down. According to this model, the difference +in score between the two teams is the most important feature for predicting win probabilities at any given +moment in an NFL game. +In addition, Yurko et al. (2019) estimate win probability in the NFL using a generalized additive model +(GAM), as part of the nflscrapR package (Horowitz et al., 2020) and nflWAR framework. This model takes +into account the estimated expected points obtained from the model described in Section 2, along with +other predictors for time, current half, and timeouts. The two win probability frameworks proposed by Lock +& Nettleton (2014) and Yurko et al. (2019) were also implemented in Yam & Lopez (2019) with minimal +modifications. Specifically, the authors combined both approaches to estimate the win probability for each +play, with an overall goal of assessing fourth down decision-making in American football. +A vital highlight of Yurko et al. (2019)’s win probability model is that it is fully reproducible and uses +publicly available data. One of Yurko et al. (2019)’s goals was also to encourage researchers to “use, explore, +and improve upon our work,” which ultimately inspired nflfastR (Carl & Baldwin (2022)), now considered +the successor to nflscrapR. +Figure 4 shows a win probability graph for the 2018 NFL Playoffs Divisional Round matchup between the +New Orleans Saints and the Minnesota Vikings on January 14, 2018. We obtain the estimated probability of +winning for each team using the nflfastR R package, which implements a gradient boosting model via the +xgboost library (Chen et al. (2022)) for estimating win probabilities. Minnesota was leading throughout +the first three quarters of the game, having win probabilities of 0.869, 0.941, and 0.742 at the end of the +first, second, and third quarters, respectively. The win probabilities get close to parity late in the fourth +quarter, when the Saints took the lead with 3:01 left in the game. The last play of this game—famously +known as the Minneapolis Miracle—resulted in a drastic swing in win probabilities for both teams. With 10 +seconds remaining in the game, the Vikings begin the final possession with a 25.3% chance of winning. Their +probability increased to a perfect 1 when Stefon Diggs scored a game-winning 61-yard receiving touchdown +as the game clock expired. +3.3 +Basketball +Stern (1994) provides an investigation of in-game win probability and the scoring process in basketball using +a Brownian motion model. Let p(l, t) represent the win probability for the home team given an l-point lead +after t seconds of game time. The model introduced by Stern (1994) is a probit regression model, which +10 + +A preprint - January 11, 2023 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +900 +1800 +2700 +3600 +Time Remaining (seconds) +Win Probability +Minnesota Vikings +New Orleans Saints +Figure 4: Win probability graph for New Orleans Saints vs. Minnesota Vikings in the 2017–18 NFL Playoffs. +provides an estimate for p(l, t). Specifically, +p(l, t) = Φ +� +l + (1 − t)µ +� +(1 − t)σ2 +� +. +Here, a Brownian motion process with drift µ points advantage for the home team and variance σ2 is used to +model the score difference between the home and away teams. +On a related note, Deshpande & Jensen (2016) extend Stern (1994)’s framework by applying it in a Bayesian +setting. Deshpande & Jensen (2016) propose a Bayesian linear regression model to assess the impact of +individual players on their team’s chance of winning at any given time of a basketball game. This model +assumes independence of observations and constant variability in win probability. +Moreover, McFarlane (2019) uses logistic regression to estimate win probability for evaluating end-of-game +decisions in the NBA. The approach takes into account the remaining game time, score difference, and point +spread. This win probability model is then applied to the calculation of the End-of-game Tactics Metric +(ETM), measuring how the chance of winning a basketball game differs between the optimal and on-court +actual decisions. +3.4 +Other sports +The idea of win probability is also applied to other sports, with a diverse range of statistical techniques being +used to estimate the probability of winning for a player or team. Brenzel et al. (2019) use three-dimensional +Markov models to estimate win probability throughout a curling match. In particular, the authors propose +both homogeneous and heterogeneous Markov models for estimating the chance of winning in curling, with +different independence assumptions on the relationship between performance and the current state of the +game. In esports, Maymin (2021) relies on logistic regression to build a well-calibrated in-game win probability +model for each specific moment during a game of League of Legends. Moreover, Guan et al. (2022) develop +an in-game win probability model for the National Rugby League using functional data analysis. In this +11 + +A preprint - January 11, 2023 +approach, the rugby play-by-play event data are treated as functional, and the win probability is expressed +as a function of the match time. +4 +Team strength +A third crucial idea in sports analytics is the estimation of team strength. First, we briefly introduce a simple +method for estimating team strength based on win-loss record. Next, we detail three other more sophisticated +methods for estimating team strength in sports through pairwise evaluations. The methods in this Section +apply equally well to multiplayer teams and single-player teams. +The impetus for all methods for estimating team strength is the realization that win-loss records are a noisy +measure of team strength. As binary outcomes, and with all sports (except perhaps chess) involving some +element of chance, wins and losses carry some signal of team strength, but we can do better. +4.1 +Expected winning percentage +A simple method for estimating team strength that has become popular in sports analytics is expected +winning percentage—often called Pythagorean expectation—developed by James (2003). Later, Miller (2007) +derived the formula as a consequence of assuming that runs (in baseball) are generated by two independent +Weibull processes. +Expected winning percentage is just: +� +wpct = +Xα +Xα + Y α , +where X is the number of points (runs) that a team has scored, and Y is the number of points (runs) that +they have allowed, over some specified time period. James’s work was originally in baseball, and he posited +the value of α = 2. The resemblance to the formula for computing the length of the hypotenuse in a right +triangle provides the nod to Pythagoras. +Subsequent analysts have tried to find the optimal value of α for various time periods. This can be done with +a few lines of code, after observing that +Xα +Xα + Y α = +1 +1 + (Y/X)α +and fitting a non-linear model (see similar discussion in Baumer et al. (2021)). Figure 5 illustrates the quality +of the fit in Major League Baseball since 1954, where the optimal value of α is 1.84. +Many authors have fit expected winning percentage models to other sports—too many to cite here. See, for +example, Hamilton (2011) for association football (soccer), Caro et al. (2013) for Division I college football, +and notably, future NBA general manager Daryl Morey for basketball (Dewan & Zminda, 1993). +4.2 +Bradley-Terry models +Perhaps the most widely-used probability model for predicting the outcome of a paired comparison is the +Bradley-Terry model (BTM) (Bradley & Terry, 1952). For a pair of players (or teams) i and j, let Πij denote +the probability that i is preferred to j. Then the BTM is a logistic regression model with parameters βi, βj +such that +log +�Πij +Πji +� += βi − βj . +Here, exp (βi) is often viewed as a representation of team i’s ability. +The BTM can be implemented in R via the BradleyTerry2 package (Turner & Firth, 2020). As an example, +we consider the data given in Agresti (2018) (page 247) on tennis results from 2014–2018 for five men’s +professional players: Novak Djokovic, Roger Federer, Andy Murray, Rafael Nadal, and Stan Wawrinka. We +fit a BTM to estimate the win probability for each pair of players and obtain a ranking for this group of five. +Table 2 shows the estimated coefficients of the fitted BTM. According to the abilities, between 2014 and 2018 +the players are ranked as follows: 1) Djokovic, 2) Federer, 3) Wawrinka, 4) Nadal, 5) Murray. In addition to +an ordering, the magnitude of the coefficients in Table 2 provide a measure of relative strength. +12 + +A preprint - January 11, 2023 +0.3 +0.4 +0.5 +0.6 +0.7 +0.75 +1.00 +1.25 +1.50 +Ratio of Runs Scored to Runs Allowed +Winning Percentage +Figure 5: Winning percentages vs. runs scored and runs allowed in baseball, 1954–2021. The navy line +represents the expected winning percentage model posited by James, with the exponent value of 2. The gold +line shows the same model with an optimal exponent of 1.84. +Table 2: The estimated abilities (with standard errors) for each tennis player, relative to Wawrinka, obtained +from the fitted Bradley-Terry model. +Player +Ability +SE +Djokovic +1.176 +0.500 +Federer +1.136 +0.511 +Wawrinka +0.000 +0.000 +Nadal +-0.062 +0.515 +Murray +-0.569 +0.568 +To obtain win probabilities, as an illustration, for the Federer-Nadal matchup, an estimate for the probability +of a Federer victory is: +ˆΠ24 = +exp(ˆβ2 − ˆβ4) +1 + exp(ˆβ2 − ˆβ4) += +exp(1.136 + 0.062) +1 + exp(1.136 + 0.062) = 0.768 . +4.3 +Elo ratings +Another widely known tool for measuring team strength is the Elo rating system (Elo, 1978), which was +originally developed for chess. Given two players i and j with unknown ratings Ri and Rj, the probability +Πij of i beating j is defined as +Πij = +1 +1 + K(Rj−Ri)/400 . +13 + +A preprint - January 11, 2023 +In this formula, K is commonly known as the K-factor, or development coefficient. The International Chess +Federation (FIDE) uses K = 10 for players with any previously achieved rating of at least 2400. Finally, +K = 40 is given to new players with under 30 games played, and players under the age of 18 with rating less +than than 2300 (FIDE, 2022). +Another interpretation for Πij is the expected score of the game for player i. The scores of 0, 0.5, and 1 are +associated with three possible game outcomes loss, tie, and win, respectively. After a game, the updated Elo +rating R∗ +i for player i is +R∗ +i = Ri + K(Si − Πij) , +where Si ∈ {0, 0.5, 1}. When a tournament concludes, a post-tournament rating is obtained for each player +based on the rating updates for all games played. +To illustrate, we consider a chess game played on June 1, 2022 on Chess.com by one of the authors, with data +obtained from the chessR package (Zivkovic, 2022) (see Section 6.1). Prior to the game, the author was +rated 1732, whereas his opponent was rated 1683. Since both ratings are below 2400, we apply a development +coefficient of K = 20 to this example. The probability of the author (a) defeating their opponent (b) was +Πab = +1 +1 + 20(1683−1732)/400 = 0.591 . +The author won the match: that outcome is associated with a score of Sa = 1. The post-game Elo rating for +the author is thus +R∗ +a = 1732 + 20(1 − 0.591) = 1740 . +Besides chess, the Elo system has also been implemented to estimate team strength in other sports. See +Koning (2017) for more information on applications of the Elo rating in soccer, and Kovalchik & Reid (2019) +and Kovalchik (2016) for Elo ratings in tennis. Furthermore, Elo ratings are used extensively for rankings of +teams in numerous sports by the data journalists at FiveThirtyEight.com. +4.4 +Bayesian state-space models +Glickman & Stern (1998) propose a Bayesian state-space model for paired comparisons for predicting NFL +games, allowing team strength parameters to vary over time. In particular, they model point differential in +the NFL by introducing week-to-week and season-to-season as the two primary sources of variation in team +strengths. See also Glickman & Stern (2017) for more discussion on estimating team strengths in American +football. +More recently, Lopez et al. (2018) extend Glickman & Stern (1998)’s state-space model to understand +randomness in the four major American sports leagues. Betting moneylines are used in place of point +differentials in order to estimate team strengths, and this framework also accounts for home advantage. Both +papers motivate the usefulness of model-based measures of team strength by demonstrating their superiority +to low-resolution win-loss records. Apart from sports gambling, having an accurate estimate of team strength +is useful to team officials, who are constantly monitoring and forecasting their team’s ability. +In a similar Bayesian setting, Koopman & Lit (2015) study English Premier League soccer match results +by assuming a bivariate Poisson distribution with time-varying team abilities. This state-space approach +appears to improve on bookmaker’s odds. +5 +Sports betting market data +Most of the research in sports analytics is fueled by the analysis of data recorded from the outcome of sports +contests. However, a growing body of literature is informed by data from sports betting markets. Since the +2018 United States Supreme Court decision in Murphy v. National Collegiate Athletic Association, sports +gambling has exploded in the U.S. The increasing interest in sports gambling has led to increasing interest in +sports gambling data, and that data has proven useful to researchers in at least two major ways. +First, betting market data is probably the best source for estimating the true probability of a team winning a +game. The efficiency of betting market data in this respect has been demonstrated time and time again. The +utility of these estimates have then informed research that has helped us learn about the sports themselves. +In this sense, data generated by sports gambling has been an important source of data useful for sports +analytics (see Section 5.2). +14 + +A preprint - January 11, 2023 +Table 3: 2023 NBA Championship odds for the top 6 and bottom 6 teams. Retrieved from FanDuel Sportsbook +on January 9, 2023. +Rank +Team +Line +Odds +Prob. +Prob. Normalized +1 +Boston Celtics +390 +4.9 +0.204 +0.163 +2 +Milwaukee Bucks +500 +6.0 +0.167 +0.133 +3 +Brooklyn Nets +800 +9.0 +0.111 +0.089 +4 +Golden State Warriors +900 +10.0 +0.100 +0.080 +5 +Los Angeles Clippers +1000 +11.0 +0.091 +0.073 +6 +Denver Nuggets +1100 +12.0 +0.083 +0.067 +25 +Oklahoma City Thunder +50000 +501.0 +0.002 +0.002 +26 +Orlando Magic +50000 +501.0 +0.002 +0.002 +27 +Charlotte Hornets +50000 +501.0 +0.002 +0.002 +28 +Houston Rockets +50000 +501.0 +0.002 +0.002 +29 +San Antonio Spurs +50000 +501.0 +0.002 +0.002 +30 +Detroit Pistons +50000 +501.0 +0.002 +0.002 +Total +- +496590 +4995.9 +1.253 +1.000 +Second, sports analytics researchers have studied various types of sports gambling outlets (including fantasy +sports). This research has estimated probabilities, evaluated common strategies, and offered optimized +approaches for a variety of different games of chance (see Section 5.3). Some researchers have then tried to +demonstrate a positive return on some of these betting strategies, with very limited success. +5.1 +Example: Win probabilities from betting market data +To see how betting market data can be used to estimate team strengths, consider the betting lines posted +on FanDuel Sportsbook for the 2023 NBA Champion on January 9, 2023 and shown in Table 3. This is +a futures market, because the actual NBA champion will not be determined until June 2023. The Boston +Celtics are the favorite to win, with a moneyline of +390, meaning that a $100 bet on the Celtics to win the +championship will pay back the original bet and an additional $390 if the Celtics win it all. This style of +odds are sometimes called American odds. The corresponding fractional odds have the Celtics at 4.9:1 to win +the championship. Conversely, six teams share the lowest odds at +50000. +These moneylines (ℓi) can be converted into an implied probability (pi) using the formula: +pi = +100 +100 + ℓi +. +The sum of those probabilities is greater than one—this is why the sportsbook makes money regardless +of who wins the championship. However, the implied probabilities can be normalized by dividing by their +sum to recover true probabilities of each team winning the championship. Many different researchers have +shown that these normalized implied probabilities are accurate, unbiased, and efficient estimates of the true +unknowable probabilities (see Lopez et al. (2018) for discussion and an extensive list of references). +In this case, the FanDuel futures market suggests that the Celtics have a 16.3% chance of winning the +championship, while the Milwaukee Bucks have the second best chance, at 13.3%. These implied probabilities +can be used to fit various models for team strength, as described in Section 4. +5.2 +The use of betting market data for sports analytics +While Lopez et al. (2018) use betting market data to model team strengths, they do not directly address +strategies for betting or inefficiencies in betting markets. Early work by Gandar et al. (1988) examine +the rationality of NFL betting markets and concludes that statistical tests fail to reject the hypothesis of +rationality. Related work such as Lacey (1990) and Boulier et al. (2006) explores the efficiency of NFL betting +markets in the mid-1980s and late-1990s, respectively. Neither paper finds strong evidence for inefficiencies +in the markets. Boulier & Stekler (2003) compare the predictive performance of power rankings and media +experts to the betting market for NFL games and found that the betting market is the best for predicting +winners. Lopez & Matthews (2015) show that betting market data was most useful in predicting men’s +college basketball tournament outcomes. +15 + +A preprint - January 11, 2023 +Sports betting market data has also been used to investigate competitive behavior within leagues. Soebbing +& Humphreys (2013) find evidence that sports bettors think tanking in the NBA is occurring, although the +evidence for whether it actually is remains mixed. +5.3 +Analytics for sports betting +Many different types of bets can be placed on sports. For individual contests, bets involve point spreads, +moneylines (see Section 5.1 for an example), odds, or other ways of handicapping which team will win. +Money can also be wagered on futures, where odds are given in advance for events that may or may not +transpire (e.g., a certain team making the playoffs, or a certain player winning the MVP award). Here, we +focus on betting pools, in which a group of people compete to predict winners in multiple contests (often a +tournament). We also address the inevitable question of whether strategies exist that will consistently beat +the market. +5.3.1 +Betting pools +One popular type of betting pool is a survivor pool, in which participants stay in the competition as long +as they continue to successfully pick winners. Bergman & Imbrogno (2017) present formal optimization +approaches for NFL survivor pools and conclude that planning for only part of the season yields optimal +results in terms of maximizing survival probability. Imbrogno & Bergman (2022) estimate the probability of +having to share the winning pot in NFL survivor pools. +Perhaps the most commonly-studied sports betting market surrounds the NCAA men’s college basketball +tournament. Breiter & Carlin (1997) use Monte Carlo methods to study the standard “office pool.” Kaplan +& Garstka (2001) consider a variety of NCAA college basketball pools, and find that the simple strategy +of picking the team with the better seed is generally, but not always, optimal. Metrick (1996) finds that +bettors overback the heaviest favorites. Niemi et al. (2008) show an improved return on investment by +picking an undervalued champion and then completing the rest of one’s bracket by using published odds. +Clair & Letscher (2007) develop and test strategies for maximizing expected return in both survivor and +tournament-style pools. +5.3.2 +Beating the market +Naturally, after studying the efficiency of sports betting markets, researchers try to find inefficiencies that +can be exploited for financial gain. Not surprisingly (given the efficiency of these markets), such gains are +hard to come by. +Sauer (1998) finds that while racetrack betting markets are generally efficient, information asymmetry plays +a role in creating inefficient markets. Nichols (2014) concludes that the impact of travel is not completely +incorporated into the betting markets, but that any effect is too small to find any profitable advantage. Paul +& Weinbach (2014) investigate the less-saturated betting market for the WNBA and fail to find strategies +for positive return on investment. Spann & Skiera (2009) show no way to beat the market in the German +premier soccer league, given the high fees associated with placing bets. +More successfully, Buttrey (2016) explores the NHL betting market and produces a model to predict win +probabilities in given games, then tests the model by placing market price bets in games where the predicted +probability differs from the market. They find that their methods were able to produce a positive return on +investment. +6 +Tools +Analytical work in sports requires facility with an ever-changing set of computational tools for working with +data. Sources of authoritative data about sports are myriad, and are too numerous to list here. Software tools +for sports analytics are similarly numerous. For R, we maintain a CRAN Task View for Sports Analytics +that catalogs R packages published on the Comprehensive R Archive Network (CRAN) and organizes them +by sport (Baumer et al., 2022). Table 4 provides an overview of the currently available sport-specific CRAN +packages. Recently, Casals et al. (2022) offer a systematic review of sport-related packages on CRAN. Further, +a more general collection of software tools is being curated by the SportsDataverse initiative (Gilani, 2022). +In the remainder of this section, we highlight a few tools for sports analytics that are of general interest and +illustrate a common paradigm for how these tools can be used in conjunction. +16 + +A preprint - January 11, 2023 +Table 4: A summary of sport-specific packages available on the Comprehensive R Archive Network (CRAN) +as of October 16, 2022. While the major North American sports dominate the list, perhaps the fastest-growing +collection is for esports. +Sport +Number of Packages +List of Packages +American Football +12 +nflverse, nflfastR, nflreadr, nfl4th, nflseedR, +nflplotR, NFLSimulatoR, fflr, ffscrapr, +ffsimulator, gsisdecoder, cfbfastR +Association Football (Soccer) +9 +worldfootballR, engsoccerdata, socceR, +ggsoccer, footballpenaltiesBL, footBayes, +itscalledsoccer, FPLdata, EUfootball +Basketball +8 +BAwiR, AdvancedBasketballStats, uncmbb, +BasketballAnalyzeR, NBAloveR, wehoop, +hoopR, toRvik +Baseball/Softball +7 +Lahman, retrosheet, pitchRx, mlbstats, +baseballDBR, baseballr, runexp +Chess +5 +chess, stockfish, bigchess, rchess, chessR +Esports +5 +CSGo, rbedrock, ROpenData, opendotaR, +RDota2 +Hockey +5 +hockeyR, NHLData, nhlapi, nhlscrape, +fastRhockey +Cricket +4 +yorkr, cricketr, cricketdata, howzatR +GPS Activity Tracking +3 +trackeR, trackeRapp, rStrava +Track and Field +2 +combinedevents, JumpeR +Australian Rules Football +1 +fitzRoy +Swimming +1 +SwimmeR +Volleyball +1 +volleystat +6.1 +Case study in how tools fit together: chess +Many tools in sports analytics provide the ability to read, write, and plot data stored in a sport-specific +format. For example, consider chess, where the sequence of moves in games is often recorded in Portable +Game Notation (PGN). Software tools can then be built around this well-defined format. The chess package +(Lente, 2020) provides R users with the ability to read, write, display, and manipulate chess data in PGN +format. +Application programming interfaces (APIs) are also a common source for data retrieval. In chess, the chessR +package (Zivkovic, 2022) allows R users to download game data from the Chess.com API. This type of +infrastructure, where one package is the “workhorse” that facilitates common generic data operations, and +other packages layer on specific functionality, is common in sports analytics. See Section 4.3 for an example +of how the chessR package can be used to compute Elo ratings. +Figure 6 shows a rendering of the starting chess board obtained via the chess package, along with the final +position in the game won by one of the authors mentioned earlier in Section 4.3 (with data downloaded via +the chessR package). We note how the contextual information provided by the chessboard is instrumental +in helping the reader understand the data (How many of us can visualize PGN directly?). In Section 6.2, we +outline a collection of graphical tools that provide similar context for different playing surfaces. +6.2 +Graphical tools +Creating effective data graphics is a key component of statistical communication, and sports is no exception. +We highlight a few packages that assist with the creation of data graphics about sports. +Each professional sports team has its own brand, most obviously identified by a team logo and set of colors. +The teamcolors R package (Baumer & Matthews, 2020) provides color palettes and logos for men’s and +women’s professional and collegiate sports teams, as well as color and fill scale functions compatible with +ggplot2 (Wickham et al., 2022). For example, the NFL teams’ colors and logos shown in Figure 4 were +provided by the teamcolors package. Figure 7 illustrates how the use of team colors, which have a natural +association for many sports fans, can help to untangle what would otherwise be messy data graphics. In Figure +17 + +A preprint - January 11, 2023 +Figure 6: At left, the starting chess board printed via the chess package. At right, the final position for one +of the authors’ recent wins (a checkmate playing Black). +7, 30 different lines are plotted on top of one another, crisscrossing and intersecting in various unpredictable +ways. However, the use of team colors to identify the lines makes it possible to follow the trajectory of most +teams over the course of the season. +nflplotR (Carl, 2022) has a similar goal to teamcolors. It also provides ggplot2 extensions but is designed +specifically for the NFL. A great feature of nflplotR is the collection of geom_*() (geometric object) functions +that enhance high-quality plotting of NFL team logos and player images with ggplot2. Figure 8 shows a +scatterplot of offensive and defensive expected points added for NFL teams in the 2021 regular season. The +logos of all 32 American football clubs are plotted in place of the usual dots, making it easier for the reader +to identify which team each data point represents. +Player tracking data contains coordinates that reveal player movement, and these coordinates are always +understood in context relative to reference points on the field, court, ice, board, or pitch for a particular sport. +Orienting these points graphically may require drawing a complex set of guidelines that provide that context +to readers. Thankfully, the sportyR package (Drucker, 2022) contains generic playing surfaces for baseball, +basketball, curling, American football, ice hockey, soccer, and tennis that can be added to ggplot graphics +with a single function call. The surfaces plotted in Figure 9 are helpful in contextualizing player tracking +data (such as those shown in Figure 3) and would be laborious for each analyst to have to create on their +own. With the increased availability of player tracking data, this particular tool should see increased usage. +6.3 +Case study in the evolution of tools and research: baseball +As the granularity of baseball data has evolved over time, so too have the statistical methodologies for +modeling that data, and the tools for working with it. +For example, before George Lindsey’s work (see Section 2), most of the baseball data that was publicly +available was seasonal: it showed only season totals for each player. These data, now available through +the Lahman package (Friendly et al., 2022), were sufficient to study broad trends in baseball, and led to +insights such as the value of expected winning percentage (see Section 4.1) and the importance of on-base +percentage relative to batting average. These relatively simple insights fueled the “Moneyball” (Lewis, 2004) +era revolution in sports analytics (B. Baumer & Zimbalist, 2014). +Over time, the resolution of baseball data has improved to include play-by-play data, pitch-by-pitch data, +and now player tracking data. +18 + +兰 +U +5A preprint - January 11, 2023 +COL +ARI +NYN +ATL +MIL +LAN +PHI +MIA +CIN +PIT +SDN +SFN +CHN +SLN +WAS +90 wins +−60 +−30 +0 +30 +0 +50 +100 +150 +Cumulative Games Played +Games above .500 +Daily MLB Standings, 2021 +Source: Retrosheet +Figure 7: The progression of National Leauge team standings during the 2021 Major League Baseball season. +Note how the use of team colors makes it possible to untangle what would otherwise be a messy jumble of +indistinguishable lines. Data provided by retrosheet and colors provided by teamcolors. +The retrosheet package (Douglas & Scriven, 2021) now provides access to the historical play-by-play data +available from Retrosheet (this is a comprehensive version of what Lindsay collected for his research). This +play-by-play data allowed researchers to learn about strategies, like those that we discussed in Section 2. In +baseball, this deepened our understanding of bunting, stolen bases, handedness, batting order, and many +other aspects of the game. Play-by-play data underlies much of the analysis in Tango et al. (2007). +The pitchRx package (Sievert, 2015) provides access to pitch-by-pitch data that fueled innovative research +into catcher framing (Deshpande & Wyner, 2017), pitch values (Healey, 2019), and pitch classification (Sidle +& Tran, 2018). Catcher framing is a notable example of a concept that scouts talked about for decades, but +that could not be quantified by analysts until data of the appropriate resolution became available. +While play-by-play data allows us to make valuations between plays, player tracking data allows us to make +valuations within plays. The baseballr package (Petti & Gilani, 2022) now provides access to player tracking +data from Statcast. These data have led to investigations into how defensive shifts affect batting performance +(Bouzarth et al., 2021), as well as how launch angles affect the probability of hitting a home run (Marchi et +al., 2018). +As we saw above with chess, the packages in baseball fit together in creative ways. In Figure 7, we showed +how teamcolors can illuminate data pulled from retrosheet to make an informative data graphic. One +could just as easily use sportyR to generate a field graphic, and then overlay player tracking data obtained +from baseballr to depict defensive shifts. +Thus, these R packages enable research by making data more easily available. Moreover, because R is +scriptable, they make it easier to share research that is reproducible. Recent conferences, such as the Carnegie +Mellon Sports Analytics Conference, have included a reproducible research competition to foster these efforts +(see Section 7). +19 + +A preprint - January 11, 2023 +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +−0.15 +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +Offensive EPA/play +Defensive EPA/play +Figure 8: Offensive and defensive expected points added per play for the 2021 NFL regular season, plotted +with nflplotR using data from nflfastR. +Figure 9: At left, an NBA basketball court drawn by sportyR. At right, an NHL hockey rink drawn by +sportyR. +20 + +GARADERSBnyNEYORK +UETSSteelersTCV +Vnu