Add files using upload-large-folder tool

-dFAT4oBgHgl3EQfqB3j/content/tmp_files/2301.08645v1.pdf.txt

@@ -0,0 +1,255 @@
+arXiv:2301.08645v1  [gr-qc]  20 Jan 2023
+Complex scalar field in κ-Minkowski spacetime
+Andrea Bevilacqua1, ∗
+1National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa, Poland
+It is often expected that one cannot treat spacetime as a continuous manifold as the Planck
+scale is approached, because of to possible effects due to a quantum theory of gravity. There
+have been several proposals to model such a deviation from the classical behaviour, one of
+which is noncommutativity of spacetime coordinates. In this context, the non-commutativity
+scale is seen as an observer-independent length scale. Of course, such a scale impose a modi-
+fication of ordinary relativistic symmetries, which now need to be deformed to accommodate
+this fundamental scale. The κ-Poincar´e algebra is an example of this deformation. In what
+follows I will briefly describe a construction of a κ-deformed complex scalar field theory,
+while at the same time shedding light on the behaviour of discrete and continuous symme-
+tries in this formalism. This in turn will open the way to the study of the application of
+this formalism to actual physical processes. I will then conclude with some comments and
+prospects for the future.
+I.
+FROM NON-COMMUTATIVE SPACETIME TO THE ACTION
+We will present a brief preview of the works in Refs. [1], [2]. For a more detailed introduction
+to the framework and the formalism, see for example Ref. [3]. The starting point is k-Minkowski
+spacetime, whose coordinates satisfy the an(3) Lie algebra defined as [ˆx0, ˆxi] = iˆxi/κ.
+Notice
+that one recovers canonical Minkowski spacetime in the formal limit κ → ∞. To have a more
+physical understanding of the above commutator, notice that 1/κ has dimensions of length, and it
+is sometimes identified with the Plank length. In order to build fields in this spacetime, we need to
+define plane waves first. To do so, one can proceed as follows. First pick an explicit representation
+of the an(3) Lie algebra in terms of matrices (it turns out that the lowest dimensional one has
+dimension 5). Then, one can define a plane wave as an element ˆek of the relative AN(3) Lie group,
+i.e. for example1 ˆek = exp(ikiˆxi) exp(ik0ˆx0). Notice that, since the elements of the matrices ˆxµ
+∗ andrea.bevilacqua@ncbj.gov.pl
+1 A different choice on the ordering results in a change of coordinates in momentum space.
+
+2
+are dimensionful, so are the parameters kµ. Written explicitly, one has
+ˆek =
+
+
+
+
+
+
+
+
+
+
+
+¯p4
+κ
+k
+κ
+p0
+κ
+p
+κ
+1
+p
+κ
+¯p0
+κ
+− k
+κ
+p4
+κ
+
+
+
+
+
+
+
+
+
+
+
+(1)
+where2
+p0 = κ sinh k0
+κ + k2
+2κek0/κ
+pi = kiek0/κ
+p4 = κ cosh k0
+κ − k2
+2κek0/κ.
+(2)
+Both kµ and pA(k) (A = 0, 1, 2, 3, 4) are two different coordinates of momentum space. Notice that
+the pA are not independent, since −p0 + p2 + p2
+4 = κ2 (notice also that p+ = p0 + p4 > 0). It is
+therefore clear that momentum space is curved. This is reflected in a deformed sum of momenta,
+which is defined through the group property ˆekˆel =: ˆek⊕l. At the same time, inverse momenta
+are defined in terms of group inverses ˆe−1
+k
+=: ˆeS(k), and S(k) is called the antipode of k. In order
+to simplify the treatment of integrals and derivatives, we can use a3 Weyl map W to send group
+elements ˆek into canonical plane waves ep := exp(ipµ(k)xµ). The group law is preserved thanks
+to the ⋆ product defined by W(ˆek⊕l) = ep(k)⊕q(l) =: ep(k) ⋆ eq(l). In general, the ⋆ product is not
+commutative. Because of this, we choose the following action for a κ-deformed complex scalar
+field.
+S = 1
+2
+�
+R4 d4x[(∂µφ)† ⋆ (∂µφ) + (∂µφ) ⋆ (∂µφ)† − m2(φ† ⋆ φ + φ ⋆ φ†)].
+(3)
+To obtain the equations of motion one usually integrates by parts, but in this deformed context
+derivatives do not satisfy the Leibniz rule. In fact, if they did, one could get the following contra-
+diction.
+i(p ⊕ q)µep⊕q = ∂µ(ep ⋆ eq) = (∂µep) ⋆ eq + ep ⋆ ∂µeq = i(p + q)ep⊕q.
+(4)
+Hence one has to use the appropriate deformations for the Leibniz rules. After some computations,
+one can verify that the equations of motion satisfied by the field are the canonical Klein-Gordon
+equations, and the field which satisfies them can be written as
+φ(x) =
+�
+d3p
+�2ωp
+ξ(p)ape−i(ωpt−px) +
+�
+d3p∗
+�
+2|ω∗p|
+ξ(p)b†
+p∗ei(S(ωp)t−S(p)x).
+(5)
+2 The explicit expression of ¯p4 and ¯p0 in terms of kµ is not relevant for the present discussion.
+3 There are several equivalent ways in which a suitable Weyl map can be chosen, see Ref.
+[1] for more details.
+
+3
+The action of the discrete transformations C, P, T can be defined in the usual way in the deformed
+context, i.e. Tφ(t, x)T −1 = φ(−t, x), Pφ(t, x)P −1 = φ(t, −x), and Cφ(t, x)C−1 = φ†(t, x). Notice
+that this is due to the presence of the antipode in the on-shell field. Furthermore, the action is
+manifestly invariant under both CPT and κ-deformed Lorentz transformations.
+II.
+CHARGES AND FEATURES OF THE MODEL
+We now need to compute the charges for our model. There are two main ways to do so. The
+first is by direct computation using the Noether theorem. However, the fact that derivatives do
+not follow the Leibniz rule makes this job a prohibitively difficult one, except for the case of
+translation charges. The second way is to use the covariant phase-space formalism. Although very
+direct, this second method needs to be carefully defined in the deformed context.
+We chose a
+hybrid approach, namely we derived the translation charges from the Noether theorem, and then
+we built a covariant phase space approach which was able to reproduce the translation charges,
+allowing then to compute the remaining charges. Given a symplectic form Ω, and a symmetry in
+spacetime described by the vector field ξ, the charge Qξ can be defined by −δξ⌟ Ω = δQξ. In this
+context δξ is a vector field in phase space describing the variation δξA of any physical quantity
+A in phase space under the action of the symmetry ξ in spacetime, δ is the exterior derivative in
+phase space, and ⌟ represents a contraction. The translation charges computed directly from the
+Noether theorem are the following4:
+Pµ =
+�
+d3p α(p)[−S(p)µa†
+pap + pµb†
+p∗bp∗].
+(6)
+The quantity α(p) is a function of momenta whose explicit expression does not concern us in this
+context. To reproduce Eq. (6) in the covariant phase space formalism, we need to assume the
+following deformation of the canonical contraction rule between vector fields and forms5:
+δv⌟ (A ∧ B) = (δvA)B + A(S(δv)B),
+(7)
+where S(δv) can be appropriately defined [2].
+Using this definition, one can compute all the
+remaining charges, and the creation/annihilation operators algebra (which turns out to be the
+canonical one). For example, the boost charge Ni is
+Ni=− 1
+2
+� d3p α
+�
+S(ωp)
+�
+∂a†
+p
+∂S(p)i ap −a†
+p
+∂ap
+∂S(p)i
+�
++ωp
+�
+bp
+∂b†
+p
+∂pi − ∂bp
+∂pi b†
+p
+��
+.
+(8)
+4 There is also a fifth charge P4, see [1] for further details.
+5 Recall that the canonical relation is δv⌟ (A ∧ B) = (δvA)B − A(δvB).
+
+4
+One can then verify that the charges satisfy the usual non-deformed Poincar´e algebra. However,
+due to the effects of κ-deformation, one can also verify that, e.g, [Ni, C] ̸= 0, meaning that CPT
+symmetry is subtly violated. More explicitly, using the definition
+C =
+�
+d3p (b†
+pap + a†
+pbp),
+(9)
+one can show that [Ni, C] is given by
+i
+2
+�
+d3p
+�
+S(ωp)
+�
+∂ap
+∂S(p)i b†
+p − ap
+∂b†
+p
+∂S(p)i +
+∂a†
+p
+∂S(p)i bp − a†
+p
+∂bp
+∂S(p)i
+�
++
++ ωp
+�
+∂b†
+p
+∂pi ap − b†
+p
+∂ap
+∂pi + ∂bp
+∂pi a†
+p − bp
+∂a†
+p
+∂pi
+� �
+.
+(10)
+Notice that in the limit κ → ∞ one recovers the canonical result [Ni, C] = 0.
+III.
+COMMENTS AND CONCLUSIONS
+The fact that [Ni, C] ̸= 0 has several important physical consequences. The most apparent one
+is a difference in the decay time between particles and antiparticles. Furthermore, we now have a
+well defined theory which will allow us to study in details the propagator and the n-point functions
+in general. From this point of view, it will be interesting to tackle the loops in this deformed
+context. Finally, what has been done for the case of the complex scalar field will be extended to
+fields of higher spins, in order to expand the discussion to more realistic phenomena.
+ACKNOWLEDGMENTS
+Parts of these works were supported by funds provided by the Polish National Science Center,
+the project number 2019/33/B/ST2/00050.
+[1] M. Arzano, A. Bevilacqua, J. Kowalski-Glikman, G. Rosati, and J. Unger, Phys. Rev. D 103, 106015
+(2021)
+[2] A. Bevilacqua, J. Kowalski-Glikman, and W. Wislicki, Phys. Rev. D 105, 105004 (2022)
+[3] M. Arzano, and J. Kowalski-Glikman, Deformations of Spacetime Symmetries: Gravity, Group-Valued
+Momenta, and Non-Commutative Fields, Lecture Notes in Physics, Springer, 2021.
+

-dFAT4oBgHgl3EQfqB3j/content/tmp_files/load_file.txt

@@ -0,0 +1,98 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf,len=97
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='08645v1  [gr-qc]  20 Jan 2023  Complex scalar field in κ-Minkowski spacetime  Andrea Bevilacqua1, ∗  1National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa, Poland  It is often expected that one cannot treat spacetime as a continuous manifold as the Planck  scale is approached, because of to possible effects due to a quantum theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' There  have been several proposals to model such a deviation from the classical behaviour, one of  which is noncommutativity of spacetime coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' In this context, the non-commutativity  scale is seen as an observer-independent length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Of course, such a scale impose a modi-  fication of ordinary relativistic symmetries, which now need to be deformed to accommodate  this fundamental scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' The κ-Poincar´e algebra is an example of this deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' In what  follows I will briefly describe a construction of a κ-deformed complex scalar field theory,  while at the same time shedding light on the behaviour of discrete and continuous symme-  tries in this formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' This in turn will open the way to the study of the application of  this formalism to actual physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' I will then conclude with some comments and  prospects for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  FROM NON-COMMUTATIVE SPACETIME TO THE ACTION  We will present a brief preview of the works in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' For a more detailed introduction  to the framework and the formalism, see for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' The starting point is k-Minkowski  spacetime, whose coordinates satisfy the an(3) Lie algebra defined as [ˆx0, ˆxi] = iˆxi/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  Notice  that one recovers canonical Minkowski spacetime in the formal limit κ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' To have a more  physical understanding of the above commutator, notice that 1/κ has dimensions of length, and it  is sometimes identified with the Plank length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' In order to build fields in this spacetime, we need to  define plane waves first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' To do so, one can proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' First pick an explicit representation  of the an(3) Lie algebra in terms of matrices (it turns out that the lowest dimensional one has  dimension 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Then, one can define a plane wave as an element ˆek of the relative AN(3) Lie group,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' for example1 ˆek = exp(ikiˆxi) exp(ik0ˆx0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Notice that, since the elements of the matrices ˆxµ  ∗ andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='bevilacqua@ncbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='pl  1 A different choice on the ordering results in a change of coordinates in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  2  are dimensionful, so are the parameters kµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Written explicitly, one has  ˆek =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  ¯p4  κ  k  κ  p0  κ  p  κ  1  p  κ  ¯p0  κ  − k  κ  p4  κ  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  (1)  where2  p0 = κ sinh k0  κ + k2  2κek0/κ  pi = kiek0/κ  p4 = κ cosh k0  κ − k2  2κek0/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  (2)  Both kµ and pA(k) (A = 0, 1, 2, 3, 4) are two different coordinates of momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Notice that  the pA are not independent, since −p0 + p2 + p2  4 = κ2 (notice also that p+ = p0 + p4 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' It is  therefore clear that momentum space is curved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' This is reflected in a deformed sum of momenta,  which is defined through the group property ˆekˆel =: ˆek⊕l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' At the same time, inverse momenta  are defined in terms of group inverses ˆe−1  k  =: ˆeS(k), and S(k) is called the antipode of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' In order  to simplify the treatment of integrals and derivatives, we can use a3 Weyl map W to send group  elements ˆek into canonical plane waves ep := exp(ipµ(k)xµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' The group law is preserved thanks  to the ⋆ product defined by W(ˆek⊕l) = ep(k)⊕q(l) =: ep(k) ⋆ eq(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' In general, the ⋆ product is not  commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Because of this, we choose the following action for a κ-deformed complex scalar  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  S = 1  2  �  R4 d4x[(∂µφ)† ⋆ (∂µφ) + (∂µφ) ⋆ (∂µφ)† − m2(φ† ⋆ φ + φ ⋆ φ†)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  (3)  To obtain the equations of motion one usually integrates by parts, but in this deformed context  derivatives do not satisfy the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' In fact, if they did, one could get the following contra-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  i(p ⊕ q)µep⊕q = ∂µ(ep ⋆ eq) = (∂µep) ⋆ eq + ep ⋆ ∂µeq = i(p + q)ep⊕q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  (4)  Hence one has to use the appropriate deformations for the Leibniz rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' After some computations,  one can verify that the equations of motion satisfied by the field are the canonical Klein-Gordon  equations, and the field which satisfies them can be written as  φ(x) =  �  d3p  �2ωp  ξ(p)ape−i(ωpt−px) +  �  d3p∗  �  2|ω∗p|  ξ(p)b†  p∗ei(S(ωp)t−S(p)x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  (5)  2 The explicit expression of ¯p4 and ¯p0 in terms of kµ is not relevant for the present discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  3 There are several equivalent ways in which a suitable Weyl map can be chosen, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  [1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  3  The action of the discrete transformations C, P, T can be defined in the usual way in the deformed  context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Tφ(t, x)T −1 = φ(−t, x), Pφ(t, x)P −1 = φ(t, −x), and Cφ(t, x)C−1 = φ†(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Notice  that this is due to the presence of the antipode in the on-shell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Furthermore, the action is  manifestly invariant under both CPT and κ-deformed Lorentz transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  CHARGES AND FEATURES OF THE MODEL  We now need to compute the charges for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' There are two main ways to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' The  first is by direct computation using the Noether theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' However, the fact that derivatives do  not follow the Leibniz rule makes this job a prohibitively difficult one, except for the case of  translation charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' The second way is to use the covariant phase-space formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Although very  direct, this second method needs to be carefully defined in the deformed context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  We chose a  hybrid approach, namely we derived the translation charges from the Noether theorem, and then  we built a covariant phase space approach which was able to reproduce the translation charges,  allowing then to compute the remaining charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Given a symplectic form Ω, and a symmetry in  spacetime described by the vector field ξ, the charge Qξ can be defined by −δξ⌟ Ω = δQξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' In this  context δξ is a vector field in phase space describing the variation δξA of any physical quantity  A in phase space under the action of the symmetry ξ in spacetime, δ is the exterior derivative in  phase space, and ⌟ represents a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' The translation charges computed directly from the  Noether theorem are the following4:  Pµ =  �  d3p α(p)[−S(p)µa†  pap + pµb†  p∗bp∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  (6)  The quantity α(p) is a function of momenta whose explicit expression does not concern us in this  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' To reproduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' (6) in the covariant phase space formalism, we need to assume the  following deformation of the canonical contraction rule between vector fields and forms5:  δv⌟ (A ∧ B) = (δvA)B + A(S(δv)B),  (7)  where S(δv) can be appropriately defined [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  Using this definition, one can compute all the  remaining charges, and the creation/annihilation operators algebra (which turns out to be the  canonical one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' For example, the boost charge Ni is  Ni=− 1  2  � d3p α  �  S(ωp)  �  ∂a†  p  ∂S(p)i ap −a†  p  ∂ap  ∂S(p)i  �  +ωp  �  bp  ∂b†  p  ∂pi − ∂bp  ∂pi b†  p  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  (8)  4 There is also a fifth charge P4, see [1] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  5 Recall that the canonical relation is δv⌟ (A ∧ B) = (δvA)B − A(δvB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  4  One can then verify that the charges satisfy the usual non-deformed Poincar´e algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' However,  due to the effects of κ-deformation, one can also verify that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='g, [Ni, C] ̸= 0, meaning that CPT  symmetry is subtly violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' More explicitly, using the definition  C =  �  d3p (b†  pap + a†  pbp),  (9)  one can show that [Ni, C] is given by  i  2  �  d3p  �  S(ωp)  �  ∂ap  ∂S(p)i b†  p − ap  ∂b†  p  ∂S(p)i +  ∂a†  p  ∂S(p)i bp − a†  p  ∂bp  ∂S(p)i  �  +  + ωp  �  ∂b†  p  ∂pi ap − b†  p  ∂ap  ∂pi + ∂bp  ∂pi a†  p − bp  ∂a†  p  ∂pi  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  (10)  Notice that in the limit κ → ∞ one recovers the canonical result [Ni, C] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  COMMENTS AND CONCLUSIONS  The fact that [Ni, C] ̸= 0 has several important physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' The most apparent one  is a difference in the decay time between particles and antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Furthermore, we now have a  well defined theory which will allow us to study in details the propagator and the n-point functions  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' From this point of view, it will be interesting to tackle the loops in this deformed  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Finally, what has been done for the case of the complex scalar field will be extended to  fields of higher spins, in order to expand the discussion to more realistic phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  Parts of these works were supported by funds provided by the Polish National Science Center,  the project number 2019/33/B/ST2/00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content='  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Arzano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Bevilacqua, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Kowalski-Glikman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Rosati, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Unger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' D 103, 106015  (2021)  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Bevilacqua, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Kowalski-Glikman, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Wislicki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' D 105, 105004 (2022)  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Arzano, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}
+page_content=' Kowalski-Glikman, Deformations of Spacetime Symmetries: Gravity, Group-Valued  Momenta, and Non-Commutative Fields, Lecture Notes in Physics, Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'}

.gitattributes

@@ -5256,3 +5256,58 @@ rdAyT4oBgHgl3EQfZvcN/content/2301.00227v1.pdf filter=lfs diff=lfs merge=lfs -tex
 XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf filter=lfs diff=lfs merge=lfs -text
 INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf filter=lfs diff=lfs merge=lfs -text
 6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf filter=lfs diff=lfs merge=lfs -text
+UdE3T4oBgHgl3EQfEgk6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+q9FST4oBgHgl3EQfPDjc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+utE3T4oBgHgl3EQfNwlp/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+OdAzT4oBgHgl3EQfzf5C/content/2301.01769v1.pdf filter=lfs diff=lfs merge=lfs -text
+cNFKT4oBgHgl3EQfpi4R/content/2301.11870v1.pdf filter=lfs diff=lfs merge=lfs -text
+DNFQT4oBgHgl3EQfPTY7/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+YdFRT4oBgHgl3EQfOje2/content/2301.13514v1.pdf filter=lfs diff=lfs merge=lfs -text
+6NFKT4oBgHgl3EQf_C4s/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+RNAzT4oBgHgl3EQfXPwH/content/2301.01313v1.pdf filter=lfs diff=lfs merge=lfs -text
+gdE2T4oBgHgl3EQfyQgz/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+YNE2T4oBgHgl3EQfEAby/content/2301.03632v1.pdf filter=lfs diff=lfs merge=lfs -text
+XtAzT4oBgHgl3EQfYvw0/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf filter=lfs diff=lfs merge=lfs -text
+E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf filter=lfs diff=lfs merge=lfs -text
+PNAyT4oBgHgl3EQfg_jD/content/2301.00370v1.pdf filter=lfs diff=lfs merge=lfs -text
+EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf filter=lfs diff=lfs merge=lfs -text
+EdE0T4oBgHgl3EQfywKq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+W9AzT4oBgHgl3EQfYfyH/content/2301.01336v1.pdf filter=lfs diff=lfs merge=lfs -text
+utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf filter=lfs diff=lfs merge=lfs -text
+69E1T4oBgHgl3EQfnASE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+wtFLT4oBgHgl3EQflS8f/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PNAyT4oBgHgl3EQfg_jD/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+W9AzT4oBgHgl3EQfYfyH/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+1dE0T4oBgHgl3EQfdgBe/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+rdAyT4oBgHgl3EQfZvcN/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+KdE2T4oBgHgl3EQfAgZ0/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+VtE5T4oBgHgl3EQfBg7N/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+wtFLT4oBgHgl3EQflS8f/content/2301.12118v1.pdf filter=lfs diff=lfs merge=lfs -text
+bNE5T4oBgHgl3EQfEA7m/content/2301.05411v1.pdf filter=lfs diff=lfs merge=lfs -text
+qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf filter=lfs diff=lfs merge=lfs -text
+YtFPT4oBgHgl3EQftzUO/content/2301.13153v1.pdf filter=lfs diff=lfs merge=lfs -text
+I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf filter=lfs diff=lfs merge=lfs -text
+YtFPT4oBgHgl3EQftzUO/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+C9AzT4oBgHgl3EQfwf5z/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+r9AyT4oBgHgl3EQfmfgv/content/2301.00470v1.pdf filter=lfs diff=lfs merge=lfs -text
+yNAzT4oBgHgl3EQfQvuD/content/2301.01204v1.pdf filter=lfs diff=lfs merge=lfs -text
+YNE3T4oBgHgl3EQf1wsI/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf filter=lfs diff=lfs merge=lfs -text
+LtE1T4oBgHgl3EQftAUX/content/2301.03371v1.pdf filter=lfs diff=lfs merge=lfs -text
+C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf filter=lfs diff=lfs merge=lfs -text
+9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf filter=lfs diff=lfs merge=lfs -text
+YNE2T4oBgHgl3EQfEAby/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+09E0T4oBgHgl3EQfdQDB/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+bNE5T4oBgHgl3EQfEA7m/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf filter=lfs diff=lfs merge=lfs -text
+ndE_T4oBgHgl3EQf7xw1/content/2301.08371v1.pdf filter=lfs diff=lfs merge=lfs -text
+EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf filter=lfs diff=lfs merge=lfs -text
+YdFRT4oBgHgl3EQfOje2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+WNE0T4oBgHgl3EQfmAF1/content/2301.02493v1.pdf filter=lfs diff=lfs merge=lfs -text
+RNAzT4oBgHgl3EQfXPwH/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+NNFJT4oBgHgl3EQfzi34/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+NNFJT4oBgHgl3EQfzi34/content/2301.11644v1.pdf filter=lfs diff=lfs merge=lfs -text
+ndE_T4oBgHgl3EQf7xw1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+INAzT4oBgHgl3EQfxv6_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+OdAzT4oBgHgl3EQfzf5C/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text

09E0T4oBgHgl3EQfdQDB/content/2301.02375v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:40e9e2adf53efa7ac46436443030a3589fe71121eb96d9e5bd64440b83376e91
+size 3410635

09E0T4oBgHgl3EQfdQDB/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:60f5dde875d8e73594f3493beb7ab9e5982101703182b4f6fadb8db3f63d25cc
+size 3670061

09E0T4oBgHgl3EQfdQDB/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bfc79d34800cd1e9b0f86e5739b604db3489858dcadd6135f5f63b00c0f11e2e
+size 147910

09FIT4oBgHgl3EQf3Sv5/content/tmp_files/2301.11381v1.pdf.txt

@@ -0,0 +1,1541 @@
+Imaging anisotropic waveguide exciton polaritons in tin sulfide 
+ 
+Yilong Luan1,2, Hamidreza Zobeiri3, Xinwei Wang3, Eli Sutter4,5, Peter Sutter6*, Zhe Fei1,2* 
+ 
+1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  
+2Ames Laboratory, U. S. Department of Energy, Iowa State University, Ames, Iowa 50011, USA 
+3Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA 
+4Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 
+68588, USA  
+5Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, 
+USA. 
+6Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, 
+USA 
+ 
+*Corresponding to: (P.S.) psutter@unl.edu, (Z.F.) zfei@iastate.edu. 
+ 
+Abstract 
+In recent years, novel materials supporting in-plane anisotropic polaritons have attracted a lot of 
+research interest due to their capability of shaping nanoscale field distributions and controlling 
+nanophotonic energy flows. Here we report a nano-optical imaging study of waveguide exciton polaritons 
+(EPs) in tin sulfide (SnS) in the near-infrared (IR) region using the scattering-type scanning near-field 
+optical microscopy (s-SNOM). With s-SNOM, we mapped in real space the propagative EPs in SnS, which 
+show sensitive dependence on the excitation energy and sample thickness. Moreover, we found that both 
+the polariton wavelength and propagation length are anisotropic in the sample plane. In particular, in a 
+narrow spectral range from 1.32 to 1.44 eV, the EPs demonstrate quasi-one-dimensional propagation, which 
+is rarely seen in natural polaritonic materials. Further analysis indicates that the observed polariton 
+anisotropy is originated from the different optical bandgaps and exciton binding energies along the two 
+principal crystal axes of SnS.  
+ 
+Key Words: Tin sulfide, waveguide, exciton polaritons, s-SNOM, anisotropy, quasi-one-dimensional 
+ 
+Main text 
+In-plane anisotropic polaritons1,2 were first studied in metasurfaces3-5 where nanostructuring of the 
+polaritonic media or substrates breaks the symmetry, thus enabling polaritonic anisotropy. Later, several 
+natural materials were predicted and/or experimentally confirmed to support in-plane anisotropic 
+polaritons.6-10 For example, anisotropic plasmon polaritons and hybrid plasmon-phonon polaritons were 
+observed in black phosphorus carbides with far-field infrared (IR) spectroscopy.8 Anisotropic phonon 
+polaritons with hyperbolic wavefronts were imaged in MoO3,9-11 which can be conveniently tailored by 
+controlling the sample thickness and by stack- and twist-engineering.12-18 Compared to nano-engineered 
+anisotropic metasurfaces, natural materials with intrinsic anisotropic polaritons are generally more 
+convenient for applications and can avoid potential material quality degradation due to complex nano-
+fabrications. Despite these advantages, natural materials supporting in-plane anisotropic polaritons are rare 
+and are so far mainly studied in the mid-IR range. New materials enabling anisotropic polaritons in other 
+technologically important spectral regions (e.g., near-IR and visible) are desired.  
+In this Letter, we report the experimental discovery of strongly-anisotropic exciton polaritons (EPs) 
+in tin sulfide (SnS) in the technologically-important near-IR region. SnS is a post-transition-metal 
+monochalcogenide and a van der Waals (vdW) layered semiconductor with an orthorhombic structure, 
+analogous to that of black phosphorous.6-7 As sketched in Figure 1b, the two in-plane axes of SnS, namely 
+the a and b axes, are along the zigzag and armchair directions, respectively. SnS has been widely studied 
+due to its unique anisotropic optoelectronic properties19-23 and potential applications related to 
+photodetection and solar energy harvesting.24-27 In particular, the energies of excitons or optical bandgaps 
+
+along the a and b axes of SnS are about Ea ≈ 1.39 eV and Eb ≈ 1.66 eV respectively,22,23 which directly 
+impact the polaritonic responses. Note that EPs have previously been studied in other vdW semiconductors 
+(e.g. WSe2, MoSe2, etc.) with imaging28-32 and spectroscopic methods,33-37 where the EPs are isotropic in the 
+sample plane. The samples studied here are SnS microcrystals supported on mica wafers (Figure S1a). As 
+introduced in detail in the earlier work,19 these microcrystals have a wrap-around layered core-shell 
+structure: the thick SnS core is coated with a thin crystalline shell (thickness ≈ 3 nm) of layered tin disulfide 
+(SnS2). A detailed characterization of the wrap-around core-shell structures and their synthesis process were 
+reported in the earlier work.19 Note that the thin SnS shell is isotropic in the sample plane38,39, so the 
+observed anisotropic properties of EPs are solely due to the SnS core. The SnS2 shell mainly serves as a 
+protection layer of the SnS core and the waveguide EPs. Detailed discussions about the effect of the SnS2 
+shell are given in the Supporting Information. 
+ 
+To excite and probe EPs in SnS, we employed a scattering-type scanning near-field optical 
+microscope (s-SNOM) that was built based on an atomic force microscope (AFM). As illustrated in Figure 
+1a, the sharp metalized tip in s-SNOM excited by a p-polarized laser beam generates strong evanescent 
+fields underneath the tip. These evanescent fields with a wide range of wavevectors40 can effectively excite 
+transverse-magnetic (TM) polaritons inside the sample.29 The excitation source used in the study is a 
+broadband (1.24-1.77 eV) Ti:sapphire laser that covers the bandgap and exciton energies of SnS (see 
+discussions below). We used a parabolic mirror to focus the laser beam at the tip apex, and the scattered 
+photons off the tip/sample system are collected by the same parabolic mirror and then counted by a 
+photodetector. More introductions about the nano-optical setup are in Supporting Information.   
+ 
+In Figure 1c, we plot the AFM topography image of a typical SnS microcrystal coated with a thin 
+SnS2 shell.19 The lateral sizes of the crystal are approximately 6-8 m, and the thickness is about 100 nm 
+including the SnS2 shell. Here the crystal has a total of eight edges, among which the four short edges are 
+along the a or b axes.19 We were able to determine the crystal axes of the sample by examining the shape 
+of the crystal and by Raman spectroscopy (see Supporting Information). Figure 1d plots the near-field 
+amplitude (s) images taken simultaneously with the topography image (Figure 1c) at the excitation energy 
+of E = 1.38 eV. Here, the in-plane wavevector of the laser (kin) is along the b axis of SnS. From Figure 1d, 
+one can see many interference fringes and oscillations inside the sample. We focus on a string of one-
+dimensional (1D) oscillations extending from the left edge to the crystal center along the b axis (marked 
+with a white arrow). According to previous studies,29,30 these oscillations are generated due to the 
+interference between two major beam paths as sketched in Figure 1a. In the first path (P1), the excitation 
+photons are scattered back directly by the tip apex. In the second path (P2), the excitation photons are first 
+transferred into waveguide EPs by the s-SNOM tip. These EPs then propagate toward the sample edge and 
+get scattered into photons. Photons collected through the two beam paths are coherent with each other, so 
+they can generate interference. When scanning the tip perpendicular to the sample edge, the distance 
+between the tip and the sample edge varies, so a string of bright and dark spots forms due to constructive 
+and destructive interferences, respectively. As sketched in Figure 1a, the left short edge of the crystal is 
+responsible for the generation of the 1D interference oscillations along the direction of the white arrow in 
+Figure 1d. Other edges can also scatter EPs into photons and generate interference patterns. For example, 
+the four long edges that are about 43º relative to the b axis are responsible for the bight fringes parallel to 
+these edges (see Figure S2b). There are other possible interference mechanisms (e.g., edge excitation of 
+polaritons), but they are not responsible for the fringes/oscillations observed in our samples. Detailed 
+discussions of different interference mechanisms are given in Section 3 of the Supporting Information. 
+From Fig. 1d,f, we also seen fringes on the substrate side, which are generated due to the excitation and 
+scattering of photons at the air/mica interface as confirmed by dispersion analysis (see Figure S10).  
+Figure 1e,f plot the AFM and corresponding s-SNOM imaging data of the same crystal as those in 
+Figure 1c,d but rotated 90º relative to the surface normal (c axis). Here the in-plane wavevector of the 
+excitation laser is along the a axis (kin // a). Interestingly, we found no interference oscillations in the interior 
+of the crystal as those seen in Figure 1d, indicating that no waveguide EPs are propagating along the a axis. 
+To further explore the anisotropic polaritonic responses, we performed energy-dependent s-SNOM imaging. 
+The results are shown in Figure 2, where we plot the s-SNOM imaging data with kin along both the b axis 
+
+(Figure 2a-e) and a axis (Figure 2f-j) at various excitation energies. Again, we focus on the 1D oscillations 
+at the crystal center (along the direction of the white arrows) that evolve systematically with E. For kin // b, 
+the oscillations are clearly seen for photon energies from 1.29 to 1.48 eV, and their periods decrease with 
+increasing energy. In the case of kin // a, the interference oscillations appear only at energies below 1.32 eV, 
+and there are no clear 1D oscillations from 1.38 to 1.48 eV.  
+ 
+The s-SNOM imaging data shown in Figures 1 and 2 provide direct evidence of in-plane anisotropic 
+EPs of SnS in the near-IR region. To support the experimental data, we performed finite-element 
+simulations of the waveguide EPs using Comsol Multiphysics. In the model, we placed a vertically 
+polarized excitation dipole (pz) right above the sample surface. The optical constants of SnS and SnS2 were 
+obtained from the literature.22,23,38,39 A detailed description of the Comsol model is given in Supporting 
+Information. The simulation results are shown in Figure 2k-o and Figure S5, where we plot the real-space 
+images of polariton field amplitude (|Ez|) and polariton field (Ez) of EPs respectively. Here the EPs are 
+launched by a vertically polarized dipole (pz) located at the center of the image. At E = 1.29 eV (Figure 2k), 
+the dipole-launched anisotropic EPs propagate at all directions with elliptic wavefronts. As E increases to 
+1.32 eV (Figure 2l), the EPs show a faster decay along the a axis while keeping a relatively long propagation 
+distance along the b axis. The propagation along the a axis is even shorter at higher energies E ≥ 1.38 eV 
+(Figure 2m-o). As a result, the EPs appear to be quasi-1D along the b axis. The polaritonic simulations are 
+consistent with s-SNOM imaging data in Figures 1 and 2.  
+ 
+With the s-SNOM imaging data and Comsol simulation results, we were able to perform a 
+quantitative analysis of the dispersion and propagation properties of the anisotropic EPs. In Figure 3a,c, we 
+plot the line profiles extracted across the 1D interference oscillations in the energy-dependent s-SNOM 
+images (Figure 2). We then performed Fourier transforms (FTs) of these profiles to accurately obtain the 
+periods () of the interference oscillations that are linked to the polariton wavelength (p) in the following 
+relationship:29,30 
+ 
+0/ ≡ k/k0 ≈ 0/p – cos .                                                            (1) 
+ 
+Here, λ0 is the excitation photon wavelength, k0 = 2π/λ0 is the free-space photon wavevector, kρ = 2π/ρ is 
+the inverse period of the interference oscillations, and  ≈ 30º is the incidence angle of the laser beam 
+relative to the sample plane (see Figure 1a). The FT profiles for kin // b and kin // a are shown respectively 
+in Figures 3b and 3d, where the peaks (marked with blue arrows) correspond to k. We then determined the 
+polariton wavevector (kp = 2π/p) using Eq. (1) for every given excitation energy, based on which we obtain 
+the energy-momentum dispersion relations of the EPs.  
+ 
+The experimental dispersion data points of EPs obtained through FT analysis (Figure 3b,d) are 
+plotted in Figure 4a,b as black squares, which are sitting on the theoretical dispersion colormaps. In the 
+colormaps, we plot the imaginary part of the reflection coefficients Im(rp) that represents the photonic 
+density of states (see Supporting Information). Here the TM waveguide modes are visualized as bright 
+curves (marked with blue dashed curves).29 In addition to the dispersion relations, the Im(rp) colormaps also 
+reveal the mode broadening (k) that corresponds to the damping (see discussions in the following 
+paragraph). This method of dispersion calculation has been widely used in the studies of polaritons in a 
+variety of materials.29,30,40,41 In the dispersion diagrams, we also plot the dispersion data points extracted 
+from Comsol simulations (Figure 2k-o and Figure S5). The dispersion relations of the EPs from 
+experimental data, Comsol simulations, and the Im(rp) colormaps are consistent with each other, which 
+validates our experimental and theoretical approaches. From the dispersion diagrams, we can examine the 
+light-exciton interactions close to the exciton energies Ea and Eb (marked with white dashed lines in Figure 
+4a,b). The waveguide mode along the b axis exhibits a clear back-bending behavior that is a signature 
+behavior of light-exciton interactions.28,29 By fitting the dispersion with the coupled oscillator model, we 
+were able to determine the Rabi splitting energy (~160 meV), which is larger than the average polariton 
+linewidth (~105 meV) (see Supporting Information). Therefore, the EPs along the b axis are in the strong 
+coupling regime. The mode coupling is much weaker along the a axis, likely due to the small exciton 
+
+binding energy. According to literature,22 excitons along the b axis are robust with a binding energy of ~ 
+50 meV. The binding energy of excitons along the a axis, on the other hand, is much smaller and Ea is close 
+to the fundamental bandgap.22 The light-exciton coupling is much stronger at lower temperatures (e.g., T = 
+27 K) with more prominent mode bending features close to the exciton energies (Figure S8).   
+ 
+In addition to the polariton dispersion, we also extracted the propagation lengths (Lep) of the EPs 
+(Lep ≡ 1/[2Im(kep)]). The extraction was done by fitting the decay trend of the polariton oscillations from 
+both the s-SNOM data (Figure 2a-j) and Comsol simulations (Figure 2k-o and Figure S5). A detailed 
+description of the fitting procedures is given in the Supporting Information. As shown in Figure 4c,d, Lep 
+along both the a and b axes are over 3 m at E = 1.29 eV. With increasing E, Lep drops systematically along 
+both directions, but the drop along the a axis is much faster. As E approaches Ea ≈ 1.39 eV, Lep along the a 
+axis drops below 1 m and becomes unmeasurable. Lep along the b axis, on the other hand, is as high as 2.5 
+m at E = 1.38 eV, where quasi-1D EPs were observed (Figure 1d,f). Lep drops to 1 m or below along the 
+b axis when E gets close to Eb ≈ 1.66 eV. The energy dependence of Lep is fully consistent with the mode 
+broadening behaviors shown in the theoretical dispersion colormaps in Figure 4. The larger the polariton 
+width (k), the smaller the propagation length.  
+ 
+Finally, we explored the dependence of EPs on the thicknesses of SnS crystals. In Figure 5a, we 
+plot the nano-optical images of SnS microcrystals with various thicknesses (d) taken at an excitation energy 
+of E = 1.38 eV. Due to the strong damping of EPs along the a axis, we only show in Figure 5 the data 
+images with the excitation along the b axis (kin // b). We focus on the 1D interference oscillations (Figures 
+1 and 2), which evolve systematically with varying thicknesses. Figure 5b plots the line profiles taken 
+directly across the 1D oscillations in Figure 5a. Using Fourier transform and Eq. (1), we extracted the 
+polariton wavelengths p at different sample thicknesses, which are plotted in Figure 5c. Here one can see 
+that p decreases systematically with increasing d, which is expected since the crystal thickness determines 
+both the out-of-plane (kz ~ 1/d) and in-plane wavevectors of the waveguide mode.42 For samples with 
+thicknesses over 150 nm, p drops below 300 nm that is 3 times smaller than the photon wavelength 0 = 
+900 nm. The mode confinement is comparable to if not better than waveguide EPs in other materials.29-32  
+ 
+In summary, we have performed a comprehensive nano-optical study of SnS microcrystals using 
+s-SNOM. We found through near-field imaging that SnS supports waveguide EPs in near-IR, which are 
+sensitively dependent on both the excitation energy and sample thickness. More interestingly, both the 
+dispersion and transport properties of the EPs are strongly anisotropic in the sample plane. In particular, in 
+the energy range from 1.35 to 1.55 eV, the EPs show quasi-1D propagation along the b axis, which has not 
+been reported in other natural polaritonic materials. Future studies with a pump-probe s-SNOM setup31,32 
+are expected for the exploration of the ultrafast dynamics of anisotropic EPs in SnS. It is also interesting to 
+study TE waveguide EPs of SnS that could be seen in atomically thin crystals, where active tunability of 
+EPs is possible with electrical gating.  
+ 
+The anisotropic EPs discovered here in SnS are promising for a variety of applications. One 
+potential application is low-pass waveguide filters for planar photonic circuits. The cut-off energies of the 
+filters can be chosen by selecting the direction of signals propagating through the waveguide (i.e., along a 
+or b axes), Another possible application is selective nanophotonic interconnection. A concept device is 
+sketched in Figure S13, where the signal source is connecting two devices through an SnS interconnector. 
+The two ports for the two devices are along the a and b axes, respectively. When the incident photonic 
+signal (e.g., on or off signals) is at energies of E ≤ 1.29 eV, EPs can propagate along all directions in SnS, 
+so both devices can receive the signal. When the incident photonic signal is at energies of 1.38 eV ≤ E ≤ 
+1.48 eV, EPs propagate only along the b axis, so device 2 will not receive the signal. Therefore, the signal 
+interconnection can be controlled selectively by choosing different energies of the incident signals. Such a 
+directional control of the flow of nanophotonic energy and signals cannot be easily realized in isotropic 
+polaritonic materials without complicated nano-fabrications. With the potential controllability or tunability 
+by chemical doping or electrical gating, SnS-based polaritonic devices could play an important role in future 
+planar optics43 in the technologically important near-IR region. 
+ 
+
+Supporting Information 
+Nano-optics setup; Determination of the crystal axes of SnS; Interference mechanisms; COMSOL 
+simulations; Dispersion calculations; Low-temperature dispersion of EPs in SnS; Effect of the SnS2 shell 
+on waveguide EPs; Photonic mode at the air/mica interface; Extraction of the propagation lengths of EPs; 
+Coupling strength of EPs 
+ 
+Corresponding Authors 
+Peter Sutter, Email: psutter@unl.edu,  
+Zhe Fei, Email: zfei@iastate.edu. 
+ 
+Notes 
+The authors declare no competing interests. 
+ 
+Acknowledgments 
+ 
+This work is supported by the National Science Foundation under Grant No. DMR-1945560. The 
+nano-optics setup used in the work is supported in part by Ames Laboratory. Ames Laboratory is operated 
+for the U.S. Department of Energy by Iowa State University under Grant No. DE-AC02-07CH11358. 
+Materials synthesis, electron microscopy, and complementary cathodoluminescence spectroscopy by E.S. 
+and P.S. were supported by the National Science Foundation, Division of Materials Research, Solid State 
+and Materials Chemistry Program under Grant No. DMR-1904843. H.Z. and X.W. are grateful for the 
+support from National Science Foundation under Grant No. CBET-1930866 and CMMI-203246.  
+ 
+References 
+(1) Ma, W.; Shabbir, B.; Ou, Q.; Dong, Y.; Chen, H.; Li, P.; Zhang, X.; Lu, Y.; Bao, Q. Anisotropic 
+polaritons in van der Waals materials. InfoMat. 2020, 2, 777-790. 
+(2) Wang, C.; Zhang, G.; Huang, S.; Xie, Y.; Yan, H. The Optical Properties and Plasmonics of Anisotropic 
+2D Materials. Adv. Optical Mater. 2020, 8, 1900996.  
+(3) Li, P.; Dolado, I.; Alfaro-Mozaz, F. J.; Casanova, F.; Hueso, L. E.; Liu, S.; Edgar, J. H.; Nikitin, A. Y.; 
+Vélez, S.; Hillenbrand, R. Infrared hyperbolic metasurface based on nanostructured van der Waals 
+materials. Science 2018, 359, 892-896. 
+(4) Chevrier, K.; Benoit, J. M.; Symonds, C.; Saikin, S. K.; Yuen-Zhou, J.; Bellessa, J. Anisotropy and 
+Controllable Band Structure in Suprawavelength Polaritonic Metasurfaces. Phys. Rev. Lett. 2019, 122, 
+173902. 
+(5) High, A. A.; Devlin, R. C.;  Dibos, A.; Polking, M.; Wild, D. S.; Perczel, J.; de Leon N. P.; Lukin, M. 
+D.; Park, H. Visible-frequency hyperbolic metasurface. Nature 2015, 522, 192-196. 
+(6) Correas-Serrano, D; Gomez-Diaz, J; Melcon, A. A.; Alù, A. Black phosphorus plasmonics: anisotropic 
+elliptical propagation and nonlocality-induced canalization. J. Optics 2016, 18, 104006. 
+(7) Lee, I.-H.; Martin-Moreno, L.; Mohr, D. A.; Khaliji, K.; Low, T.; Oh, S. H. Anisotropic acoustic 
+plasmons in black phosphorus. ACS Photon. 2018, 5, 2208-2216. 
+(8) Huang, X.; Cai, Y.; Feng, X.; Tan. W. C.; Hansan, D. M. N.; Chen, L.; Chen, N.; Wang, L.; Huang, L.; 
+Duffin, T. J.; Nijhuis, C. A.; Zhang, Y.-W.; Lee, C.; Ang, K.-W. Black Phosphorus Carbide as a 
+Tunable Anisotropic Plasmonic Metasurface. ACS Photon. 2018, 5, 3116-3123.  
+(9) Ma, W.; Alonso-González, P; Li, S.; Nikitin, A. Y.; Yuan, J.; Martín-Sánchez, J.; Taboada-Gutiérrez, 
+J.; Amenabar, I.; Li, P.; Vélez, S.; Tollan, C.; Dai, Z.; Zhang, Y.; Sriram, S.; Kalantar-Zadeh, K.; Lee, 
+S.-T.; Hillenbrand, R.; Bao, Q. In-plane anisotropic and ultra-low-loss polaritons in a natural van der 
+Waals crystal. Nature 2018, 562, 557-562. 
+(10) 
+Zheng, Z.; Chen, J.; Wang, Y.; Wang, X.; Chen, X.; Liu, P.; Xu, J.; Xie, W.; Chen, H.; Deng, S.; 
+Xu, N. Highly Confined and Tunable Hyperbolic Phonon Polaritons in Van Der Waals Semiconducting 
+Transition Metal Oxides. Adv. Mater. 2018, 30, 1705318.  
+
+(11) 
+Zheng Z.; Xu N.; Oscurato S. L.; Tamagnone, M.; Sun, F.; Jiang, Y.; Ke, Y.; Chen, J.; Huang, W.; 
+Wilson, W. L. Ambrosio, A.; Deng, S.; Chen, H. A mid-infrared biaxial hyperbolic van der Waals 
+crystal. Sci. Adv. 2019, 5, eaav8690. 
+(12) 
+Duan, J.; G. Álvarez-Pérez, View ProfileK. V. Voronin, I. Prieto, J. Taboada-Gutiérrez, V. S. 
+Volkov, J. Martín-Sánchez, A. Y. Nikitin, and P. Alonso-González. Enabling propagation of 
+anisotropic polaritons along forbidden directions via a topological transition. Sci. Adv. 2021, 7, 
+eabf2690.  
+(13) 
+Zhang, Q.; Ou, Q.;  Hu, G.; Liu, J.; Dai, Z.; Fuhrer, M. S.; Bao, Q.; Qiu, C.-W. Hybridized 
+Hyperbolic Surface Phonon Polaritons at α‑MoO3 and Polar Dielectric Interfaces. Nano Lett. 2021, 21, 
+3112-3119. 
+(14) 
+Hu, G.; Krasnok, A.; Mazor, Y.; Qiu, C.-W.; Alù, A. Moire Hyperbolic Metasurfaces. Nano Lett. 
+2020, 20, 3217−3224. 
+(15) 
+Hu, G.; Ou, Q.; Si. G.; Wu, Y.; Wu, J.; Dai, Z.; Krasnok, A.; Mazor, Y.; Zhang, Q.; Bao, Q.; Qiu, 
+C.-W.; Alù, A. Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers. Nature 
+2020, 582, 209−213. 
+(16) 
+Zheng, Z.; Su, F.; Huang, W.; Jiang, J.; Zhan, R.; Ke, Y.; Chen, H.; Deng S. Phonon Polaritons in 
+Twisted Double-Layers of Hyperbolic van der Waals Crystals. Nano Lett. 2020, 20, 5301−5308. 
+(17) 
+Duan, J.; Capote-Robayna, N.; Taboada-Gutierrez, J.; Álvarez-Pérez, G.; Prieto, I.; Martín-Sanchez, 
+J.; Nikitin, A. Y.; Alonso-González, P. Twisted Nano-Optics: Manipulating Light at the Nanoscale with 
+Twisted Phonon Polaritonic Slabs. Nano Lett. 2020, 20, 5323−5329. 
+(18) 
+Chen, M.; Lin, X.; Dinh, T. H.; Zheng, Z.; Shen, J.; Ma, Q.; Chen, H.; Jarillo-Herrero, P.; Dai, S. 
+Configurable phonon polaritons in twisted α-MoO3. Nat. Mater. 2020, 19, 1−5. 
+(19) 
+Sutter, P.; Wang, J.; Sutter, E. Wrap-Around Core–Shell Heterostructures of Layered Crystals. Adv. 
+Mater. 2019, 31, 1902166.  
+(20) 
+Lin. S.; Carvalho, C.; Yan, S.; Li, R.; Kim, S.; Rodin, A.; Carvalho, L.; Chan, E. M.; Wang, X.; 
+Castro Neto, A. H.; Yao, J. Accessing valley degree of freedom in bulk Tin(II) sulfide at room 
+temperature. Nat. Commun. 2018, 9, 1455.  
+(21) 
+Banai, R. E.; Burton, L. A.; Choi, S. G.; Hofherr, F.; Sorgenfrei, T.; Walsh, A.; To, B.; Gröll, A.; 
+Brownson, J. R. S. Ellipsometric characterization and density-functional theory analysis of anisotropic 
+optical properties of single-crystal α-SnS. J. Appl. Phys. 2014, 116, 013511. 
+(22) 
+Nguyen, H. T.; Le, Y. L.; Nguyen, T. M. H.; Kim, T. J.; Nguyen, X. A.; Kim, B.; Kim, K.; Lee, 
+W.; Cho, S.; Kim, Y. D. Temperature dependence of the dielectric function and critical points of α‑SnS 
+from 27 to 350 K. Sci. Rep. 2020, 10, 18396.  
+(23) 
+Le, V. L.; Cuong, D. D.; Nguyen, H. T.; Nguyen, X. A.; Kim, B.; Kim, K.; Lee, W.; Hong, S. C.; 
+Kim, T. J.; Kim, Y. D. Anisotropic behavior of excitons in single-crystal α-SnS. AIP Adv. 2020, 10, 
+105003.  
+(24) 
+Noguchi, H.; Setiyadi, A.; Tanamura, H.; Nagatomo, T.; Omoto. Characteriztion of vacuum-
+evaporated tin sulfide film for solar cell materials. Sol. Energy Mater. Sol. Cells 1994, 35, 325-331.  
+(25) 
+Steinmann, V.; Jaramillo, R.; Hartman, K.; Chakraborty, R.; Brandt, R. E.; Poindexter, J. R.; Lee, 
+Y. S.; Sun, L.; Polizzotti, A.; Park, H. H.; Gordon, R. G.; Buonassisi, T. 3.88% Efficient Tin Sulfide 
+Solar Cells using Congruent Thermal Evaporation. Adv. Mater. 2014, 26, 7488-7492.  
+(26) 
+Deng, Z.; Cao, D.; He, J. Lin, S.; Lindsay, S. M. Liu, Y. Solution Synthesis of Ultrathin Single-
+Crystalline SnS Nanoribbons for Photodetectors via Phase Transition and Surface Processing. ACS 
+Nano 2012, 6, 6197-6207.  
+(27) 
+Krishnamurthi, V.; Khan, H.; Ahmed, T.; Zavabeti, A.; Tawfik, S. A.; Jain, S. K.; Spencer, M. J. 
+S.; Balendhran, S.; Crozier, K. B.; Li, Z.; Fu, L.; Mohiuddin, M.; Low, M. X.; Shabbir, B.; Boes, A.; 
+Mitchell, A.; McConville, C. F.; Li, Y.; Kalantar-Zadeh, K.; Mahmood, N.; Walia, S. Liquid-Metal 
+Synthesized Ultrathin SnS Layers for  High-Performance Broadband Photodetectors. Adv. Mater. 2020, 
+32, 2004247.  
+(28) 
+Hu, F.; Fei, Z. Recent progress on exciton polaritons in layered transition-metal dichalcogenides, 
+Adv. Optical Mater. 2019, 1901003. 
+
+(29) 
+Hu, F.; Luan, Y.; Scott, M. E.; Yan, J.; Mandrus, D. G.; Xu, X.; Fei, Z. Imaging exciton-polariton 
+transport in MoSe2 waveguides, Nat. Photon. 2017, 11, 356-360. 
+(30) 
+Hu, F.; Luan, Y.; Speltz, J.; Zhong, D.; Liu, C. H.; Yan, J.; Mandrus, D. G.; Xu, X.; Fei, Z. "Imaging 
+propagative exciton polaritons in atomically-thin WSe2 waveguides", Phys. Rev. B 2019, 100, 
+121301(R).  
+(31) 
+Mrejen, M.; Yadgarov, L.; Levanon, A.; Suchowski, H. Transient exciton-polariton dynamics in 
+WSe2 by ultrafast near-field imaging. Sci. Adv. 2019, 5, eaat9618.  
+(32) 
+Sternbach, A. J.; Latini, S.; Chae, S.; Hübener, H.; Giovannini, U. D.; Shao, Y.; Xiong, L.; Sun, Z.; 
+Shi, N.; Kissin, P.; Ni, G.-X.; Rhodes, D.; Kim, B.; Yu, N.; Millis, A. J.; Fogler, M. M.; Schuck, P. J.; 
+Lipson, M.; Zhu, X.-Y.; Hone, J.; Averitt, R. D.; Rubio, A.; Basov, D. N. Femtosecond exciton 
+dynamics in WSe2 optical waveguides. Nat. Commun. 2020, 11, 3567. 
+(33) 
+Liu, X.; Galfsky, T.; Sun, Z.; Xia, F.; Lin, E.-C.; Lee, Y.-H.; Kéna-Cohen, S.; Menon, V. M. Nat. 
+Photonics Strong light–matter coupling in two-dimensional atomic crystals. 2015, 9, 30-34.  
+(34) 
+Dufferwiel, S.; Schwarz, S.; Withers, F.; Trichet, A. A. P.; Sich, M.; Pozo-Zamudio, O. D.; Clark, 
+C.; Nalitov, A.; Solnyshkov, D. D.; Malpuech, G.; Novoselov, K. S.; Smith, J. M.; Skolnick, M. S.; 
+Krizhanovskii, D. N.; Tartakovskii, A. I. Exciton–polaritons in van der Waals heterostructures 
+embedded in tunable microcavities. Nat. Commun. 2015, 6, 8579.  
+(35) 
+Lundt, N.; Klembt, S.; Cherotchenko, E.; Betzold, S.; Iff, O.; Nalitov, A. V.; Klaas, M.; Dietrich, 
+C. P.; Kavokin, A. V.; Höfling, S.; Schneider, C. Room-temperature Tamm-plasmon exciton-polaritons 
+with a WSe2 monolayer. Nat. Commun. 2016, 7, 13328.  
+(36) 
+Flatten, L.C.; He, Z.; Coles, D. M.; Trichet, A. A. P.; Powell, A. W.; Taylor, R. A.; Warner, J. H.; 
+Smith, J. M. Room-temperature exciton polaritons with two-dimensional WS2. Sci. Rep. 2016, 6, 33134. 
+(37) 
+Wang, Q.; Sun, L.; Zhang, B.; Chen, C.; Shen, X.; Lu, W. Direct observation of strong light-exciton 
+coupling in thin WS2 flakes. Opt. Express 2016, 24, 7151-7157. 
+(38) 
+Bertrand, Y.; Leveque, G.; Raisin, C.; Levy, F. Optical properties of SnSe2 and SnS2. J. Phys. C: 
+Solid State Phys. 1979, 12, 2907-2916.  
+(39) 
+Lucovsky, G. Mikkelsen, J. C. Jr. Liang, W. Y.; White, R. M.; Martin, R. M. Optical phonon 
+anisotropies in the layer crystals SnS2 and SnSe2. Phys. Rev. B 1976, 14, 1663-1669.  
+(40) 
+Fei, Z.; Andreev, G. O.; Bao, W.; Zhang, L. M.; McLeod, A. S.; Wang, C.; Stewart, M. K.; Zhao, 
+Z.; Dominguez, G.; Thiemens, M.; Fogler, M. M.; Tauber, M. J.; Castro-Neto, A. H.; Lau, C. N.; 
+Keilmann, F.; Basov, D. N. Infrared Nanoscopy of Dirac Plasmons at the Graphene-SiO2 interface 
+Nano Lett. 2011, 11, 4701-4705. 
+(41) 
+Dai, S.; Fei, Z.; Ma, Q.; Rodin, A. S.; Wagner, M.; McLeod, A. S.; Liu, M.; Gannett, W.; Regan, 
+W.; Watanabe, K.; Taniguchi, T.; Thiemens, M.; Dominguez, G.; Castro Neto, A. H.; Zettl, A.; 
+Keilmann, F.; Jarillo-Herrero, P.; Fogler, M. M.; Basov, D. N. Tunable phonon polaritons in atomically 
+thin van der Waals crystals of boron nitride. Science 2014, 343, 1125-1129.  
+(42) 
+Hunsperger, R. G. Integrated Optics. 6th ed.; Springer, New York, NY, 2009.  
+(43) 
+Genevet, P., Capasso, F., Aieta, F., Khorasaninejad & Devlin, R. Recent advances in planar optics: 
+from plasmonic to dielectric metasurfaces. Optica 2017, 4, 139-152. 
+ 
+ 
+ 
+Figure captions 
+ 
+
+ 
+Figure 1. (a) Illustration of the experimental setup and the two beam paths (labeled as ‘P1’ and ‘P2’) 
+responsible for the formation of the observed interference oscillations. (b) Sketch the crystal structure of 
+SnS. The Sn and S atoms are in silver and yellow, respectively. (c)-(f) The AFM topography and the 
+simultaneously-taken nano-IR images of an SnS microcrystal (thickness = 100 nm) with in-plane 
+wavevector (k//) of the excitation laser along the b (c,d) and a (e,f) axes, respectively. The red arrows in (c) 
+and (e) mark the direction of k//. The white arrow in (d) marks the 1D oscillations discussed in the main 
+text.   
+ 
+ 
+Figure 2. (a)-(e) Energy-dependent imaging data of EPs in SnS with the in-plane laser wavevector kin along 
+the b axis. (f)-(j) Energy-dependent imaging data of EPs in SnS with kin along the a axis. Here the sample 
+
+a
+focusing
+cantilever
+c
+AFM
+d
+s-SNOM, 1.38 eV
+mirror
+mica
+Kin
+Ra
+tip
+kin // b
+sample
+sample
+EP
+d (nm)
+s (a.u.)
+100
+13.0
+mica
+2um
+2μm
+AFM
+s-SNOM, 1.38 eV
+b
+e
+f
+mica
+-10
+10.5
+Kin//a
+sample
+a
+SnS crystal
+2μum
+2uma
+ki. Il b, E=1.29 eV
+b
+C
+k, II b, E=1.38 eV
+d
+kn // b, E=1.32 eV
+k. I/ b, E=1.44 eV
+e
+k.// b, E=1.48 eV
+s (a.u.)
+2um
+max
+f
+k.// a, E=1.29eV
+g
+k, // a, E=1.32 eV
+h
+kn // a, E=1.38 eV
+kin // a, E=1.44 eV
+1
+kn // a, E=1.48 eV
+2um
+k
+[E], E=1.29 eV
+[E,], E=1.32 eV
+w
+IEzl, E=1.38 eV
+n
+IE21, E=1.44 eV
+[Ez, E=1.48 eV
+[E2]
+(a.u.)
+4umis the 100-nm-thick SnS microcrystal shown in Figure 1. (k)-(o) Simulated polariton field amplitude (|Ez|) 
+maps of waveguide EPs in 100-nm-thick SnS at various energies. The EPs are excited by a point dipole (pz) 
+located at the center of the image right above the sample surface. 
+ 
+ 
+Figure 3. Real-space line profiles along the direction of the 1D interference oscillations in Figure 2 and 
+their Fourier-transformed (FT) profiles with the in-plane laser wavevector kin along both the b axis (a,b) 
+and the a axis (c,d), respectively. The unit for the horizontal axes of the FT profiles is k0 = 2π/λ0.  
+ 
+ 
+Figure 4. (a),(b) Dispersion diagrams of the EPs along the b and a axes, respectively. The colormaps plot 
+the imaginary part of the reflection coefficient Im(rp) that represents the photonic density of states. The blue 
+dashed curves mark the dispersion of the waveguide EPs. (c),(d) The propagation lengths of EPs along both 
+b axis and the a axis, respectively. The red curves are drawn to guide the eye. The data points in all panels 
+were obtained from s-SNOM imaging data (squares) and Comsol simulations (triangles).  
+ 
+
+a
+profiles (kin ll b )
+b
+FT (kin Il b )
+profiles (kin Il a )
+d
+FT (kin ll α)
+1.53eV
+.53ev
+1.50eV
+1.50eV
+44ev
+'n:
+'n'
+.41ev
+41eV
+a
+1.38eV
+1.38eV
+s
+1.35eV
+1.32eV
+29eV
+0
+2
+4
+1
+2
+3
+0
+2
+4
+1
+2
+3
+x (μm)
+k (ko)
+x (μm)
+k (ko)a
+kin Il b
+b
+kin Il a
+c
+kin // b
+1.8
+1.8
+6
+口
+Data
+Data
+Simulation
+5
+Simulation
+1.7
+1.7
+(un)
+4
+3
+Eb
+2
+1.6
+1.6
+(eV)
+(eV)
+0
+山
+1.5
+W1.5
+d
+kin /l α
+口
+Data
+△
+Simulation
+1.4
+1.4
+4
+Im(rp)
+xew
+3
+1.3
+1.3
+2
+AH
+0
+1.2
+1.2
+0
+1.5
+2.5
+1.3
+1.4
+2
+2.5
+3
+1.5
+2
+3
+1.5
+k (ko)
+k (ko)
+E(eV) 
+Figure 5. (a) Nano-optical images of SnS microcrystals with various thicknesses. Here the excitation 
+energy E = 1.38 eV and the in-plane wavevector is along the b axis. (b) Line profiles taken perpendicular 
+to the interference oscillations in (a). (c) Extracted polariton wavelength p versus the thickness of SnS 
+crystals.  
+ 
+Supporting Information for 
+ 
+Imaging anisotropic waveguide exciton polaritons in tin sulfide 
+ 
+Yilong Luan1,2, Hamidreza Zobeiri3, Xinwei Wang3, Eli Sutter4,5, Peter Sutter6*, Zhe Fei1,2* 
+ 
+1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  
+2Ames Laboratory, U. S. Department of Energy, Iowa State University, Ames, Iowa 50011, USA 
+3Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA 
+4Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 
+68588, USA  
+5Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, 
+USA. 
+6Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, 
+USA 
+ 
+*Corresponding to: (P.S.) psutter@unl.edu, (Z.F.) zfei@iastate.edu. 
+ 
+ 
+ 
+List of contents: 
+ 
+1. Nano-optics setup 
+ 
+2. Determination of the crystal axes of SnS 
+ 
+3. Interference mechanisms 
+ 
+4. COMSOL simulations 
+ 
+5. Dispersion calculations 
+ 
+6. Low-temperature dispersion of EPs in SnS 
+ 
+
+a
+d = 160 nm
+b
+c
+700
+口
+Data
+Simulation
+d = 140 nm
+600
+Theory
+s (a.u.)
+max
+台贝
+d = 120 nm
+500
+('n'e)
+= 120 nm
+(nm)
+d = 100 nm
+400
+d = 100 nm
+0
+d = 91 nm
+300
+d = 91 nm
+E = 1.38 eV
+1
+1
+1
+200
+a
+2 μm
+0
+1
+2
+3
+4
+5
+80
+100
+120
+140
+160
+180
+kin // B
+x (μum)
+d (nm)7. Effect of the SnS2 shell on waveguide EPs 
+ 
+8. Photonic mode at the air/mica interface 
+ 
+9. Extraction of the propagation lengths of EPs 
+ 
+10. Coupling strength of EPs 
+ 
+ 
+References for the Supporting Information 
+ 
+Supporting Figures: Figures S1-S13 
+ 
+ 
+ 
+ 
+ 
+1. Nano-optics setup  
+To image the propagative waveguide EPs in SnS, we applied the s-SNOM from Neaspec GmbH 
+The s-SNOM was built based on a tapping-mode Atomic Force Microscope (AFM). The AFM tips used in 
+the study were Pt/Ir-coated silicon tips (Arrow NCPT from Nanoandmore GmbH) with a tapping frequency 
+of ~270 kHz. The tapping amplitude of the tip was set to be about 50 nm. For optical excitations, we used 
+a Ti:sapphire laser (Spectra-Physics, Tsunami) operating at the continuous-wave mode with a photon 
+energy tunable from 1.3 to 1.8 eV that covers the exciton and bandgap energies of SnS. The main observable 
+of the s-SNOM is the complex near-field scattering signal that is modulated due to the tip tapping. 
+Demodulating the signal at the nth harmonics (n ≥ 2) of the tapping frequency can effectively suppress the 
+background signal (n = 2 in the current work). In addition, the pseudo-heterodyne interferometric detection 
+method is used to extract both the amplitude (s) and phase () of the near-field scattering signal. In the 
+current work, we mainly discuss the amplitude part of the signal that is sufficient for describing propagative 
+EPs. All s-SNOM measurements were performed at ambient conditions. 
+ 
+2. Determination of the crystal axes of SnS  
+As introduced in the main text, our samples are SnS microcrystals coated with a thin shell of SnS2. 
+In Figure S1a, we show an optical photo of the sample, where tens of microcrystals are sitting on the mica 
+substrate. Prior to the s-SNOM imaging measurements, we first determined the in-plane crystal axes (i.e., 
+a and b axes) of SnS. The most convenient way to determine the crystal orientation is by inspecting the 
+crystal shape.1,2 As shown in Figure S1b, the crystal is slightly elongated along the b axis. As a result, the 
+crystal corner angles are 85º and 95º, respectively. Therefore, by measuring the corner angle, we can 
+determine the crystal axes. Alternatively, we also applied polarization-dependent Raman spectroscopy to 
+confirm the crystal axes. Here the incident laser beam is polarization-controlled, and the detector collects 
+photons from all polarizations. Due to the strong anisotropy of SnS, Raman spectra show sensitive 
+dependence on the polarization direction of the incident laser.1 In Figure S1c, we plot the Raman spectra of 
+a SnS microcrystal with laser polarization along the a and b axes. When polarization is parallel with the a 
+axis (the black curve in Figure S1c), there are two outstanding peaks at 159 cm-1 (B3g) and 188 cm-1 (Ag), 
+respectively. When polarization parallel with b axis (the red curve in Figure S1c), one more prominent peak 
+at around 93 cm-1 (Ag) emerges in addition to the two peaks at 159 cm-1 and 188 cm-1. The Raman peak at 
+93 cm-1 was used to distinguish the a and b axes of SnS.1 
+ 
+3. Interference mechanisms 
+ 
+As discussed in the main text, the real-space fringes or oscillations of EPs observed in our s-SNOM 
+imaging data were formed due to the interference between tip-back-scattered photons and edge-scattered 
+
+EPs (termed as “Mechanism I”). In the main text, we focus on the discussions of the string of 1D oscillations 
+at the center of the SnS microcrystal. The optical paths responsible for the formation of these interference 
+oscillations are sketched in Figure 1a of the main text and Figure S2a. In this case, the incident laser beam 
+is perpendicular to the short edge (labeled as edge I in Figure S2a) and has an incident angle of  ≈ 30º 
+relative to the sample plane. Upon tip illumination, part of the laser beam is scattered directly by the tip to 
+the detector (labeled as path P1). The tip also excites in-plane EPs and propagate perpendicular to the short 
+edge along the a axis (e.g., edge I in Figure S2a). When reaching edge I, the EPs are scattered to be photons 
+(labeled as path P2) that are collected by the detector. The photons collected from the two paths interfere 
+with each other and thus generating the 1D interference oscillations at the crystal center. In addition to these 
+1D oscillations, there are also interference fringes parallel to the relatively long edges (e.g., edge II in Figure 
+S2b) that are about 43º relative to the b axis. The general mechanism for the fringe formation is similar to 
+those of the 1D oscillations. The main difference is that the EPs responsible for the fringe formation parallel 
+to edge II are propagating along a direction off the two principal crystal axes (i.e., a and b axes). With 
+careful boundary-condition analysis, we found that the propagation direction of the EPs has an angle of  
+≈ 20º relative to the b axis to form interference fringes parallel to edge II (see Figure S2b).  
+ 
+In addition to “mechanism I” discussed above and in the main text, there are also other possible 
+interference mechanisms. One possible mechanism (termed as “mechanism II”) is related to edge excitation 
+of EPs followed by tip scattering, and the other involves tip excitation of EPs followed by edge reflection 
+and tip scattering (termed as “mechanism III”). As discussed below, both mechanisms II and III are not 
+responsible for the interference oscillations/fringes in the current work. 
+ 
+Mechanism II (edge excitation → tip scattering) requires that the focused laser spot (radius ~ 1 m) 
+is at the sample edge, which is only possible when the tip is very close to the sample edge because the laser 
+is always focused on the tip apex. Therefore, for interference fringes or oscillations 1 m away from the 
+sample edge, edge excitation has little contributions. In addition, edge excitation followed by tip scattering 
+is exactly the reverse process of tip excitation followed by edge scattering, so the distance of the optical 
+paths and hence the interference fringes are expected to be the same in the two cases. Finally, the s-SNOM 
+tip is in principle more efficient in polariton excitation than the sample edge due to its metallicity. 
+Considering the above three factors, we believe Mechanism II is not responsible for the interference fringes 
+or oscillations observed in our work. 
+ 
+Mechanism III (tip excitation → edge reflection → tip scattering) plays important role when 
+polaritons or plasmons are highly confined (confinement factor kp/k0 >> 1), which is typical for plasmons 
+and polaritons in the mid-infrared region (e.g., graphene plasmons).3,4 Here, highly confined polaritons or 
+plasmons are efficiently reflected at the sample edge due to the large impedance mismatch. In the case of 
+polaritons or plasmons in the near infrared or visible range (e.g., metal plasmons or exciton polaritons), the 
+confinement is weaker. As a result, only a small portion of polaritons or plasmons can be reflected. Take 
+EPs of SnS for example, the confinement factor kp/k0 is in the range of 1.6-2.5 (see Figure 4a in the main 
+text), therefore the polariton reflectance at the sample edge R ≈ |(kp – k0)/( kp + k0)|2 is in the range of 5% to 
+18%. Moreover, additional geometric and intrinsic damping during the round-trip propagation (from the tip 
+to the edge and back to the tip) further weakens the reflected EPs. Therefore, Mechanism III also does not 
+play an important role in the fringes/oscillations observed in SnS.  
+ 
+4. COMSOL simulations  
+To support the experimental study, we performed finite-element simulations of waveguide EPs in 
+SnS with COMSOL Multiphysics. To excite EPs, we placed a z-polarized point dipole (pz) right above the 
+sample surface. We used two types of models to simulate the SnS microcrystal sample. The first one is a 
+realistic model, where the sample was set to be a four-layer heterostructure (SnS2/SnS/SnS2/mica). Due to 
+the ultra-thin SnS2 shell layer (thickness ~ 3 nm), the realistic model requires ultra-fine messing, so it is 
+time-consuming and not suitable for the simulations of large samples. The second one is an effective model, 
+where the SnS layer is set to be in a homogeneous dielectric environment. The effective permittivity (eff) 
+of the homogeneous dielectric environment can be considered as an average value of air, SnS2, and mica. 
+
+We treated eff as a fitting parameter that was determined by comparing the two models. As shown in Figure 
+S3, the simulated polariton field (out-of-plane Ez field) maps of the two models are consistent with each 
+other when using eff = 2.2 for the 100-nm-thick microcrystal sample. The consistency is also checked at all 
+other excitation energies. Due to the simplicity of the effective model, the simulations are much more 
+efficient. Moreover, we can simulate very large samples with a size of tens of microns, which is necessary 
+for the extraction of the propagation lengths of EPs. The main simulation results in this work were produced 
+with the effective model. The permittivity of SnS along the a and b axes (plotted in Figure S4) used in the 
+Comsol simulations and dispersion calculations is from previous literature.5,6 The c-axis permittivity of SnS 
+is adopted from Ref. 7. In Figure S4, we also mark the optical bandgap or exciton energies (blue arrows), 
+which are 1.66 eV and 1.39 eV along the b and a axes, respectively. The permittivity of SnS2 is set to be 
+about 10 in the ab plane and 6 along the c axis in our spectral range.8,9 The permittivity of mica is set to be 
+2.5 according to Ref. 10. 
+In Figure 2k-o and Figure S5a-e, we plot respectively the polariton field amplitude (|Ez|) and 
+polariton field (Ez) maps at various excitation energies. Based on these field maps, we were able to 
+determine the polariton wavelength and propagation length, which match well the experimental results (see 
+Figure 4, Figure S6 and Figure S7). To better visualize the anisotropy of the EP modes, we performed 2D 
+Fourier transform of the Ez field maps in Figure S5a-e to generate the isofrequency contours. The results 
+are shown in Figure S5f-j, where one can see that the EP mode has a clear elliptic shape at E = 1.29 eV and 
+E = 1.32 eV. As E increases to 1.38 eV and above, the top part of the ellipses (i.e., corresponding to the 
+mode along the a axis) becomes strongly weakened due to the high damping, so EPs prefer propagating 
+along the b axis. 
+ 
+5. Dispersion calculations 
+In Figure 4 of the main text, we plot the dispersion diagrams of SnS along both the a and b axes, 
+where the data points obtained from s-SNOM experiments and Comsol simulations are overlaid on the 
+theoretical dispersion colormaps. In the energy-momentum dispersion colormaps, we plot the imaginary 
+part of the p-polarization reflection coefficient Im(rp), which represents the photonic density of states. The 
+bright curves shown in the dispersion colormaps correspond to transverse magnetic (TM) waveguide 
+modes.11 The transverse-electric waveguide modes, on the other hand, can be revealed when plotting the 
+imaginary part of the s-polarization reflection coefficient Im(rs).12 In the transfer-matrix calculations, we 
+considered the entire SnS2/SnS/SnS2/Mica heterostructure. Take the microcrystal sample in Figs. 1 and 2 
+in the main text for example, the crystal thickness is ~100 nm. Considering a 3-nm-thick SnS2 shell at the 
+top and bottom,1 the SnS core has a thickness of ~94 nm.  
+ 
+6. Low-temperature dispersion of EPs in SnS 
+ 
+To explore theoretically the low-temperature behavior of waveguide EPs in SnS, we plot in Figure 
+S8a,b the calculated polariton dispersion of the 100-nm-thick SnS microcrystal at T = 27 K. The low-
+temperature permittivity of SnS is from Refs. 5 and 6.  For comparison, we plot in Figure S8c,d the 
+dispersion diagrams of EPs at T = 300 K (replotted from Figure 4 in the main text). Compared to room-
+temperature dispersion color plots, the waveguide polariton mode at T = 27 K (Figure S8) is sharper due to 
+the smaller damping at the lower temperature. Besides, the exciton energies slightly increase at the lower 
+temperature. Similar temperature dependence of exciton energies has also been seen in other van der Waals 
+semiconductors.13,14 Furthermore, the light-exciton coupling is much stronger at T = 27 K with more 
+prominent mode bending features close to the exciton energies.  
+ 
+7. Effect of the SnS2 shell on waveguide EPs  
+ 
+As discussed in the main text, the SnS microcrystals studied in this work are coated with a thin 
+SnS2 shell that has a thickness of ~ 3 nm at the top and bottom surfaces.1 Here we wish to evaluate the 
+effect of the thin SnS2 shell on the waveguide EPs. In Figure S9, we plot the calculated dispersion diagrams 
+of waveguide EPs along the two principal axes of SnS with (Figure S9a,c) and without (Figure S9b,d) 
+consideration of the SnS2 shell. From Figure S9, one can see that the SnS2 shell has a very limited effect on 
+
+the EPs. Close examination indicates that the SnS2 shell only induces a slight (~3-4%) decrease of polariton 
+wavelengths of EPs propagating along both the a and b axes. The polaritonic anisotropy along the a and b 
+axes, on the other hand, is solely due to the SnS core.  
+ 
+8. Photonic mode at the air/mica interface 
+ 
+The fringes are also seen on the mica substrate (Figures 1,2 and Figure S10a), which are generated 
+due to the interference of photonic modes propagating at or close to the surface of the mica substrate. The 
+interference mechanism for the substrate fringes is sketched in Figure S10b. To verify that, we performed 
+a dispersion analysis of the substrate mode. In Figure S10c, we show the excitation-energy-dependent fringe 
+profiles extracted along the blue dashed line in Figure S10a. With Fourier transform (Figure S10d), we 
+were able to determine the fringe period, which can be converted directly into the mode wavevector of the 
+substrate ks using the following equation:  
+ 
+0/ ≡ ks/k0 ≈ 0/s + cos ≡ ks/k0 + cos .                                            (S1) 
+ 
+Note the difference between Eq. S1 with Eq. 1 in the manuscript (‘+’ sign instead of ‘-’ sign). Following 
+the Fourier transform, we obtained the dispersion data points, which match well the theoretical dispersion 
+colormap (Figure S10e). From the dispersion diagram, one can see that the wavevector of the substrate 
+mode is roughly proportional to the free-space photon wavevector k0 indicating their photonic nature (note 
+that the k axis in the dispersion diagram is normalized to k0). The mode wavevector is between k0 to nk0, 
+where n ≈ 1.6 is the refractive index of mica. Therefore, we believe the substrate mode measured here 
+corresponds to in-plane photons at the air/mica interface.   
+ 
+9. Extraction of the propagation lengths of EPs 
+ 
+In this section, we describe the extraction processes of the propagation lengths Lep ≡ 1/(2q2), where 
+q2 is the imaginary component of the polariton wavevector q = q1 + iq2. We extracted Lep from both the 
+experimental data and COMSOL simulations. The experimental Lep was extracted from the polariton fringe 
+profiles shown in Figure 3a,c in the main text. We first subtracted the baseline signal of the sample to obtain 
+the pure EP fringe oscillations. The baseline signal comes mainly from the background signal of the sample 
+without the generation of the propagative EPs. Detailed introductions about baseline subtraction can be 
+found in Ref. 12. The baseline-corrected fringe profiles are plotted as black curves in Figs. S6a and S7a, 
+which show clear decay with distance. We then performed envelop fitting of the profiles with a radial 
+exponential decay function x-1/2exp(-x/2Lep), which is expected for radially propagating 2D waves. Note 
+that not all experimental profiles can be fitted due to the high damping. As shown in Figs. S6a, the profile 
+at E = 1.54 eV cannot be fitted for kin // b.  In the case of kin // a (Figure S7a), the profiles at E ≥ 1.38 eV 
+cannot be fitted. Similar fitting procedures were also applied to extract Lep from the simulated EP 
+oscillations (see Figs. S6b and S7b).  
+ 
+10. Coupling strength of EPs 
+ 
+In this section, we estimate the coupling strength of EPs propagating along the b axis of SnS. The 
+criterion for strong coupling is that the Rabi splitting energy R ≈ 2hg is larger than the average EP 
+linewidth (ex + ph)/2, where hg is the coupling energy, ex and ph are the linewidths (full width at half 
+maximum) of exciton and waveguide photon mode.15 To determine the Rabi splitting energy, we fit the 
+dispersion relationship of the EP mode along the b axis of SnS using the equation below:15,16  
+2
+2
+1
+(
+)
+4(
+)
+2
+2
+ph
+ex
+ph
+ex
+E
+E
+E
+E
+E
+hg
+
++
+
+
+−
++
+.                                        (S2) 
+Note that the fitting is mainly based on the bottom-branch of the EP mode that was verified experimentally. 
+Similar approach has been adopted in Ref. 16. The fitting result is shown in Figure S11, where the fitting 
+curves with Eq. S2 match well the dispersion relation of EPs revealed by the colormap. Note that the 
+dispersion colormap is a replot of Figure 4a without normalization of the k axis to k0. Through the fit, we 
+
+obtain the Rabi splitting energy to be about 160 meV, which is comparable to or even bigger than those 
+reported in other materials. The exciton linewidth of SnS along the b axis is about 140 meV at room 
+temperature by fitting the dielectric function from previous literature (see Figure S12a,b).5 The linewidth 
+of the bare waveguide photon mode is estimated to be 70 meV at the exciton energy (see Figure S12c), so 
+the average polariton linewidth is about 105 meV, which is smaller than the Rabi splitting energy. Therefore, 
+we conclude that the EPs of SnS along the b axis is in the strong coupling regime.  
+ 
+References for the Supporting Information 
+(44) 
+Sutter, P.; Wang, J.; Sutter, E. Wrap-Around Core–Shell Heterostructures of Layered Crystals. Adv. 
+Mater. 2019, 31, 1902166.  
+(45) 
+Lin. S.; Carvalho, C.; Yan, S.; Li, R.; Kim, S.; Rodin, A.; Carvalho, L.; Chan, E. M.; Wang, X.; 
+Castro Neto, A. H.; Yao, J. Accessing valley degree of freedom in bulk Tin(II) sulfide at room 
+temperature. Nat. Commun. 2018, 9, 1455.  
+(46) 
+Fei, Z.; Rodin, A. S.; Andreev, G. O.; Bao, W.; McLeod, A. S.; Wagner, M.; Zhang, L. M.; Zhao, 
+Z.; Thiemens, M.; Dominguez, G.; Fogler, M. M.; Castro Neto, A. H.; Lau, C. N.; Keilmann, F.; Basov, 
+D. N. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 2012, 487, 82−85. 
+(47) 
+Chen, J.; Badioli, M.; González, P. A.; Thongrattanasiri, S.; Huth,  F.; Osmond, J.; Spasenović, M.; 
+Centeno, A.; Pesquera, A.; Godignon,  P.; Elorza, A. Z.; Camara, N.; García de Abajo, F. J.; Hillenbrand, 
+R.; Koppens, F. H. L. Optical nano-imaging of gate-tunable graphene plasmons. Nature 2012, 487, 
+77−80. 
+(48) 
+Nguyen, H. T.; Le, Y. L.; Nguyen, T. M. H.; Kim, T. J.; Nguyen, X. A.; Kim, B.; Kim, K.; Lee, 
+W.; Cho, S.; Kim, Y. D. Temperature dependence of the dielectric function and critical points of α‑SnS 
+from 27 to 350 K. Sci. Rep. 2020, 10, 18396.  
+(49) 
+Le, V. L.; Cuong, D. D.; Nguyen, H. T.; Nguyen, X. A.; Kim, B.; Kim, K.; Lee, W.; Hong, S. C.; 
+Kim, T. J.; Kim, Y. D. Anisotropic behavior of excitons in single-crystal α-SnS. AIP Adv. 2020, 10, 
+105003.  
+(50) 
+Banai, R. E.; Burton, L. A.; Choi, S. G.; Hofherr, F.; Sorgenfrei, T.; Walsh, A.; To, B.; Gröll, A.; 
+Brownson, J. R. S. Ellipsometric characterization and density-functional theory analysis of anisotropic 
+optical properties of single-crystal α-SnS. J. Appl. Phys. 2014, 116, 013511. 
+(51) 
+Bertrand, Y.; Leveque, G.; Raisin, C.; Levy, F. Optical properties of SnSe2 and SnS2. J. Phys. C: 
+Solid State Phys. 1979, 12, 2907-2916.  
+(52) 
+Lucovsky, G. Mikkelsen, J. C. Jr. Liang, W. Y.; White, R. M.; Martin, R. M. Optical phonon 
+anisotropies in the layer crystals SnS2 and SnSe2. Phys. Rev. B 1976, 14, 1663-1669.  
+(53) 
+Nitsche, R. & Fritz, T. Precise determination of the complex optical constant of mica. Appl. Opt. 
+43, 3263-3270 (2004). 
+(54) 
+Hu, F.; Luan, Y.; Scott, M. E.; Yan, J.; Mandrus, D. G.; Xu, X.; Fei, Z. Imaging exciton-polariton 
+transport in MoSe2 waveguides, Nat. Photon. 2017, 11, 356-360. 
+(55) 
+Hu, F.; Luan, Y.; Speltz, J.; Zhong, D.; Liu, C. H.; Yan, J.; Mandrus, D. G.; Xu, X.; Fei, Z. "Imaging 
+propagative exciton polaritons in atomically-thin WSe2 waveguides", Phys. Rev. B 100, 121301(R) 
+(2019).  
+(56) 
+Liu, X.; Bao, W.; Li, Q.; Ropp, C.; Wang, Y.; Zhang, X. Control of Coherently Coupled Exciton 
+Polaritons in Monolayer Tungsten Disulphide. Phys. Rev. Lett. 2017, 119, 027403.  
+(57) 
+Christopher, J. W.; Goldberg, B. B.; Swan, A. K. Long tailed trions in monolayer MoS2: 
+Temperature dependent asymmetry and resulting red-shift of trion photoluminescence spectra. Sci. Rep. 
+2017, 7, 14062.  
+(58) 
+Hu, F.; Fei, Z.; Recent progress on exciton polaritons in layered transition‐metal dichalcogenides. 
+Adv. Opt. Mater. 2020, 8, 1901003. 
+(59) 
+Dovzhenko, D.; Lednev, M.; Mochalov, K.; Vaskan, I.; Samokhvalov, P.; Rakovich, Y.; Nabiev, I. 
+Strong excitonphoton coupling with colloidal quantum dots in a tunable microcavity. Appl. Phys. Lett. 
+2021, 119, 011102.  
+ 
+
+ 
+ 
+ 
+ 
+Supporting Figures 
+ 
+ 
+Figure S1. (a) A large-area optical photo of SnS microcrystals on a mica substrate. (b) The AFM image of 
+an SnS microcrystal. The marked angle of the crystal corner is 85º. (c) Polarization-dependent Raman 
+spectra of SnS along both the a and b axes. Detailed discussions are given in Section 2 of the Supporting 
+Information. 
+ 
+ 
+Figure S2. Illustrations of the formation mechanism of 1D interference oscillations at the crystal center (a) 
+and interference fringes parallel to the long edges (b). Detailed discussions are given in Section 3 of the 
+Supporting Information. 
+ 
+ 
+Figure S3. The simulated polariton field (Ez field) maps with the realistic model (a) and the effective model 
+(b). Detailed discussions are given in Section 4 of the Supporting Information. 
+ 
+
+a
+b
+C
+50
+mica
+Raman intensity (a.u.)
+aaxis
+baxis
+40
+30
+SnS
+850
+20
+10
+50μm
+12μm
+0
+50
+100
+150
+200
+250
+300
+Ramanshift(cm)a
+b
+Laser
+Laser
+P1
+tip
+EPs
+tp
+EPs
+edge I
+edge II6
+Ez (a.u.)
+800nm 
+Figure S4. The real part (red) and the imaginary part (black) of the permittivity along b axis (a) and a axis 
+(b). The blue arrow marks the optical bandgap or exciton energy. 
+ 
+ 
+Figure S5. (a-e) Simulated out-of-plane field (Ez) maps of waveguide EPs in 100-nm-thick SnS at various 
+energies. The EPs are excited by a point dipole (pz) located at the center of the image right above the sample 
+surface. (f-j) Energy-dependent isofrequency contour maps of waveguide EPs generated by the Fourier 
+transform of the Ez maps in (a-e). Detailed discussions are given in Section 4 of the Supporting Information. 
+ 
+ 
+ 
+
+a 
+b axis
+25
+b
+25
+aaxis
+20
+20
+15
+15
+3
+3
+10
+10
+5
+5
+0
+0
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+E (eV)
+E (eV)a
+Ez, E=1.29 eV
+b
+Ez,E=1.32eV
+c
+Ez, E=1.38 eV
+p
+Ez, E=1.44 eV
+e
+Ez, E=1.48 eV
+Ez
+(a.u.)
+4μm
+2DFFT,E=1.29eV
+g
+2DFFT.E=1.32eVh
+2DFFT,E=1.38eV
+-
+2D FFT, E=1.44 eV
+2D FFT, E=1.48 eV
+3
+FT
+ky (105 cm-1)
+(a.u.)
+max
+0
+a
+ta
+ta
+Aa
+4a
+min
+3
+-3
+0
+3
+-3
+0
+3
+-3
+0
+3
+-3
+0
+3
+-3
+0
+3
+kx (105 cm-1)
+kx (105 cm-1)
+kx (105 cm-1)
+kx (105 cm-1)
+kx (105 cm-1) 
+Figure S6. Fitting of the propagation length (Lep) of EPs along the b-axis based on the experimental data 
+(a) and Comsol simulations (b).  
+ 
+ 
+ 
+ 
+Figure S7. Fitting of the propagation length (Lep) of EPs along the a-axis based on the experimental data 
+(a) and Comsol simulations (b).  
+ 
+ 
+
+a
+Experimental, kin // b
+b
+Simulation,ki // b
+E=1.29eV,Lep=4.5um
+E=1.29eV.Len=5um
+E=1.32eV, Lep=4um
+E=1.32eV.Len=4um
+E=1.35eV,Lep=3.2um
+E=1.35eV,Lep=3.5um
+E=1.38eV,Lep=2.9um
+E=1.38eV,Lep=3.2um
+'n'
+E=1.41eV,Lep=2.2um
+a
+S
+E
+E=1.44eV, Lep=2um
+T Z1Zz E=1.44eV Lep=2.1um
+E=1.48eV,Len=1.3um
+E=1.48eV, Lep=1.5um
+E=1.51eV,Lep=1.0um
+AZ z= =151eV Lep=0.8um
+E=1.54eV
+E=1.54eV, Lep=0.4um
+0
+2
+4
+0
+2
+4
+x (μm)
+x (μm)a
+Experimental, kin // a
+b
+Simulation, k// a
+E=1.29eV,Lep=3.2um
+E=1.29eV,Lep=3.5um
+E=1.32eV,L
+-ep=1.7um
+=1.32eV,Lep=2.2um
+E=1.35eV.L
+:1.1um
+E=1.35eV,Lep=1.4um
+E=1.38eV
+E=1.38eV.Len=1um
+(a.u.)
+E=1.41eV
+_E=1.41eV,Lep=0.75um
+S
+EN
+E=1.44eV
+E=1.44eV, Lep=0.5um
+E=1.48eV
+E=1.48eV,Lep=0.45um
+E=1.51eV
+E=1.51eV,Lep=0.35um
+E=1.54eV
+E=1.54eV,Lep=0.2um
+0
+2
+4
+0
+2
+4
+x (μm)
+x (μm) 
+Figure S8. The dispersion colormaps of SnS along the two principal axes at T = 27 K (a,b) and T = 300 K 
+(c,d). Detailed discussions are given in Section 6 of the Supporting Information. 
+ 
+ 
+Figure S9. The calculated dispersion colormaps of waveguide EPs along the two principal axes of SnS 
+with (a,c) and without (b,d) consideration of the thin SnS2 shell.  
+ 
+ 
+ 
+
+a
+kin // b, T = 27 K
+b
+kin /l a, T = 27 K
+c
+kin Il b, T = 300K
+d
+kin // a, T= 300K
+1.8
+1.8
+1.8
+1.8
+1.7
+1.7
+1.7
+1.7
+Eb
+Eb
+1.6
+1.6
+1.6
+1.6
+E
+E
+E
+E
+1.4
+1.4
+1.4
+1.4
+Im(p)
+max
+Ea
+Ea
+1.3
+1.3
+1.3
+1.3
+0
+1.2
+1.2
+1.2
+1.2
+1.5
+2
+2.5
+3
+3.5
+1.5
+2
+2.5
+3
+3.5
+1.5
+2
+2.5
+3
+3.5
+1.5
+2
+2.5
+3
+3.5
+k (ko)
+k (ko)
+k (ko)
+k (ko)a
+kin /l b, with SnS
+b
+kin /l b, w/o SnS
+c
+kin /l a, withSnS
+d
+kin /l a, w/o SnS
+1.8
+1.8
+1.8
+1.8
+1.7
+1.7
+1.7
+1.7
+Eb
+Eb
+1.6
+1.6
+1.6
+1.6
+E
+E
+E
+E
+1.4
+1.4
+1.4
+1.4
+Im(rp)
+max
+Ea
+1.3
+1.3
+1.3
+1.3
+0
+1.2
+1.2
+1.2
+1.2
+1.5
+2
+2.5
+3
+3.5
+1.5
+2
+2.5
+3
+3.5
+1.5
+2
+2.5
+3
+3.5
+1.5
+2
+2.5
+3
+3.5
+k (ko)
+k (ko)
+k (ko)
+k (ko) 
+Figure S10. (a) Nano-optical imaging data of the SnS crystal on mica substrate at E = 1.38 eV. (b) 
+Illustration of the interference mechanism for the formation of fringes on the substrate. (c) Fringe profiles 
+at various excitation energies taken along the blue dashed line shown in (a). (d) Fourier transform of the 
+energy-dependent fringe profiles shown in (c). (e) Experimental and theoretical dispersion diagram of the 
+substrate photon mode of mica. The two vertical dashed lines mark the photon dispersion in air (k = k0) and 
+in bulk mica (k=1.6k0). Detailed discussions are given in Section 8 of the Supporting Information.  
+ 
+ 
+ 
+Figure S11. (a) Fitting the dispersion relation of the EP mode along the b axis of SnS. The white dotted 
+curves were produced with Eq. S2. (b) Calculated dispersion relation of the bare waveguide photon mode 
+without coupling with excitons along the b axis of SnS. Detailed discussions are given in Section 10 of the 
+Supporting Information. 
+ 
+ 
+ 
+
+(a)
+s-SNOM, 1.38 eV
+(c)
+substrate mode profiles
+(d)
+Fouriertransform
+(e)
+substrate mode dispersion
+1.8
+Ik=1.6ko
+mica
+1.53 eV
+1.53eV
+1.50 eV
+1.50eV
+1.7
+1.47 eV
+1.47 eV
+1.6
+1.44 eV
+1.44 eV
+(Cn
+(a.u.)
+(eV)
+口
+200nm
+1.41eV
+1.41 eV
+口
+1.5
+(b)
+S
+1.38eV
+1.38eV
+E
+P2
+1.35 eV
+1.35eV
+1.4
+1.32 eV
+口口口
+1.32 eV
+substrate
+1.29eV
+1.3
+Sns
+1.29 eV
+photons
+mica
+1.2
+0.0
+0.5
+1.0
+1.5
+0
+1
+2
+3
+4
+5
+0.5
+1.5
+2.5
+3.5
+x (um)
+k (10° cm)
+k (ko)a
+2
+b
+2
+1.8
+1.8
+日1.4
+日1.4
+1.2
+1.2
+1
+0.5
+1
+1.5
+2
+2.5
+3
+0.5
+1
+1.5
+2
+2.5
+3
+k(105cm-1)
+(105cm-1)a
+b
+200
+c
+2.0
+0
+T = 27K
+T=100K
+exciton linewidth (meV)
+T=200K
+T=300K
+150
+1.5
+(cn'e) ()ul
+6
+100
+1.0
+50
+0.5
+2
+0
+0
+0.0
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+0
+50
+100
+150
+200
+250
+300
+1.2
+1.4
+1.6
+1.8
+2.0
+E (eV)
+T (K)
+E (eV)Figure S12. (a) Fitting the exciton linewidth from the b-axis permittivity of SnS from Ref. 5. (b) 
+Temperature-dependent linewidth of excitons along the b axis of SnS based on the fitting in (a). (c) Line 
+profile of the photonic mode taken along the vertical dashed line in Figure S11b. Detailed discussions are 
+given in Section 10 of the Supporting Information. 
+ 
+ 
+Figure S13. Illustration of a selective nanophotonic interconnector based on anisotropic EPs in SnS. When 
+the incident photonic signal is at energies of E ≤ 1.29 eV, both devices 1 and 2 can receive the signal.  When 
+the incident photonic signal is at energies of E ≥ 1.38 eV, only device 1 can receive the signal.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+to device
+Incident signal
+Sns
+interconnector
+s

1dE0T4oBgHgl3EQfdgBe/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:578a38fed8de70238324b171903852fe60f11badc513317493b4297419225b0b
+size 1179693

1dFRT4oBgHgl3EQfmDfL/content/tmp_files/2301.13600v1.pdf.txt

@@ -0,0 +1,2199 @@
+arXiv:2301.13600v1  [cs.GT]  31 Jan 2023
+CONSTRAINED PHI-EQUILIBRIA
+ARXIV PREPRINT
+Martino Bernasconi
+Politecnico di Milano
+martino.bernasconideluca@polimi.it
+Matteo Castiglioni
+Politecnico di Milano
+matteo.castiglioni@polimi.it
+Alberto Marchesi
+Politecnico di Milano
+alberto.marchesi@polimi.it
+Francesco Trov`o
+Politecnico di Milano
+francesco1.trovo@polimi.it
+Nicola Gatti
+Politecnico di Milano
+nicola.gatti@polimi.it
+February 1, 2023
+ABSTRACT
+The computational study of equilibria involving constraints on players’ strategies has been largely
+neglected. However, in real-world applications, players are usually subject to constraints ruling
+out the feasibility of some of their strategies, such as, e.g., safety requirements and budget caps.
+Computational studies on constrained versions of the Nash equilibrium have lead to some results
+under very stringent assumptions, while finding constrained versions of the correlated equilibrium
+(CE) is still unexplored. In this paper, we introduce and computationally characterize constrained
+Phi-equilibria—a more general notion than constrained CEs—in normal-form games. We show
+that computing such equilibria is in general computationally intractable, and also that the set of the
+equilibria may not be convex, providing a sharp divide with unconstrained CEs. Nevertheless, we
+provide a polynomial-time algorithm for computing a constrained (approximate) Phi-equilibrium
+maximizing a given linear function, when either the number of constraints or that of players’ actions
+is fixed. Moreover, in the special case in which a player’s constraints do not depend on other players’
+strategies, we show that an exact, function-maximizing equilibrium can be computed in polynomial
+time, while one (approximate) equilibrium can be found with an efficient decentralized no-regret
+learning algorithm.
+1
+Introduction
+Over the last years, equilibrium computation problems have received a terrific attention from AI and ML re-
+search (Brown and Sandholm, 2019; Bakhtin et al., 2022), as game-theoretical equilibrium notions provide a prin-
+cipled framework to deal with multi-player decision-making problems. Most of the works on equilibrium computation
+problems focus on classical solution concepts—such as the well-known Nash equilibrium (NE) (Nash, 1951) and cor-
+related equilibrium (CE) (Aumann, 1974)—, thus neglecting the presence of constraints entirely. However, in most
+of the real-world applications, the players are usually subject to constraints that rule out the feasibility of some of
+their strategies, such as, e.g., safety requirements and budget caps. Thus, addressing equilibrium notions involving
+constraints is a crucial step needed for the operationalization of game-theoretic concepts into real-world settings.
+The study of equilibrium notions involving constraints was initiated by Arrow and Debreu (1954), who define the
+concept of generalized NE (GNE). The GNE can be interpreted as an NE of a game where players’ strategies are
+subject to some constraints, which must be satisfied at the equilibrium and also determine which are the feasible
+players’ deviations. However, given that computing a GNE is clearly PPAD-hard (Daskalakis et al., 2009), all the
+works dealing with the computation of GNEs (see, e.g., (Facchinei and Kanzow, 2010)) provide efficient algorithms
+only in specific settings that require very stringent assumptions.
+Most of the computationally challenges in finding GNEs are inherited from the NE. In settings in which constrained are
+not involved, the computational issues of NEs are usually circumvented by considering weaker equilibrium notions.
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Among them, those that have received most of the attention in the literature are the CE and its variations, which
+have been shown to be efficiently computable in several settings of interest (Papadimitriou and Roughgarden, 2008;
+Celli et al., 2020). Surprisingly, with the only exception of (Chen et al., 2022) (see Section 1.2 for a detailed discussion
+on it), no work has considered the problem of computing CEs in constrained settings. Thus, investigating whether the
+CE retains its nice computational properties when adding constraints on players’ strategies is an open interesting
+question.
+1.1
+Original Contributions
+In this paper, we introduce and computationally characterize constrained Phi-equilibria, starting, as it is customary,
+from the setting of normal-form games. Our equilibria include the constrained versions of the classical CE and all of its
+variations as special cases, by generalizing the notion of Phi-equilibria introduced by Greenwald and Jafari (2003) to
+constrained settings. In particular, constrained Phi-equilibria are defined as Phi-equilibria, but in games where players
+are subject to some constraints. Such constraints must be satisfied at the equilibrium, and, additionally, players are
+only allowed to undertake safe deviations, namely those that are feasible according to the constraints. Crucially, the
+set of safe deviations of a player does not only depend on the strategy of that player, but also on those of the others.
+We start by showing that one of the most appealing computational properties of Phi-equilibria, namely that the set
+of the equilibria of a game is convex, is lost when moving to their constrained version. This raises considerable
+computational challenges in computing constrained Phi-equilibria. Indeed, we formally prove a strong intractability
+result: for any factor α > 0, it is not possible, unless P = NP, to find in polynomial time a constrained (approximate)
+Phi-equilibrium which achieves a multiplicative approximation α of the optimal value of a given linear function. Then,
+in the rest of the paper, we show several ways in which such a negative result can be circumvented.
+We prove that a constrained approximate Phi-equilibrium which maximizes a given linear function can be found in
+polynomial time, when either the number of constraints or that of players’ actions is fixed. Our results are based on a
+general algorithm that employs a non-standard “Lagrangification” of the constraints defining the set of safe deviations
+of a player. Moreover, the algorithm needs a way of dealing with the non-convexity of the set of the equilibria, which
+we provide in the form of a clever discretization of the space of the Lagrange multipliers.
+Finally, we focus on the special case in which the constraints defining the safe deviations of a player do not depend
+on the the strategies of the other players, but only on the strategy of that player. This includes constrained Phi-
+equilibria identifying a particular constrained version of the coarse CE by Moulin and Vial (1978a), in which the
+players’ strategies are subject to marginal cost constraints. These arise in several real-world applications in which
+the players have bounded resources, such as, e.g., budget-constrained bidding in auctions. In such a special case, we
+prove that a constrained (exact) Phi-equilibrium maximizing a given linear function can be computed in polynomial
+time, and we provide an efficient decentralized no-regret learning algorithm for finding one constrained (approximate)
+Phi-equilibrium.
+1.2
+Related Works
+GNEs
+Rosen (1965) initiated the study of the computational properties of GNEs. After that, several other works
+addressed the problem of computing GNEs by mainly exploiting techniques based on quasi-variational inequalities
+(see (Facchinei and Kanzow, 2010) for a survey). More recently, some works (Kanzow and Steck, 2016; Bueno et al.,
+2019; Jordan et al., 2022; Goktas and Greenwald, 2022) also studied the convergence of iterative optimization algo-
+rithms to GNEs. In order to provide efficient algorithms, all these works need to introduce very stringent assumptions,
+which are even stronger than those required for the efficient computation of NEs.
+Constrained Markov Games
+Equilibrium notions involving constraints have also been addressed in the literature
+on Markov games, with (Altman and Shwartz, 2000; Alvarez-Mena and Hern´andez-Lerma, 2006) being the first works
+introducing GNEs in such a field. More recently, Hakami and Dehghan (2015) defined a notion of constrained CE in
+Markov games. However, the incentive constraints in their notion of equilibrium only predicate on “pure” deviations,
+which, in presence of constraints, may lead to empty sets of safe deviations. Very recently, Chen et al. (2022) gener-
+alize the work of Hakami and Dehghan (2015) by considering “mixed” deviations. However, their algorithm provides
+rather weak convergence guarantees, as it only ensures that the returned solution satisfies incentive constraints in ex-
+pectation. Indeed, as we show in Proposition 3.1, the set of constrained equilibria may not be convex (it is easy to see
+that Example 1 also applies to the setting studied by Chen et al. (2022)), and, thus, the fact that incentive constraints are
+only satisfied in expectation does not necessarily imply that the “true” incentive constraints defining the equilibrium
+are satisfied. We refer the reader to Appendix A for additional details on these aspects.
+2
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+2
+Preliminaries
+In this section, we introduce all the preliminary definitions and results that are needed in the rest of the paper.
+2.1
+Cost-constrained Normal-form Games
+In a normal-form game, there is a finite set N := {1, . . . , n} of n players. Each player i ∈ N has a finite set Ai of
+actions available, with s := |Ai| for i ∈ N being the number of players’ actions.1 We denote by a ∈ A :=×i∈N Ai
+an action profile specifying an action ai for each player i ∈ N. Moreover, for i ∈ N, we let a−i ∈ A−i :=
+×j̸=i∈N Ai be an action profile of all players other than i, while (a, a−i) is the action profile obtained by adding
+a ∈ Ai to a−i. Finally, we let ui : A → [0, 1] be the utility function of player i ∈ N, with ui(a) being the utility
+perceived by that player when the action profile a ∈ A is played.
+We extend classical normal-form games by considering the case in which each player i ∈ N has mi cost functions,
+namely ci,j : A → [−1, 1] for j ∈ [mi].2 Each player i ∈ N is subject to mi constraints, which require that all player
+i’s costs are less than or equal to zero.3 For ease of notation, we assume w.l.o.g. that all players have the same number
+of constraints, namely m := mi for all i ∈ N. Moreover, we encode the costs of player i ∈ N by a vector-valued
+function ci : A → [−1, 1]m such that, for every a ∈ A, the j-th component of the vector ci(a) is ci,j(a).
+Correlated Strategies
+In this paper, we deal with solution concepts defined by correlated strategies. A correlated
+strategy z ∈ ∆A is a probability distribution defined over the set of actions profiles, with z[a] denoting the probability
+assigned to a ∈ A.4 With an abuse of notation, for every player i ∈ N, we let ui(z) be player i’s expected utility
+when the action profile played by the players is drawn from z ∈ ∆A. In particular, it holds ui(z) := �
+a∈A ui(a)z[a].
+Similarly, we let ci(z) := �
+a∈A ci(a)z[a] be the vector of player i’s expected costs, so that player i’s constraints
+can be compactly written as ci(z) ⪯ 0. Finally, we define S ⊆ ∆A as the set of safe correlated strategies, which are
+those satisfying the cost constraints of all players. Formally:
+S := {z ∈ ∆A | ci(z) ⪯ 0
+∀i ∈ N} .
+In the following, we assume w.l.o.g. that S ̸= ∅.
+2.2
+Constrained Phi-equilibria
+We generalize the notion of Phi-equilibria (Greenwald and Jafari, 2003) to cost-constrained normal-form games. Such
+equilibria are defined as correlated strategies z ∈ ∆A that are robust against a given set Φ of players’ deviations, in
+the sense that, if a mediator draws an action profile a ∈ A according to z and recommends to play action ai to each
+player i ∈ N, then no player has an incentive to deviate from their recommendation by selecting a deviation in Φ.
+For every i ∈ N, we let Φi be the set of player i’s deviations, i.e., linear transformations φi : Ai → ∆Ai that prescribe
+a probability distribution over player i’s actions for every possible action recommendation. For ease of notation, we
+encode a deviation φi by means of its matrix representation. Formally, an entry φi[b, a] of the matrix represents the
+probability assigned to action a ∈ Ai by φi(b). We denote the set of all the possible deviations by Φ := {Φi}i∈N .
+Given a correlated strategy z ∈ ∆A and a deviation φi ∈ Φi, we define φi ⋄ z as the modification of z induced by φi,
+which is a linear transformation that can be expressed as follows in terms of matrix representation:
+(φi ⋄ z)[ai, a−i] :=
+�
+b∈Ai
+φi[b, ai]z[b, a−i],
+for every ai ∈ Ai and a−i ∈ A−i. Moreover, given a set Φi of deviations of player i ∈ N, in the following we denote
+by ΦS
+i (z) := {φi ∈ Φi | φi ⋄ z ∈ S} the set of safe deviations for player i at a given correlated strategy z ∈ ∆A.
+We are now ready to provide our definition of constrained Phi-equilibria in cost-constrained normal-form games.
+Definition 2.1 (Constrained ǫ-Phi-equilibria). Given a set Φ := {Φi}i∈N of deviations and an ǫ > 0, a constrained
+ǫ-Phi-equilibrium is a safe correlated strategy z ∈ S such that, for all i ∈ N, the following holds:
+ui(z) ≥ ui(φi ⋄ z) − ǫ
+∀φi ∈ ΦS
+i (z).
+1For ease of presentation, in this paper we assume that all the players have the same number of actions. All the results can be
+easily generalized to the case of different numbers of actions.
+2In this paper, given some x ∈ N>0, we let [x] := {1, . . . , x} be the set of the first x natural numbers.
+3Since z ∈ ∆A, we can assume w.l.o.g. that all the constraints are of the form ≤ 0, as any constraint can always be cast in such
+a form by suitably manipulating the cost function ci,j.
+4In this paper, given a finite set X, we denote by ∆X the set of all the probability distributions defined over the elements of X.
+3
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+A constrained Phi-equilibrium is defined for ǫ = 0.
+2.3
+Computing Constrained Phi-equilibria
+In the following, we formally introduce the computational problem that we study in the rest of the paper.
+We denote by I := (Γ, Φ) an instance of the problem, where the tuple Γ := (N, A, {ui}i∈N , {ci,j}i∈N,j∈[m]) is a
+cost-constrained normal-form game and Φ := {Φi}i∈N is a set of deviations. Moreover, we let |I| be the size (in
+terms of number of bits) of the instance I. We assume that the number n of players is fixed, so that |I| does not grow
+exponentially in n.5 We also make the following assumption on how the sets of deviations are represented:
+Assumption 1. For every i ∈ N, the set Φi is a polytope encoded by a finite of linear inequalities.6
+Let us remark that, in games without constraints, this assumption is met by all the sets Φ which determine the classical
+notions of Phi-equilibria (Greenwald and Jafari, 2003).
+Next, we formally define our computational problem:
+Definition 2.2 (APXCPE(α, ǫ)). For any α, ǫ > 0, we define APXCPE(α, ǫ) as the problem of finding, given an
+instance I := (Γ, Φ) and a linear function ℓ : ∆A → R as input, a constrained ǫ-Phi-equilibrium z ∈ ∆A such that
+ℓ(z) ≥ αℓ(z′) for all constrained Phi-equilibria z′ ∈ ∆A.
+Intuitively, APXCPE(α, ǫ) asks to compute a constrained ǫ-Phi-equilibrium whose value for the linear function ℓ is at
+least a fraction α of the maximum value which can be achieved by an (exact) constrained Phi-equilibrium.
+In order to ensure that an instance of our problem is well defined, we make the following “Slater-like” assumption on
+how the players’ cost constraints are defined.
+Assumption 2. For every correlated strategy z ∈ ∆A, player i ∈ N, and index j ∈ [m], there exists φ◦
+i ∈ ΦS
+i (z):
+ci,j(φ◦
+i ⋄ z) ≤ −ρ,
+where ρ > 0 and 1/ρ is O(poly(|I|)), with poly(|I|) being a polynomial function of the instance size |I|.
+In Assumption 2, the condition ρ > 0 is required to guarantee the existence of a constrained Phi-equilibrium (see
+Theorem 2.1) and that the sets ΦS
+i (z) are non-empty (otherwise our solution concept would be ill defined). Moreover,
+the second condition on ρ in Assumption 2 is equivalent to requiring that our algorithms run in time polynomial in 1
+ρ.
+Assumption 2 also allows us to prove the existence of our equilibria, by showing that the constrained Nash equilibria
+introduced by Altman and Shwartz (2000), which always exist under Assumption 2, are also constrained Phi-equilibria.
+Theorem 2.1. Given a cost-constrained normal-form game Γ and a set Φ of deviations, if Assumption 2 is satisfied,
+then Γ admits a constrained Phi-equilibrium.
+2.4
+Relation with Unconstrained Phi-equilibria
+We conclude the section by discussing the relation between our constrained Phi-equilibria and classical equilibrium
+concepts for unconstrained games.
+Correlated Equilibrium
+When there are no constraints, the correlated equilibrium (CE) (Aumann, 1974) is a spe-
+cial case of Phi-equilibrium. As shown by Greenwald and Jafari (2003), the CE is obtained when the sets Φi contain
+all the possible deviations. Formally, the CE is defined by the set ΦALL := {Φi,ALL} of deviations such that:
+Φi,ALL :=
+�
+φi
+���
+�
+a∈Ai
+φi[b, a] = 1
+∀b ∈ Ai
+�
+.
+5Notice that the size of the representation of a normal-form game is O(sn), and, thus, exponential in n. Any algorithm that
+runs in time polynomial in such instance size is not computationally appealing, as even its input has size exponential in n. For this
+reason, we focus on the case in which n is fixed, and, thus, the instance size does not grow exponentially with n.
+6Notice that, since each φi ∈ Φi is represented as a matrix, a linear inequality is expressed as �
+b,a∈Ai M[b, a]φi[b, a] ≤ d,
+for some matrix M and scalar d.
+4
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Coarse Correlated Equilibrium
+The coarse correlated equilibrium (CCE) (Moulin and Vial, 1978b) is a special
+(unconstrained) Phi-equilibrium whose set of deviations is ΦCCE := {Φi,CCE}i∈N such that:
+Φi,CCE :=
+�
+φi
+��� ∃h ∈ ∆Ai : φi[b, a] = h[a] ∀b, a ∈ Ai
+�
+.
+Intuitively, such sets contain all the possible deviations that prescribe the same probability distribution independently
+of the received action recommendation.
+Thus, our constrained Phi-equilibria include the generalization of CEs and CCEs to cost-constrained games.
+Our definition of constrained Phi-equilibrium needs to employ “mixed” deviations that map action recommendations
+to probability distributions over actions. This is necessary in presence of constraints. Instead, without them, one
+can simply consider “pure” deviations that map recommendations to actions deterministically Greenwald and Jafari
+(2003).
+3
+Challenges of Constrained Phi-equilibria
+In this section, we show that, in cost-constrained normal-form games, Phi-equilibria loose the nice computational
+properties that they exhibit in unconstrained settings. This is crucially determined by the fact that the set of constrained
+Phi-equilibria may not be convex in general.
+Proposition 3.1. Given any instance I := (Γ, Φ), the set of constrained Phi-equilibria may not be convex.
+Proposition 3.1 is proved by the following example.
+Example 1. Let ΦALL be the set of all the possible deviations in a two-player game in which each player has two
+actions, namely A1 = A2 = {a0, a1}. The first player’s utility is such that u1(a, a′) = 0 for all a ∈ A1 and a′ ∈ A2,
+while the second player’s utility is such that u2(a0, a1) = 1, and 0 otherwise. Both players share the same single
+cost constraint (m = 1). Their cost functions are defined as ci(a0, a1) = 1, ci(a0, a0) = − 1
+2, and ci(a1, a) = −1
+for all a ∈ A2. Notice that the instance defined above satisfies Assumption 2 for ρ = 1/2. It is easy to see that the
+correlated strategy z1 ∈ ∆A such that z1[a0, a0] = 2
+3 and z1[a0, a1] = 1
+3 is a constrained Phi-equilibrium. Moreover,
+the “pure” correlated strategy z2 ∈ ∆A such that z2[a1, a0] = 1 is also a constrained Phi-equilibrium. However, the
+combination z3 = 1
+2(z1 + z2) is not a constrained Phi-equilibrium. Indeed, the second player has an incentive to
+deviate by using a deviation φ2 such that φ2[a0, a1] = 1 and φ2[a1, a1] = 1. Such a deviation prescribes to play action
+a1 when a0 is recommended, and to play action a1 when the recommendation is a1. Straightforward calculations show
+that, for every a ∈ A1:
+(φ2 ⋄ z3)[a, a′] =
+� 1
+2
+if
+a′ = a1
+0
+otherwise,
+and u2(φ2 ⋄ z3) = 1
+2 > u2(z3) = 1
+6. Moreover, the deviation is safe, since φ2 ∈ ΦS
+2 (z3) as c2(φ2 ⋄ z3) = 0.
+In order to formally asses the computational challenges of computing constrained Phi-equilibria, we prove the follow-
+ing strong inapproximability result:
+Theorem 3.1 (Hardness). For any constant α > 0, the problem APXCPE(α, (α/s)2) is NP-hard, where s is the
+number of players’ actions in the instance given as input.
+Intuitively, Theorem 3.1 states that, for every multiplicative approximation factor α > 0, it is not possible to find
+a constrained ǫ-Phi-equilibrium having value of ℓ at least a fraction α of its optimal value in time polynomial in 1
+ǫ.
+Moreover, as a byproduct of Theorem 3.1, we also get the inapproximability up to within any factor of the problem of
+computing an optimal constrained (exact) Phi-equilibrium.
+Notice that the hardness result in Theorem 3.1 cannot hold for values of ǫ that are independent from the instance size.
+Indeed, as we prove in Corollary 4.3 in Section 4, problem APXCPE(1, ǫ) can be solved in quasi-polynomial time
+in the instance size whenever ǫ > 0 is a given constant. Thus, any NP-hardness result for APXCPE(α, ǫ) would
+contradict the exponential-time hypothesis.7
+4
+Computing Optimal Constrained ǫ-Phi-equilibria Efficiently
+In this section, we show how to circumvent the negative result established by Theorem 3.1. In particular, we prove
+that, when the number of cost constraints is fixed, problem APXCPE(1, ǫ) can be solved in time polynomial in the
+7The exponential-time hypothesis conjectures that solving 3SAT requires at least exponential time.
+5
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+instance size and 1
+ǫ for ǫ > 0 (Corollary 4.2). Moreover, we also prove that, in general, for any constant ǫ > 0 problem
+APXCPE(1, ǫ) admits a quasi-polynomial-time algorithm, whose running time becomes polynomial when the number
+of players’ actions is fixed (Corollary 4.3).
+First, in Section 4.1, we provide a general algorithm that is at the core of the two main results of this section. Then, in
+Section 4.1, we show how the algorithm can be suitably instantiated in order to prove each result. In the rest of this
+section, unless stated otherwise, we always assume that an ǫ > 0 has been fixed, and that I := (Γ, Φ) and ℓ : ∆A → R
+are the inputs of a given instance of problem APXCPE(1, ǫ).
+4.1
+General Algorithm
+The main technical tool that we employ in order to design our algorithm is a “Lagrangification” of the constraints
+defining the sets ΦS
+i (z) of safe deviations. First, we prove the following preliminary result, which shows that strong
+duality holds for the problem maxφi∈ΦS
+i (z) ui(φi ⋄ z) of finding the best safe deviation for player i ∈ N at z ∈ ∆A.
+Lemma 4.1. For every z ∈ ∆A and i ∈ N, it holds
+max
+φi∈ΦS
+i (z) ui(φi ⋄ z) = sup
+φi∈Φi
+inf
+ηi∈Rm
++
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+=
+inf
+ηi∈Rm
++
+sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+.
+Then, by exploiting Lemma 4.1, we can prove that, under Assumption 2, strong duality continues to hold even when
+restricting the Lagrange multipliers ηi to have ℓ1-norm less than or equal to 1/ρ. Formally:
+Lemma 4.2. Let D :=
+�
+η ∈ Rm
++ | ||η||1 ≤ 1/ρ
+�
+. Then, for every z ∈ ∆A and i ∈ N, it holds:
+max
+φi∈ΦS
+i (z) ui(φi ⋄ z) = max
+φi∈Φi min
+ηi∈D
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+= min
+ηi∈D max
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+.
+Lemma 4.2 allows us to write player i’s incentive constraints in the definition of constrained ǫ-Phi-equilibria as
+ui(z) ≥ min
+ηi∈D max
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+− ǫ.
+(1)
+This crucially allows us to show the following result: solving problem APXCPE(1, ǫ) is equivalent to computing
+max(η1,...,ηn)∈Dn Fǫ(η1, . . . , ηn), where Fǫ(η1, . . . , ηn) is the optimal value of a suitable maximization problem
+parameterized by tuples of Lagrange multipliers ηi ∈ D, one per player i ∈ N. Such a problem asks to compute a safe
+correlated strategy maximizing the linear function ℓ subject to players’ incentive constraints that are re-formulated by
+means of Lemma 4.2. Formally, we define Fǫ(η1, . . . , ηn) as the maximum of ℓ(z) over those z ∈ S that additionally
+satisfy the following constraint for every i ∈ N:
+ui(z) ≥ max
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+− ǫ.
+(2)
+Notice that the min operator that appears in the right-hand side of Constraints (1) is dropped by adding the outer
+maximization over the tuples (η1, . . . , ηn) ∈ Dn, as the maximum of ℓ is always achieved when the right-hand side
+of such constraints is as small as possible.
+Next, we show that Fǫ(η1, . . . , ηn) can be computed in polynomial time by means of the ellipsoid algorithm.
+Lemma 4.3. For every tuple (η1, . . . , ηn) ∈ Dn, the value of Fǫ(η1, . . . , ηn) can be computed in time polynomial in
+the instance size |I| and 1
+ǫ.
+Proof. We show that Fǫ(η1, . . . , ηn) can be solved in polynomial time by means of the ellipsoid algorithm. Let us
+notice that Constraints (2) can be equivalently encoded by a set of linear inequalities, one for each player i ∈ N
+and deviation φi ∈ vert(Φi), where vert(Φi) denotes the set of vertexes of the polytope Φi (recall Assumption 1).
+Thus, solving Fǫ(η1, . . . , ηn) is equivalent to solving an LP with a (possibly) exponential number of constraints, but
+polynomially-many variables. Such an LP can be solved in polynomial time by means of the ellipsoid algorithm,
+provided that a polynomial-time separation oracle for the linearized version of Constraints (2) is available. Such an
+oracle can be implemented by solving the maximization in the right-hand side of Constraints (2) for a correlated
+strategy z ∈ ∆A given as input. Formally, the separation oracle solves the following problem for each player i ∈ N:
+φ⋆
+i ∈ arg max
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+,
+6
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+which can be done efficiently thanks to Assumption 1. Then, if the separation oracle finds any φ⋆
+i such that:
+ui(z) ≥ ui(φ⋆
+i ⋄ z) − η⊤
+i ci(φ⋆
+i ⋄ z),
+it outputs the above inequality as a separating hyperplane to be used in the ellipsoid algorithm.
+Lemma 4.3 is not enough to complete our algorithm, since we need an efficient way of optimizing Fǫ(η1, . . . , ηn)
+over all the tuples of Lagrange multipliers. This problem is non-trivial, since Fǫ(η1, . . . , ηn) is non-concave in ηi.
+Nevertheless, we show that, by restricting the domain D of the Lagrange multipliers to a suitably-defined finite “small”
+subset, we can still find a constrained ǫ-Phi-equilibrium whose value of ℓ is at least as large as that of any constrained
+(exact) Phi-equilibrium. This is enough to solve APXCPE(1, ǫ). In particular, we need a finite subset of “good”
+Lagrange multipliers, in the sense of the following definition.
+Definition 4.1. Given any δ > 0, a set ˜D ⊆ D is δ-optimal if, for every z ∈ ∆A and i ∈ N, the following holds:
+min
+ηi∈ ˜
+D
+max
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+≤
+max
+φi∈ΦS
+i (z) ui(φi ⋄ z) + δ.
+Intuitively, thanks to Lemma 4.2, if we optimize the Lagrange multipliers over a δ-optimal set ˜D ⊆ D, instead of
+optimizing them over D, then we are allowing the players to violate incentive constraints by at most δ.
+In the following, we assume that a finite δ-optimal set ˜D ⊆ D is available. In Section 4.2, se show how to design two
+particular δ-optimal sets that allow to prove our main results. For ease of presentation, we let
+L ˜
+D,ǫ :=
+max
+(η1,...,ηn)∈ ˜
+Dn Fǫ(η1, . . . , ηn)
+be the optimal value of Fǫ(η1, . . . , ηn) when the Lagrange multipliers are constrained to be in a δ-optimal set ˜D ⊆ D.
+Next, we show that, given any δ-optimal set ˜D with δ ≤ ǫ, the value of L ˜
+D,ǫ is at least that achieved by constrained
+(exact) Phi-equilibria, namely LD,0. Formally:
+Lemma 4.4. Given any 0 < δ ≤ ǫ and a δ-optimal set ˜D ⊆ D, the following holds: L ˜
+D,ǫ ≥ LD,0.
+Intuitively, Lemma 4.4 is proved by noticing that, provided that δ ≤ ǫ, the incentive constraints violation introduced
+by using ˜D instead of D is at most ǫ. Moreover, the set of feasible correlated strategies can only expand by allowing
+incentive constraints to be violated, and, thus, the value of the objective ℓ can only increase.
+Lemma 4.4 suggests a way of solving APXCPE(1, ǫ). Indeed, given a finite δ-optimal set ˜D ⊆ D with δ ≤ ǫ, by
+enumerating over all the tuples of Lagrange multipliers ηi ∈ ˜D, one per player i ∈ N, we can find the desired
+constrained ǫ-Phi-equilibrium. The following theorem shows that this procedure gives an algorithm for APXCPE(1, ǫ)
+that runs in time polynomial in the instance size, | ˜D|, and 1
+ǫ.
+Theorem 4.1. Given a finite δ-optimal set ˜D ⊆ D with δ ≤ ǫ, there exists an algorithm that solves APXCPE(1, ǫ)
+and runs in time polynomial in the instance size |I|, the number | ˜D| of elements in ˜D, and 1
+ǫ for every ǫ > 0.
+Proof. The algorithm works by enumerating over all the possible tuples of Lagrange multipliers ηi ∈ ˜D, one per
+player i ∈ N. These are polynomially many in the size | ˜D| when the number of players n is fixed. For every tuple
+(η1, . . . , ηn) ∈ ˜Dn, the algorithm solves Fǫ(η1, . . . , ηn), which can be done in time polynomial in |I| and 1
+ǫ thanks
+to Lemma 4.3. Finally, the algorithm returns the correlated strategy z ∈ ∆A with the highest value of ℓ among those
+computed while solving Fǫ(η1, . . . , ηn). It is easy to see that the returned solution solves problem APXCPE(1, ǫ) by
+applying Lemma 4.4. This concludes the proof.
+4.2
+Instantiating the General Algorithm
+Next, we show how to build δ-optimal sets ˜D that, when they are plugged in the algorithm in Theorem 4.1, allow us
+to derive our results. In particular, we consider the set:
+Dτ :=
+�
+η ∈ D
+��� ηj = kτ, k ∈ {0, . . . , ⌊1/τρ⌋} ∀j ∈ [m]
+�
+,
+which is a discretization of D with a regular lattice of step τ ∈ R+ (notice that ηj is the j-th component of η).
+By a simple stars and bars combinatorial argument, we have that |Dτ| =
+�⌊1/τρ⌋+m
+m
+�
+. Thus, since it holds that
+7
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+|Dτ| = O((1/τρ)m), if the number of constraints m is fixed, |Dτ| is bounded by a polynomial in 1/τρ. Moreover,
+simple combinatorial arguments show that |Dτ| ≤ (1 + m)⌊1/τρ⌋.8 Thus, it also holds that |Dτ| = O(m
+1/τρ). Notice
+that the two bounds on |Dτ| are non-comparable, and, thus, they give rise to two distinct results, as we show in the
+following.
+By using the first bound on |Dτ|, we can show that the set Dτ is δ-optimal for δ = mτ. Formally:
+Lemma 4.5. For any τ > 0, the set Dτ is (τm)-optimal.
+Thus, whenever the number m of cost constraints is fixed, Lemma 4.5, together with Theorem 4.1, allows us to provide
+a polynomial-time algorithm. Indeed, it is sufficient to apply Theorem 4.1 for the (τm)-optimal set Dτ with τ := ǫ/m
+to obtain the following first main result:
+Corollary 4.2. There exists an algorithm that solves problem APXCPE(1, ǫ) in time polynomial in |I| and 1
+ǫ for every
+ǫ > 0, when the number m of cost constraints is fixed.
+On the other hand, by using the second bound on |Dτ|, we can show that Dτ is δ-optimal for δ depending logarithmi-
+cally on the number of players’ actions. Formally:
+Lemma 4.6. For any τ > 0, the set Dτ is δ-optimal for δ = 2
+�
+2τ log s/ρ, where s is the number of players’ actions.
+Lemma 4.6 (together with Theorem 4.1) immediately gives us a quasi-polynomial-time for solving APXCPE(1, ǫ) for
+a given constant ǫ > 0. Moreover, its running time becomes polynomial when the number of players’ actions is fixed.
+Corollary 4.3. For any constant ǫ > 0, there exists an algorithm that solves APXCPE(1, ǫ) in time O(|I|log s).
+Moreover, when the number s of players’ actions is fixed, the algorithm runs in time polynomial in |I|.
+Notice that it is in general not possible to design an algorithm that runs in time polynomial in 1
+ǫ, since this would
+contradict the hardness result in Theorem 3.1.
+5
+A Special Case: Deviation-dependent Costs
+We complete our computational study of constrained Phi-equilibria by considering a special case in which player i’s
+costs associated to a deviation φi only depend on φi and not on the (overall) modified correlated strategy φi ⋄ z.
+We consider instances satisfying the following assumption.
+Assumption 3. For every player i ∈ N and player i’s deviation φi ∈ Φi, there exists a function ˜ci : Φi → [−1, 1]m
+such that ˜ci(φi) := ci(φi ⋄ z) for every z ∈ ∆A.
+Notice that, whenever Assumption 3 holds, the set ΦS
+i (z) of safe deviations does not depend on z. Thus, in the rest of
+this section, we write w.l.o.g. ΦS
+i rather than ΦS
+i (z).
+A positive effect of Assumption 3 is that it recovers the convexity of the set of constrained Phi-equilibria, rendering
+them more akin to unconstrained ones. Formally:
+Proposition 5.1. For instances I := (Γ, Φ) satisfying Assumption 3, the set of constrained ǫ-Phi-equilibria is convex.
+Proposition 5.1 suggests that constrained Phi-equilibria are much more computationally appealing under Assumption 3
+than in general, as we indeed show in the rest of this section.
+First, in Section 5.1, we show that APXCPE(1, 0) admits a polynomial-time algorithm under Assumption 3. Then, in
+Section 5.2, we design a no-regret learning algorithm that efficiently computes one constrained ǫ-Phi equilibrium with
+ǫ = O(1/
+√
+T) as the number of rounds T grows. Finally, in Section 5.3, we provide a natural example of constrained
+Phi-equilibria satisfying Assumption 3.
+5.1
+A Poly-time Algorithm for Optimal Equilibria
+We prove that, whenever Assumption 3 holds, the problem of computing an (exact) Phi-equilibrium maximizing a
+given linear function can be solved in polynomial time. This is done by formulating the problem as an LP with
+polynomially-many variables and exponentially-many constraints, which can be solved by means of the ellipsoid
+method, similarly to how we compute Fǫ(η1, . . . ηn) in Section 4 (see the proof of Lemma 4.3). Formally:
+Theorem 5.1. Restricted to instances I := (Γ, Φ) which satisfy Assumption 3, APXCPE(1, 0) admits a polynomial-
+time algorithm.
+8See Appendix D for a formal proof.
+8
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+5.2
+An Efficient No-regret Learning Algorithm
+Next, we show how Assumption 3 allows us to find a constrained ǫ-Phi-equilibrium by means of a polynomial-time
+decentralized no-regret learning algorithm. Our algorithm is based on the Phi-regret minimization framework intro-
+duced by Greenwald and Jafari (2003), which needs to be extended in order to be able to work with polytopal sets ΦS
+i
+of safe deviations, rather than finite sets of “pure” deviations.
+Algorithm 1 Learning a Constrained ǫ-Phi-equilibria
+Require: Regret minimizers Ri for the sets ΦS
+i , for i ∈ N
+1: Initialize the regret minimizers Ri
+2: for t = 1, . . . , T do
+3:
+for each player i ∈ N do
+4:
+φi,t ← Ri.RECOMMEND()
+5:
+Play according to a distribution xi,t ∈ ∆Ai s.t.
+xi,t[a] =
+�
+b∈Ai
+φi,t[b, a]xi,t[b]
+∀a ∈ Ai
+6:
+end for
+7:
+zt ← ⊗i∈N xi,t
+8:
+Ri.OBSERVE(φi �→ ui(φi ⋄ zt))
+9: end for
+10: return ¯zT := 1
+T
+�T
+t=1 zt
+Algorithm 1 outlines our no-regret algorithm. It instantiates a regret minimizer Ri for the polytope ΦS
+i for each i ∈ N.
+Ri is an object that, at each round t ∈ [T ], recommends a safe deviation φi,t ∈ ΦS
+i to player i (Line 4 of Algorithm 1),
+and, then, observes a function φi �→ ui(φi ⋄ zt) that specifies the utility that would have been obtained by selecting
+any safe deviation φi ∈ ΦS
+i at round t (Line 8 of Algorithm 1). Ri guarantees that the regret RT
+i cumulated by player
+i over [T ] grows sublinearly, i.e., RT
+i = o(T ), where:
+RT
+i := max
+φi∈Φi
+T
+�
+t=1
+ui(φi ⋄ zt) −
+T
+�
+t=1
+ui(φi,t ⋄ zt),
+which is how much player i loses by selecting φi,t at each t rather than choosing the same best-in-hindsight deviation
+at all rounds. Notice that, by taking inspiration from the Phi-regret framework (Greenwald and Jafari, 2003), given
+a recommended deviation φi,t, player i actually plays according to a probability distribution xi,t ∈ ∆Ai, which
+is a stationary distribution of the matrix representing φi,t. This is crucial in order to implement the algorithm in a
+decentralized fashion and to provide convergence guarantees to constrained ǫ-Phi-equilibria (see Theorem 5.2). All the
+distributions xi,t jointly determine a correlated strategy zt ∈ ∆A at each round t ∈ [T ], defined as zt := ⊗i∈N xi,t,
+where ⊗ denotes the product among distributions; formally, zt[a] := �
+i∈N xi,t[ai] for all a ∈ A.
+Algorithm 1 provides the following guarantees:
+Theorem 5.2. Given an instance I := (Γ, Φ) satisfying Assumption 3, after T ∈ N>0 rounds, Algorithm 1 returns a
+correlated strategy ¯zT ∈ ∆A that is a constrained ǫT -Phi-equilibrium with ǫT = O(1/
+√
+T). Moreover, each round of
+Algorithm 1 runs in polynomial time.
+Let us remark that the crucial property which allows us to design Algorithm 1 is that the sets ΦS
+i of safe deviations do
+not depend on players other than i. Finally, from Theorem 5.2, the following result follows:
+Corollary 5.3. In instances I := (Γ, Φ) satisfying Assumption 3, a constrained ǫ-Phi-equilibrium can be computed
+in time polynomial in the instance size and 1
+ǫ by means of a decentralized learning algorithm.
+5.3
+Marginally-constrained CCE
+We conclude the section by introducing a particular (natural) notion of constrained ǫ-Phi-equilibrium for which As-
+sumption 3 is satisfied. This is a constrained version of the classical CCE in cost-constrained normal-form games
+where a player’s costs only depend on the action of that player. We call it marginally-constrained ǫ-CCE. Formally,
+such an equilibrium is defined for games in which, for every player i ∈ N, it holds ci(a) = ci(a′) for all a, a′ ∈ A
+such that ai = a′
+i, and for the set ΦCCE of CCE deviations that we have previously introduced in Section 2.4. Next,
+we prove that, with the definition above, Assumption 3 is satisfied.
+9
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Theorem 5.4. For instances I := (Γ, ΦCCE) such that ci(a) = ci(a′) for every player i ∈ N and action profiles
+a, a′ ∈ A : ai = a′
+i, Assumption 3 holds.
+Thanks to Theorem 5.4, we readily obtain the two following corollaries of Theorems 5.1 and 5.1.
+Corollary 5.5. The problem of computing a marginally-constrained (exact) CCE that maximizes a linear function
+ℓ : ∆A → R can be solved in polynomial time.
+Corollary 5.6. A marginally-constrained ǫ-CCE can be computed in time polynomial in the instance size and 1
+ǫ by
+means of a decentralized learning algorithm.
+6
+Discussion and Open Problems
+The main positive results that we provide in this paper (Corollaries 4.2 and 4.3) show that a constrained ǫ-Phi equilib-
+rium maximizing a given linear function can be computed in time polynomial in the instance size and 1
+ǫ, when either
+the number of constraints or that of players’ actions is fixed. Clearly, this implies that, under the same assumptions,
+a constrained ǫ-Phi-equilibrium can be found efficiently. Moreover, in Section 5, we designed an efficient no-regret
+learning algorithm that finds a constrained ǫ-Phi-equilibrium in settings enjoying special properties (Corollary 5.3).
+However, the problem of efficiently computing a constrained ǫ-Phi-equilibrium remains open in general. Formally:
+Definition 6.1 (Open Problem). Given any instance I := (Γ, Φ), find a constrained ǫ-Phi-equilibrium in time polyno-
+mial in the instance size and 1
+ǫ.
+Solving the problem above is non-trivial. Proposition 3.1 in Section 3 proves that the set of constrained ǫ-Phi-equilibria
+is non-convex, and, thus, solving the problem in Definition 6.1 is out of scope for most of the known equilibrium
+computation techniques. On the other hand, it is unlikely that such a problem is NP-hard. Indeed, a constrained
+ǫ-Phi-equilibrium always exists and, given any z ∈ ∆A, it is possible to verify whether z is an equilibrium or not
+in polynomial time. Formally, such a problem is said to belong to the TFNP complexity class, and, thus, standard
+arguments show that, if the problem is NP-hard, then NP = coNP (Megiddo and Papadimitriou, 1991). Thus, one
+should try to reduce the problem in Definition 6.1 to problems in TFNP, such as that of computing a Nash equilibrium.
+However, while the problem in Definition 6.1 shares some properties with that of computing a Nash equilibrium, such
+as the non-convexity of the set of the equilibria, the former is fundamentally different from the latter, since it exhibits
+correlation among the players. Thus, a reduction from such a problem to that of computing Nash equilibria would
+require a gadget to break the correlation among the players, and doing that is highly non-trivial as cost constraints are
+expressed by linear functions.
+10
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+References
+Eitan Altman and Adam Shwartz. 2000. Constrained markov games: Nash equilibria. In Advances in dynamic games
+and applications. Springer, 213–221.
+Jorge Alvarez-Mena and On´esimo Hern´andez-Lerma. 2006. Existence of Nash equilibria for constrained stochastic
+games. Mathematical Methods of Operations Research 63, 2 (2006), 261–285.
+Kenneth J Arrow and Gerard Debreu. 1954. Existence of an equilibrium for a competitive economy. Econometrica:
+Journal of the Econometric Society (1954), 265–290.
+Robert J Aumann. 1974. Subjectivity and correlation in randomized strategies. Journal of mathematical Economics 1,
+1 (1974), 67–96.
+A Bakhtin, N Brown, E Dinan, G Farina, C Flaherty, D Fried, A Goff, J Gray, H Hu, AP Jacob, et al. 2022. Human-
+level play in the game of Diplomacy by combining language models with strategic reasoning. Science (New York,
+NY) (2022), eade9097–eade9097.
+Noam Brown and Tuomas Sandholm. 2019. Superhuman AI for multiplayer poker. Science 365, 6456 (2019), 885–
+890.
+Luis Felipe Bueno, Gabriel Haeser, and Frank Navarro Rojas. 2019. Optimality conditions and constraint qualifications
+for generalized Nash equilibrium problems and their practical implications. SIAM Journal on Optimization 29, 1
+(2019), 31–54.
+Andrea Celli, Alberto Marchesi, Gabriele Farina, and Nicola Gatti. 2020. No-regret learning dynamics for extensive-
+form correlated equilibrium. Advances in Neural Information Processing Systems 33 (2020), 7722–7732.
+Ziyi Chen, Shaocong Ma, and Yi Zhou. 2022. Finding Correlated Equilibrium of Constrained Markov Game: A
+Primal-Dual Approach. In Advances in Neural Information Processing Systems.
+Constantinos Daskalakis, Paul W Goldberg, and Christos H Papadimitriou. 2009. The complexity of computing a
+Nash equilibrium. SIAM J. Comput. 39, 1 (2009), 195–259.
+Ivar Ekeland and Roger Temam. 1999. Convex analysis and variational problems. SIAM.
+Francisco Facchinei and Christian Kanzow. 2010. Generalized Nash equilibrium problems. Annals of Operations
+Research 175, 1 (2010), 177–211.
+Denizalp Goktas and Amy Greenwald. 2022. Exploitability Minimization in Games and Beyond. Advances in Neural
+Information Processing Systems (2022).
+Amy Greenwald and Amir Jafari. 2003. A general class of no-regret learning algorithms and game-theoretic equilibria.
+In Learning theory and kernel machines. Springer, 2–12.
+Vesal Hakami and Mehdi Dehghan. 2015. Learning stationary correlated equilibria in constrained general-sum stochas-
+tic games. IEEE Transactions on Cybernetics 46, 7 (2015), 1640–1654.
+Johan H˚astad. 1999. Clique is hard to approximate within n1−ǫ. Acta Mathematica 182, 1 (1999), 105–142.
+Elad Hazan et al. 2016. Introduction to online convex optimization. Foundations and Trends® in Optimization 2, 3-4
+(2016), 157–325.
+Michael I Jordan, Tianyi Lin, and Manolis Zampetakis. 2022. First-Order Algorithms for Nonlinear Generalized Nash
+Equilibrium Problems. arXiv preprint arXiv:2204.03132 (2022).
+Christian Kanzow and Daniel Steck. 2016. Augmented Lagrangian methods for the solution of generalized Nash
+equilibrium problems. SIAM Journal on Optimization 26, 4 (2016), 2034–2058.
+Nimrod Megiddo and Christos H Papadimitriou. 1991. On total functions, existence theorems and computational
+complexity. Theoretical Computer Science 81, 2 (1991), 317–324.
+H. Moulin and J-P Vial. 1978a. Strategically zero-sum games: the class of games whose completely mixed equilibria
+cannot be improved upon. INT J GAME THEORY 7, 3 (1978), 201–221.
+Herv´e Moulin and J-P Vial. 1978b.
+Strategically zero-sum games: the class of games whose completely mixed
+equilibria cannot be improved upon. International Journal of Game Theory 7, 3 (1978), 201–221.
+John Nash. 1951. Non-cooperative games. Annals of mathematics (1951), 286–295.
+Christos H Papadimitriou and Tim Roughgarden. 2008. Computing correlated equilibria in multi-player games. Jour-
+nal of the ACM (JACM) 55, 3 (2008), 1–29.
+J Ben Rosen. 1965. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica:
+Journal of the Econometric Society (1965), 520–534.
+David Zuckerman. 2007. Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number.
+Theory of Computing 3, 6 (2007), 103–128.
+11
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+A
+On the Weaknesses of the Guarantees of the Algorithm of Chen et al. (2022)
+The Algorithm of Chen et al. (2022) finds a distribution µ over correlated strategies ∆A such that:
+Ez∼µ
+�
+max
+φi∈ΦS
+i (z) ui(φi ⋄ z) − ui(z)
+�
+≤ 0.
+(3)
+However, here we claim that this solution concept inherits some weaknesses from the non-convexity of the equilibria
+set that we proved in Theorem 5.1. Indeed, consider the same instance of Theorem 5.1 and consider the uniform
+distribution µ over {z1, z2}. In Theorem 5.1 we proved that maxφi∈ΦS
+i (z1) ui(φi ⋄z1)−ui(z1) ≤ 0 for all i ∈ {1, 2}
+and maxφi∈ΦS
+i (z2) ui(φi ⋄ z2) − ui(z2) ≤ 0 for all i ∈ {1, 2} and thus Equation (3) holds over the distribution µ.
+However we show that the expected correlated strategy z3 derived from distribution µ, i.e., z3 = Ez∼µ[z] = 1
+2z1 +
+1
+2z2, it is not a feasible equilibrium, or an approximate one.
+Indeed, in Theorem 5.1, we proved that maxφ2∈ΦS
+2 (z3) u2(φ2 ⋄z3)−u2(z3) ≥ 1
+3, showing that the average correlated
+strategies returned by their Algorithm is not an equilibrium nor close to it.
+This comes from the peculiar fact about Constrained Phi-equilibria that exhibit non-convex set of solutions, which is
+in striking contrast with the unconstrained case. Indeed the guarantees of Equilibria (3) would imply that Ez∼µ[z] is
+a equilibrium in the unconstrained case in which the set of equilibria is convex.
+B
+Proofs Omitted from Section 2
+Theorem 2.1. Given a cost-constrained normal-form game Γ and a set Φ of deviations, if Assumption 2 is satisfied,
+then Γ admits a constrained Phi-equilibrium.
+Proof. With assumption 2 Altman and Shwartz (2000, Theorem 2.1) proves the existence of a constrained Nash equi-
+librium. In our setting this is equivalent to a product distribution z = ⊗i∈[N]xi so that it is a Constrained Phi-
+equilibrium for any set of deviations Φi.9 This is easily seen by observing that a Constrained Nash Equilibria is
+defined as:
+�
+a∈A
+ui
+
+ �
+j∈[N]
+xj(aj)
+
+ ≥
+�
+a∈A
+ui
+
+˜xi(aj)
+�
+j∈[N]\{i}
+xi(ai)
+
+
+for all ˜xi ∈ ∆(Ai) s.t. xi ⊗ x−i ∈ S.
+On the other hand it easily seen that for all φi ∈ Φi(z) there exists some ˜xi ∈ ∆(Ai) such that
+φi ⋄
+�
+⊗j∈[N]xj
+�
+= ˜xi ⊗ x−i
+and ˜xi ⊗ x−i ∈ S.
+This is proved by the following calculations:
+φi ⋄
+�
+⊗j∈[N]xj
+�
+[ai, a−i] :=
+�
+b∈Ai
+φi[b, ai]xi(b)x−i(a−i)
+(4)
+= ˜xi(ai) ⊗ x−i(a−i),
+(5)
+where ˜xi(ai) := �
+b∈Ai φi[b, ai]xi(b) and ˜xi ∈ ∆(Ai) since, by definition, �
+ai∈Ai φi[b, ai] = 1 for all b ∈ Ai.
+This proves that a Constrained Nash Equilibrium is a Phi-Constrained Equilibrium for all Φ.
+C
+Proofs Omitted from Section 3
+Theorem 3.1 (Hardness). For any constant α > 0, the problem APXCPE(α, (α/s)2) is NP-hard, where s is the
+number of players’ actions in the instance given as input.
+9As common in the normal form game literature, for any distribution x ∈ ∆(X) and y ∈ ∆(Y ), x ⊗ y ∈ ∆(X × Y ) is the
+product distribution defined as (x ⊗ y)[a, b] = x[a]y[b] for a ∈ X and b ∈ Y .
+12
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Proof. We reduce from GAP-INDEPENDENT-SET, which is a promise problem that formally reads as follows:
+given an δ > 0 and a graph G = (V, E), with set of nodes V and set of edges E, determine whether G admits
+an independent set of size at least |V |1−δ or all the independent sets of G have size smaller than |V |δ.
+GAP-
+INDEPENDENT-SET is NP-hard for every δ > 0 (H˚astad, 1999; Zuckerman, 2007).
+Let ℓ = |V | and α > 0 be the desired approximation factor. Given an instance of GAP-INDEPENDENT-SET,
+we build an instance such that if there exists an independent set of size ℓ1−δ, then there exists a Constrained Phi-
+equilibrium with social welfare 1. Otherwise, if all the independent sets have size at most ℓδ, all the Constrained
+ǫ-Phi-equilibria have social welfare at most α/2. We can use any δ > 0, since we simply need ℓδ < ℓ1−δ. Moreover,
+we take ǫ =
+α2
+128ℓ2 . As we will see, ℓ will be smaller than the number of action of the players, satisfying the condition
+in the statement.
+Construction.
+The first player has a set of actions A1 that includes actions a0, a1, a2 and an action av for each
+v ∈ V . Moreover, the first player has an action aF .10 The second player has a set of actions A2 that includes actions
+av and ¯av for each v ∈ V . Moreover, the second player has an action aF . Let γ = η = α/8. The utility of the first
+agent is as follows:
+• u1(a0, a) = γ + 1
+2η for all a ∈ A2 \ {aF},
+• u1(a1, av) = γ + η and u1(a2, av) = γ for all v ∈ V .
+• u1(a1, ¯av) = γ and u1(a2, ¯av) = γ + η for all v ∈ V .
+• u1(av, av) = u1(av, ¯av) = γ for all v ∈ V
+• u1(av, av′) = γ and u1(av, ¯av′) = γ +
+ℓ−ℓ1−δ
+ℓ−ℓ1−δ−1η for all v′ ̸= v.
+• u1(aF , a) = 0 for each a ∈ A2.
+• u1(a, aF ) = 0 for each a ∈ A1.
+The utility of the second agent is u2(a0, a) = 1 for each a ∈ A2 \ {aF} and 0 otherwise.
+There is a cost function cv for each v ∈ V , which is common to both the agents. For each v ∈ V , the cost function cv
+is such that
+• cv(av, av′) = −1 for each v′ ̸= v, (v, v′) ∈ E,
+• cv(av, av′) = 0 for each v′ ̸= v, (v, v′) /∈ E,
+• cv(av, av) = 1 for each v ∈ V .
+• cv(aF , a) = − 1
+4ℓ2 for each a ∈ A2.
+• cv(a, aF ) = − 1
+4ℓ2 for each a ∈ A1.
+• For every other action profile the cost is 0.
+We dropped the player index from the cost functions c as they are equal to both players.
+Moreover, we set of deviations Φi = Φi,ALL for both players i ∈ {1, 2}.
+Notice that the instance satisfies Assumption 2. Indeed, the deviation φi such that φi[a, aF ] = 1 for all a ∈ Ai for
+i ∈ {1, 2}, that deviates deterministically to aF is always strictly feasible for both player 1 and player 2. Moreover, its
+cost is polynomial in the instance size.
+Completeness.
+We show that if there exists an independent set of size ℓ1−δ, then the social welfare of an optimal
+Constrained Phi-equilibria is at least 1. Let V ∗ be an independent set of size ℓ1−δ. We build a Constrained Phi-
+equilibria z with social welfare at least 1. Consider the correlated strategy such that z[a0, av] =
+1
+2ℓ1−δ for all v ∈ V ∗,
+while z[a0, ¯av] =
+1
+2(ℓ−ℓ1−δ) for all v /∈ V ∗. All the other action profiles have probability 0.
+10This action is needed only to satisfy the strictly feasibility assumption.
+13
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+It is easy to see that the correlated strategy has social welfare at least 1 since player 1 always plays action a0 and
+u2(a0, a) = 1 for all a ∈ A2. Moreover, it is easy to verify that it is safe since cv(a0, a) ≤ 0 for each a ∈ A2. Hence,
+to show that z is an Constrained Phi-equilibria we only need to prove that it satisfies the incentive constraints. The
+incentive constraints of the second player are satisfied since they obtain the maximum possible utility, i.e., 1.
+Consider now a possible deviation of the first player φ1 ∈ Φ1. As a first step, we compute the expected utility of a
+strategy φ1. Let us define the following quantities:
+• T 1 = �
+v∈V ∗ φ1[a0, av]
+��
+z[a0, av] + z[a0, ¯av] + �
+v′̸=v z[a0, av′]
+�
+γ +
+�
+γ +
+ℓ−ℓ1−δ
+ℓ−ℓ1−δ−1η
+� �
+v′̸=v z[a0, ¯av]
+�
+• T 2 = �
+v /∈V ∗ φ1[a0, av]
+��
+z[a0, av] + z[a0, ¯av] + �
+v′̸=v z[a0, av′]
+�
+γ +
+�
+γ +
+ℓ−ℓ1−δ
+ℓ−ℓ1−δ−1η
+� �
+v′̸=v z[a0, ¯av]
+�
+• T 3 =
+�
+γ + η
+2
+�
+φ1[a0, a0] + γ+η
+2 (φ1[a0, a1] + φ1[a0, a2]) + γ
+2(φ1[a0, a1] + φ1[a0, a2])
+We bound each component individually.
+T 1 =
+�
+v∈V ∗
+φ1[a0, av]
+
+
+
+z[a0, av] + z[a0, ¯av] +
+�
+v′̸=v
+z[a0, av′]
+
+ γ +
+�
+γ + η
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1
+� �
+v′̸=v
+z[a0, ¯av]
+
+
+=
+�
+v∈V ∗
+φ1[a0, av]
+�1
+2γ + 1
+2
+�
+γ + η
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1
+��
+=
+�
+v∈V ∗
+φ1[a0, av]
+�
+γ + η
+2
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1
+�
+≤
+�
+v∈V ∗
+φ1[a0, av](γ + η),
+where in the last inequality we use
+ℓ−ℓ1−δ
+ℓ−ℓ1−δ−1 ≤ 2 for ℓ large enough. while
+T 2 =
+�
+v /∈V ∗
+φ1[a0, av]
+
+
+
+z[a0, av] + z[a0, ¯av] +
+�
+v′̸=v
+z[a0, av′]
+
+ γ +
+�
+γ + η
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1
+� �
+v′̸=v
+z[a0, ¯av]
+
+
+=
+�
+v /∈V ∗
+φ1[a0, av]
+��1
+2 +
+1
+2(ℓ − ℓ1−δ)
+�
+γ +
+�1
+2 −
+1
+2(ℓ − ℓ1−δ)
+� �
+γ + η
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1
+��
+=
+�
+v /∈V ∗
+φ1[a0, av]
+�
+γ + η
+2
+�
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1 −
+1
+ℓ − ℓ1−δ − 1
+��
+=
+�
+v /∈V ∗
+φ1[a0, av]
+�
+γ + η
+2
+�
+.
+Finally,
+T 3 = [a0, a0]
+�
+γ + η
+2
+�
++ γ + η
+2
+([a0, a1] + [a0, a2]) + γ
+2 ([a0, a1] + [a0, a2])
+=
+�
+γ + η
+2
+�
+([a0, a0] + φ1[a0, a1] + φ1[a0, a2])
+Finally, the utility of a deviation φ1 is
+�
+a1∈A1,a2∈A2
+�
+a∈A1
+φ1[a1, a]z[a1, a2]u1(a, a2)
+=
+�
+a∈A1,a2∈A2
+φ1[a0, a]z[a0, a2]u1(a, a2)
+= T 1 + T 2 + T 3
+≤ (γ + η)
+�
+v∈V ∗
+φ1[a0, av] +
+�
+γ + η
+2
+� �
+v /∈V ∗
+φ1[a0, av] +
+�
+γ + η
+2
+�
+(φ[a0, a0] + φ1[a0, a1] + φ1[a0, a2])
+14
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+= η
+2
+�
+v∈V ∗
+φ1[a0, av] +
+�
+γ + η
+2
+�
+(1 − φ1[a0, aF])
+Now, we show that no deviation φ1 ∈ Φ1 is both safe and increases player 1 utility. In particular, we show that if a
+strategy φ1 increases the utility than it is not safe. Indeed, if φ1 increases the utility, then
+�
+a1∈A1,
+a2∈A2
+�
+a∈A1
+φ1[a1, a]z[a1, a2]u1(a, a2) > γ + η
+2
+This implies that
+η
+2
+�
+v∈V ∗
+φ1[a0, av] +
+�
+γ + η
+2
+�
+(1 − φ1[a0, aF ]) > γ + η
+2
+and
+�
+v∈V ∗
+φ1[a0, av] > 1
+2φ1[a0, aF ]
+(6)
+Next, we show that any φ1 that increases the utility (and hence that satisfies Eq (6)) is not a feasible deviation. First,
+notice that equation (6) implies that there is a ¯v ∈ V ∗ such that
+φ1[a0, a¯v] > 1
+2ℓφ1[a0, aF ].
+(7)
+Then, we show that the deviation φ1 violates the constraint c¯v. In particular,
+�
+a1∈A1,a2∈A2
+�
+a∈A1
+φ1[a1, a]z[a1, a2]cv(a, a2) = φ1[a0, a¯v]z[a0, a¯v]1 − 1
+4ℓ2 φ1[a0, aF ] −
+�
+v∈V ∗:(v,¯v)∈E
+φ1[a0, a¯v]z[a0, av]1
+= φ1[a0, a¯v]z[a0, a¯v] − 1
+4ℓ2 φ1[a0, aF ]
+=
+1
+2ℓ1− 1
+ℓ φ1[a0, a¯v] − 1
+4ℓ2 φ1[a0, aF ]
+> φ1[a0, a¯v]
+�
+1
+2ℓ1− 1
+ℓ − 1
+2ℓ
+�
+≥ 0,
+where the second inequality holds since V ∗ is an independent set, and the second-to-last inequality by Equation (7).
+Hence, there is no deviation φ1 that increases players 1 utility and that is safe. This concludes the first part of the
+proof.
+Soundness. We show that if there exists a Constrained w-Phi-equilibria with social welfare α/2, then there exists an
+independent set of size strictly larger than ℓδ, reaching a contradiction. Suppose by contradiction that there exists a
+Constrained ǫ-Phi-equilibrium z with social welfare strictly greater than α/2. Thus,
+�
+a′∈A2\{aF }
+z[a0, a′] · 1 +
+�
+a∈A1,a′∈A2
+(γ + η) ≥
+�
+a∈A1,a′∈A2
+z[a, a′](u1(a, a′) + u2(a, a′)) ≥ α/2,
+where the first inequality comes from u2(a0, a′) = 1 for each a′ ∈ A2 \ {aF } and 0 otherwise, and u1(a, a′) ≤ γ + η
+for each a ∈ A1 and a′ ∈ A2. This implies
+�
+a′∈A2
+z[a0, a′] ≥ α/4.
+(8)
+Then, we show that z assigns similar probabilities on the set of action profiles {a0, av}v∈V and {a0, ¯av}v∈V Given
+an a ∈ A1, let φa ∈ Φ1 be a deviation of the first player such that φa[a0, a] = 1 and φa[a′, a′] = 1 for each a′ ̸= a0.
+Since z is an Constrained ǫ-Phi-equilibrium there is no feasible deviation φa that increases the utility of player 1 by
+more than ǫ. This implies that
+�����
+�
+v∈V
+z[a0, av] −
+�
+v∈V
+z[a0, ¯av]
+����� ≤ 2ǫ
+η .
+(9)
+15
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Indeed, if
+�
+v∈V
+z[a0, av] >
+�
+v∈V
+z[a0, ¯av] + 2ǫ
+η ,
+(10)
+then the deviation φa1 has utility at least
+�
+v∈V
+z[a0, av]φa1[a0, a1](γ + η) + z[a0, ¯av]φa1[a0, a1]γ +
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]φa1[a, a]ui(a, a′)
+= η
+�
+v∈V
+z[a0, av] + γ
+�
+v∈V
+(z[a0, av] + z[a0, ¯av]) +
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]φa1[a, a]ui(a, a′)
+> η
+2
+�
+2ǫ
+η +
+�
+v∈V
+(z[a0, av] + z[a0, ¯av])
+�
++ γ
+�
+v∈V
+(z[a0, av] + z[a0, ¯av])
++
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]φa1[a, a]ui(a, a′)
+≥ ǫ +
+�η
+2 + γ
+� �
+v∈V
+(z[a0, av] + z[a0, ¯av]) +
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]φa1[a, a]ui(a, a′)
+≥ u1(z) + ǫ,
+where the first inequality comes from adding �
+v∈V z[a0, av] to both sides of Equation (10). Moreover, φa1 is feasible
+since for each constraint c¯v, ¯v ∈ V , it has cost
+�
+v∈V
+(z[a0, av]φa1[a0, a1]c¯v(a1, av) + z[a0, ¯av]φa1[a0, a1]c¯v(a1, av))
++
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]φa1[a, a]c¯v(a, a′)
+=
+�
+v∈V
+(z[a0, av]φa1[a0, a1]c¯v(a0, av) + z[a0, ¯av]φa1[a0, a1]c¯v(a0, ¯av))
++
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]φa1[a, a]c¯v(a, a′)
+= c¯v(z) ≤ 0.
+A similar argument shows that if �
+v∈V z[a0, av] < �
+v∈V z[a0, ¯av] − 2ǫ
+η then the deviation φa2 is safe and increases
+the utility. As a consequence of Equation (9), it holds
+2
+�
+v∈V
+z[a0, ¯av] ≥
+�
+v∈V
+(z[a0, av] + z[a0, ¯av]) − ǫ
+η =
+�
+a∈A2\{aF }
+z[a0, a] − 2ǫ
+η ,
+(11)
+where the first inequality comes from adding �
+v∈V z[a0, ¯av] to both sides of �
+v∈V z[a0, ¯av]| ≥ �
+v∈V z[a0, av]− 2ǫ
+η
+The next step is to show that it is if there is no safe deviation φav, v ∈ V , that increases the utility, then there exists
+an independent set of size larger than ℓδ. Since z is an Constrained ǫ-Phi-equilibrium, for each av, v ∈ V one of the
+following two conditions holds: i) φav /∈ ΦS
+1 (z) or ii) u1(φav ⋄ z) ≤ u1(z) + ǫ. Let V 1 ⊆ V be the set of vertexes
+v such that φav is not safe, i.e., φav /∈ ΦS
+1 (z), and V 2 = V \ V 1 be the set of v such that φav does not increase the
+utility by more than ǫ and are not in V 1, i.e., u1(φav ⋄ z) ≤ u1(z) and φav ∈ ΦS
+1 (z). We show that |V 2| ≤ ℓ − ℓ1−δ.
+Indeed, for each v ∈ V 2, deviation φav does not increase the utility and hence it holds:
+γ
+�
+a∈A2\{aF }
+z[a0, a] + η
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1
+�
+v′̸=v
+z[a0, ¯av′] +
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]φa1[a, a]ui(a, a′)
+=
+� �
+v′∈V
+z[a0, a] + z[a0, ¯av]
+�
+φ[a0, av]γ +
+�
+v′̸=v
+φ[a0, av]z[a0, ¯av′]
+�
+γ + η
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1
+�
++
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]φa1[a, a]ui(a, a′)
+16
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+≤ u1(z) + ǫ
+=
+�
+γ + η
+2
+�
+�
+a∈A2\{aF }
+z[a0, a] +
+�
+a∈A1\{a0}
+�
+a′∈A2
+z[a, a′]ui(a, a′) + ǫ,
+where the inequality holds since the lhs is the utility of the deviation φav.
+This implies
+��
+v′
+z[a0, ¯av′] − z[a0, ¯av]
+�
+η
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1 ≤ η
+2
+�
+a∈A2\{aF }
+z[a0, a] + ǫ ≤ η
+�
+v∈V
+z[a0, ¯av] + 2ǫ,
+where the last inequality holds by Equation (11). Hence,
+z[a0, ¯av]
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1 ≥
+�
+ℓ − ℓ1−δ
+ℓ − ℓ1−δ − 1 − 1
+� �
+v′
+z[a0, ¯av′] − 2ǫ/η,
+and
+¯z[a0, av] ≥
+1
+ℓ − ℓ1−δ
+�
+v′
+z[a0, ¯av′] − 2ǫ/η.
+(12)
+Suppose that |V 2| > ℓ − ℓ1−δ, and hence Equation (12) is satisfied by at least |V 2| ≥ ℓ − ℓ1−δ + 1 vertexes. We need
+the following inequality.
+1
+ℓ
+�
+v′
+z[a0, ¯av′] ≥ 1
+ℓ
+�
+a∈A2\{aF }
+z[a0, a] − 2ǫ
+ℓη ≥ α
+4ℓ − 2ǫ
+ℓη = α
+4ℓ − α
+8ℓ3 ≥ α
+8ℓ = 2ℓ
+η
+� α2
+16ℓ2
+�
+= 2ℓ
+η ǫ
+(13)
+where the first inequality comes from Equation (11), and the second one by Equation (8). Then, summing over the
+|V 2| inequalities we get
+�
+v∈V 2
+¯z[a0, av] ≥ (ℓ − ℓ1−δ + 1)
+�
+1
+ℓ − ℓ1−δ
+�
+v′
+z[a0, ¯av′] − 2ǫ/η
+�
+≥
+�
+v′
+z[a0, ¯av′] + 1
+ℓ
+�
+v′
+z[a0, ¯av′] − 2ℓ ǫ
+η
+>
+�
+v′
+z[a0, ¯av′],
+where the last inequality follows from equation (13). Hence, we reach a contradiction and |V 2| ≤ ℓ − ℓ1−δ.
+To conclude the proof, we show that V 1 is an independent set. Since |V 1| ≥ |V | − |V 2| = ℓ1−δ we reach a
+contradiction. Let v and v′ be two nodes in V 1 and w.l.o.g. let z[a0, av] ≥ z[a0, av′]. We show that (v, v′) /∈ E.
+Since v′ ∈ V 1, φav is not a safe deviation for player 1 with respect to constraint cv′. if (v, v′) ∈ E, then
+�
+a1∈A1,a2∈A2
+�
+a∈ A1
+φ[a1, a]z[a1, a2]cv(a, a2)
+= z[a0, a′
+v] −
+�
+v′′:(v′′,v′)∈E
+z[a0, av′′] − 1
+4ℓz[a0, aF]cv(a, a2)
++
+�
+a1∈A1\{a0},a2∈A2
+�
+a∈A1
+φ[a1, a]z[a1, a2]cv(a, a2)
+≤ z[a0, a′
+v] − z[a0, av] − 1
+4ℓz[a0, aF ]cv(a, a2)+
++
+�
+a1∈A1\{a0},a2∈A2
+�
+a∈A1
+φ[a1, a]z[a1, a2]cv(a, a2)
+≤ − 1
+4ℓz[a0, aF ]cv(a, a2) +
+�
+a1∈A1\{a0},a2∈A2
+�
+a∈A1
+φ[a1, a]z[a1, a2]cv(a, a2)
+17
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+= cv(z) ≤ 0.
+Hence, (v, v′) /∈ E. Since V 1 is an independent set of size at least ℓ1−δ we reach a contradiction. This concludes the
+proof.
+D
+Proofs Omitted from Section 4
+Lemma 4.1. For every z ∈ ∆A and i ∈ N, it holds
+max
+φi∈ΦS
+i (z) ui(φi ⋄ z) = sup
+φi∈Φi
+inf
+ηi∈Rm
++
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+=
+inf
+ηi∈Rm
++
+sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+.
+Proof. First, it is easy to see that
+sup
+φi∈ΦS
+i (z)
+ui(φi ⋄ z) = sup
+φi∈Φi
+inf
+ηi∈Rm
++
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+.
+Indeed, for every φi /∈ ΦS
+i (z), it holds that the vector ci(φi ⋄ z) has at least one positive component, and, thus, the
+vector of Lagrange multipliers ηi can be selected so that ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z) goes to −∞. This implies that
+the supremum over Φi cannot be attained in ΦS
+i (z). On the other hand, for every φi ∈ ΦS
+i (z), all the components of
+ci(φi ⋄ z) are negative, and, thus, the inf is achieved by ηi = 0. This proves the first equality.
+Then, the second equality directly follows from the generalization of the max-min theorem for unbounded domains
+(see (Ekeland and Temam, 1999, Proposition 2.3)), which allows us to swap the sup and the inf.
+Lemma D.1. For any two real-valued functions f(x) and g(x) with g(x) ≤ c then min(f(x), g(x)) ≤ min(f(x), c).
+Proof. We can identify three sets I1, I2 and I3 defined as follows:
+I1 := {x s.t. f(x) ≥ c}
+I2 := {x s.t. g(x) ≤ f(x) ≤ c}
+I3 := {x s.t. f(x) ≤ g(x) ≤ c}.
+Then for all x ∈ I1 we have that min(f(x), c) = c ≥ min(f(x), g(x)) = g(x), while for all x ∈ I2 we have
+that min(f(x), c) = f(x) ≥ min(f(x), g(x)) = g(x). Finally for all x ∈ I3 we have min(f(x), c) = f(x) =
+min(f(x), g(x)) = f(x). In all three sets we have that min(f(x), c) ≥ min(f(x), g(x)).
+Lemma D.2. For all ηi ∈ Dc it holds that
+sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+≥ 1
+Proof. Thanks to Assumption 2 we have that for all z ∈ ∆(A) we have that there exists ˜φi ∈ ΦS
+i (z) such that
+ci(˜φi ⋄ z) ⪯ −ρ1. Then, for all ηi ∈ Dc we have:
+η⊤
+i ci(˜φi ⋄ z) ≤ −ρ∥ηi∥1 ≤ −1.
+This easily concludes the proof of the statement
+sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+≥ ui(˜φi ⋄ z) − η⊤
+i ci(˜φi ⋄ z) ≥ 1,
+as ui is positive.
+Lemma D.3. For all ηi ∈ D we have
+inf
+ηi∈D sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+≤ 1
+18
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Proof. Since ui ≤ 1 we have that
+inf
+ηi∈D sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+≤ 1 − sup
+ηi∈D
+inf
+φi∈Φi η⊤
+i ci(φi ⋄ z).
+Next we claim that sup
+ηi∈D
+inf
+φi∈Φi η⊤
+i ci(φi ⋄ z) ≥ 0. This follows from the fact that for all negative components of
+ci(φi ⋄ z) then the corresponding components of ηi will be 0. This concludes the statement.
+Lemma 4.2. Let D :=
+�
+η ∈ Rm
++ | ||η||1 ≤ 1/ρ
+�
+. Then, for every z ∈ ∆A and i ∈ N, it holds:
+max
+φi∈ΦS
+i (z) ui(φi ⋄ z) = max
+φi∈Φi min
+ηi∈D
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+= min
+ηi∈D max
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+.
+Proof. In Lemma 4.1 we already showed that:
+sup
+φi∈ΦS
+i (z)
+ui(φi ⋄ z) = sup
+φi∈Φi
+inf
+ηi∈Rm
++
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+=
+inf
+ηi∈Rm
++
+sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+.
+Note that to prove the statement it is enough to prove that:
+inf
+ηi∈Rm
++
+sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+= inf
+ηi∈D sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+and more specifically that:
+inf
+ηi∈Rm
++
+sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+≥ inf
+ηi∈D sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+since the reverse inequality holds trivially. We can show this by the following inequalities:
+inf
+ηi∈Rm
++
+sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+= min
+�
+inf
+ηi∈D sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+, inf
+ηi∈Dc sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+�
+≥ min
+�
+inf
+ηi∈D sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+, 1
+�
+= inf
+ηi∈D sup
+φi∈Φi
+�
+ui(φi ⋄ z) − η⊤
+i ci(φi ⋄ z)
+�
+,
+where the first inequality hold thanks to Lemma D.1 and Lemma D.2, while that last equation follows from Lemma D.3.
+Lemma 4.4. Given any 0 < δ ≤ ǫ and a δ-optimal set ˜D ⊆ D, the following holds: L ˜
+D,ǫ ≥ LD,0.
+Proof. By definition we have that: L ¯
+D,ǫ = ℓ(˜z⋆), where ˜z⋆ is a solution to the problem
+P1 :=
+
+
+
+
+
+
+
+
+
+˜z⋆ ∈ arg max
+z∈S
+ℓ(z) s.t.
+ǫ + ui(˜z⋆) ≥ max
+φi∈Φi
+�
+ui(φi ⋄ ˜z⋆) − ˜η⋆,⊤
+i
+ci(φi ⋄ ˜z⋆)
+�
+˜η⋆
+i ∈ arg inf
+ηi∈ ¯
+D sup
+φi∈Φi
+�
+ui(φi ⋄ ˜z⋆) − η⊤
+i ci(φi ⋄ ˜z⋆)
+�
+On the other hand, call z⋆ the optimal Constrained Phi-equilibrium. This is a solution to the problem:
+P2 :=
+
+
+
+
+
+
+
+
+
+z⋆ ∈ arg max
+z∈S
+ℓ(z) s.t.
+ui(z⋆) ≥ max
+φi∈Φi
+�
+ui(φi ⋄ z⋆) − η⋆,⊤
+i
+ci(φi ⋄ z⋆)
+�
+η⋆
+i ∈ arg inf
+ηi∈D sup
+φi∈Φi
+�
+ui(φi ⋄ z⋆) − η⊤
+i ci(φi ⋄ z⋆)
+�
+19
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+which has value LD,0 = ℓ(z⋆).
+Moreover, thanks to Lemma 4.2 and since ¯D is δ-optimal we have that:
+max
+φi∈Φi
+�
+ui(φi ⋄ ˜z⋆) − ˜η⋆,⊤
+i
+ci(φi ⋄ ˜z⋆)
+�
+≤ max
+φi∈Φi
+�
+ui(φi ⋄ z⋆) − η⋆,⊤
+i
+ci(φi ⋄ z⋆)
+�
++ δ
+which implies that feasible correlated strategies of problem P2 are feasible correlated strategies of problem P1, and
+thus problem P1 as long as δ ≥ ǫ. Thus problem P1 is the problem of maximizing the same objective function over a
+larger set then problem P2 and thus L ¯
+D,ǫ ≥ LD,0.
+Lemma 4.5. For any τ > 0, the set Dτ is (τm)-optimal.
+Proof. By Lemma 4.2, we know that for each player there exists an η⋆
+i ∈ D such that maxφ∈ΦS
+i (z) ui(φi ⋄ z) =
+max
+φi∈Φi
+�
+ui(φi ⋄ z) − η⋆,⊤
+i
+ci(φi ⋄ z)
+�
+. By construction of Dǫ there exists a ¯ηi ∈ Dǫ such that ||¯ηi − η⋆
+i ||∞ ≤ ǫ. Thus
+max
+φ∈ΦS
+i (z) ui(φi ⋄ z) = max
+φi∈Φi
+�
+ui(φi ⋄ z) − η⋆,⊤
+i
+ci(φi ⋄ z)
+�
+≤ max
+φi∈Φi
+�
+ui(φi ⋄ z) − ¯η⊤
+i ci(φi ⋄ z)
+�
++ mǫ,
+where the last inequality comes the fact that:
+|(η⋆
+i − ¯ηi)⊤ci(φi ⋄ z)| ≤ ∥ci(φi ⋄ z)∥1∥η⋆
+i − ¯ηi∥∞ ≤ mǫ
+as ci ∈ [−1, 1]m.
+Lemma 4.6. For any τ > 0, the set Dτ is δ-optimal for δ = 2
+�
+2τ log s/ρ, where s is the number of players’ actions.
+Proof. The proof exploits a probability interpretation of the Lagrange multipliers. Let η⋆ be the optimal multipliers,
+i.e., , η⋆ ∈ argminη∈D maxφi∈Φi
+�
+ui(φi ⋄ z) − η⊤ci(φi ⋄ z)
+�
+. Now consider a basis Γ = { 1
+ρej}j∈[m] ∪ {0} for D.
+By Carathoedory’s theorem there exists a distribution γ ∈ ∆(Γ) such that η⋆ = �
+η∈Γ γηη. Assume that ǫ and ρ are
+such that 1/ǫρ is an integer and take 1/ρǫ samples from the distribution γ and call ˜η the resulting empirical mean.
+First, we argue that ˜η ∈ Dǫ. Indeed ˜ηj =
+kj
+1/ρǫ
+1
+ρ = ǫ
+�
+kj
+1/ρǫ
+1
+ρǫ
+�
+= ǫkj where kj ∈ N and thus we have that ˜η ∈ Dǫ.11
+Now we show that with high probability ˜η ∈ Dǫ is close (in terms of utilities) to the optimal multiplier η⋆. First
+observe that:
+η⋆,⊤
+i
+ci(φi ⋄ z) :=
+�
+ai∈Ai,bi∈Ai
+
+φi[b, ai]
+�
+a−i∈A−i
+η⋆,⊤ci(ai, a−i)z[b, a−i]
+
+
+(14a)
+≤
+�
+ai∈Ai,bi∈Ai
+
+φi[b, ai]
+
+δai,b +
+�
+a−i∈A−i
+˜η⊤ci(ai, a−i)z[b, a−i]
+
+
+
+
+(14b)
+=
+�
+ai∈Ai,bi∈Ai
+
+φi[b, ai]
+�
+a−i∈A−i
+˜η⊤ci(ai, a−i)z[b, a−i]
+
+ +
+�
+ai∈Ai,bi∈Ai
+φi[b, ai]δai,b
+(14c)
+= ˜η⊤
+i ci(φi ⋄ z) +
+�
+ai∈Ai,bi∈Ai
+φi[b, ai]δai,b
+(14d)
+where the inequality comes from applying the Hoeffeding’s inequality to every ai, b ∈ Ai:
+������
+�
+a−i∈A−i
+(˜η − η⋆)⊤ ci(ai, a−i)z[b, a−i]
+������
+≤ δai,b
+11If ǫ if not such that 1/ρǫ ∈ N then the one can take ⌈1/ρǫ⌉ samples from γ ∈ ∆(Γ) and then the statement hold for a slightly
+smaller ǫ′ < ǫ defined as ǫ′ :=
+1
+⌈1/ρǫ⌉
+1
+ρ.
+20
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+where
+δai,b
+=
+2
+ρ
+�
+2
+1/ρǫ log
+�
+2
+pai,b
+� �
+�
+a−i∈A−i z[b, a−i]
+�
+since
+the
+range
+of
+the
+each
+sample
+is
+1
+ρ
+��
+a−i∈A−i z[b, a−i]
+�
+.
+Moreover, for Hoeffeding’s inequality, for every ai, b ∈ Ai the above inequality holds with probability at least
+1 − pai,b and thus holds for all the ai, b ∈ Ai simultaneously, with probability at least p := �
+ai,b∈Ai pai,b.
+If then we take pai,b :=
+1
+2|Ai|2 for all ai, b ∈ Ai, then we have that p = 1/2 > 0 and δ := δai,b =
+2
+ρ
+�
+2
+1/ρǫ log (|Ai|)
+�
+�
+a−i∈A−i z[b, a−i]
+�
+Now the following holds with probability at lest 1/2:
+������
+�
+a−i∈A−i
+(˜η − η⋆)⊤ ci(ai, a−i)z[b, a−i]
+������
+≤ δ
+
+
+�
+a−i∈A−i
+z[b, a−i]
+
+ ,
+∀ai, b ∈ Ai
+The proof is concluded by observing plugging this definition of δ
+=
+δai,b in Equation (14) yields
+�
+ai∈Ai,bi∈Ai φi[b, ai]δai,b = δ, and we can conclude that:
+η⋆,⊤
+i
+ci(φi ⋄ z) ≤ ˜η⊤
+i ci(φi ⋄ z) + δ.
+This holds with positive probability, and thus shows the existence of such ˜η ∈ Dǫ for which the above inequality holds
+and thus Dǫ is
+�
+2
+�
+2ǫ
+ρ log(|Ai|)
+�
+-optimal.
+E
+Proofs Omitted from Section 5
+Proposition 5.1. For instances I := (Γ, Φ) satisfying Assumption 3, the set of constrained ǫ-Phi-equilibria is convex.
+Proof. Let z′ and z′′ be Constrained ǫ-Phi-equilibria that is for all i ∈ [N]:
+ǫ + ui(z′) ≥ ui(φ′
+i ⋄ z′)
+for φ′ ∈ arg max
+φi∈ΦS
+i
+ui(φi ⋄ z′). Equivalently it holds for all i ∈ [N] that:
+ǫ + ui(z′′) ≥ ui(φ′′
+i ⋄ z′′)
+where φ′′ ∈ arg max
+φi∈ΦS
+i
+ui(φi ⋄ z′′). For any z := αz′ + (1 − α)z′′ we have that:
+ǫ + ui(z) = α (ǫ + ui(z′)) + (1 − α) (ǫ + ui(z′′))
+≥ αui(φ′
+i ⋄ z′) + (1 − α)ui(φ′′
+i ⋄ z′′)
+≥ max
+φi∈ΦS
+i
+ui(φi ⋄ z),
+where the inequality holds for the linearity of ui, the first inequality as both z′ and z′′ are Constrained ǫ-Phi-equilibria
+and the last inequality holds since the max is a convex operator.
+Theorem 5.1. Restricted to instances I := (Γ, Φ) which satisfy Assumption 3, APXCPE(1, 0) admits a polynomial-
+time algorithm.
+Proof. APXCPE(1, 0) amounts to solving the following problem:
+max
+z∈S ℓ(z)
+s.t.
+(15a)
+ui(z) ≥ max
+φi∈ΦS
+i
+ui(φi ��� z)
+∀i ∈ N,
+(15b)
+which can be written as an LP with (possibly) exponentially-many constraints, by writing a constraint for each vertex
+of ΦS
+i . We can find an exact solution to such an LP in polynomial time by means of the ellipsoid algorithm that uses
+suitable separation oracle. Such an oracle solves the following optimization problem for every i ∈ N:
+φ⋆
+i ∈ arg max
+φi∈ΦS
+i
+ui(φi ⋄ z).
+21
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Then, the oracle returns as a separating hyperplane the incentive constraint corresponding to a φ⋆
+i (if any) such that
+ui(z) ≥ ui(φ⋆
+i ⋄ z). Since all the steps of the separation oracle can be implemented in polynomial time, the ellipsoid
+algorithm runs in polynomial time, concluding the proof.
+Theorem 5.2. Given an instance I := (Γ, Φ) satisfying Assumption 3, after T ∈ N>0 rounds, Algorithm 1 returns a
+correlated strategy ¯zT ∈ ∆A that is a constrained ǫT -Phi-equilibrium with ǫT = O(1/
+√
+T). Moreover, each round of
+Algorithm 1 runs in polynomial time.
+Proof. Any regret minimizer Ri for ΦS
+i guarantees that, for every φi ∈ ΦS
+i :
+T
+�
+t=1
+ui(φi ⋄ zt) −
+T
+�
+t=1
+ui(φi,t ⋄ zt) ≤ ǫi,T T,
+(16)
+where ǫi,T = o(T ). Since xi,t[a] = �
+b∈Ai φi,t[b, a]xi,t[b] for all a ∈ Ai, for every t ∈ [T ] and a = (ai, a−i) ∈ A:
+(φi,t ⋄ zt)[ai, a−i] =
+�
+b∈Ai
+φi,t[b, ai]z[b, a−i]
+=
+�
+b∈Ai
+φi,t[b, ai]
+�
+xi,t[b] ⊗ x−i,t[a−i]
+�
+=
+� �
+b∈Ai
+φi,t[b, ai]xi,t[b]
+�
+⊗ x−i,t[a−i]
+= xi,t[ai] ⊗ x−i,t[a−i]
+= zt[ai, a−i],
+Plugging the equation above into Equation (16), we get:
+T
+�
+t=1
+ui(φi ⋄ zt) −
+T
+�
+t=1
+ui(zt) ≤ ǫi,T T.
+Now, since ¯zT := �T
+t=1 zt and ui(z) is linear in z, we can conclude that, for every i ∈ N and φi ∈ ΦS
+i :
+ui(zT ) ≥ ui(φi ⋄ ¯zT ) − ǫi,T ,
+and, thus, by letting ǫT := maxi∈N ǫi,T we get that ¯zT satisfies the incentivize constrained for being a constrained
+ǫT -Phi-equilibrium. We are left to verify that ¯zT ∈ S, namely ci(¯zT ) ≤ 0 for all i ∈ N. This readily proved as:
+ci(¯zT ) = 1
+T
+T
+�
+t=1
+ci(zt)
+= 1
+T
+T
+�
+t=1
+ci(φi,t ⋄ zt)
+= 1
+T
+T
+�
+t=1
+˜ci(φi,t)
+≤ 0,
+where the first equality holds since ci is linear, the second equality holds thanks to zt = φi,t ⋄ zt, the third one by
+Assumption 3, while the inequality holds since φi,t ∈ ΦS
+i . This concludes the proof of the first part of the statement.
+In conclusion, Algorithm 1 runs in polynomial time as finding xi,t[a] = �
+b∈Ai φi,t[b, ai]xi,t[b] for all a ∈ Ai is
+equivalent to finding a stationary distribution of a Markov Chain, which can be done in polynomial time. Moreover,
+we can implement the regret minimizers Ri over the polytopes ΦS
+i so that their operations run in polynomial time,
+such as, e.g., online gradient descent; see (Hazan et al., 2016).
+Theorem 5.4. For instances I := (Γ, ΦCCE) such that ci(a) = ci(a′) for every player i ∈ N and action profiles
+a, a′ ∈ A : ai = a′
+i, Assumption 3 holds.
+22
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Proof. Since the costs ci(a) of player i ∈ N only depends on player i’s action ai and not on the actions of other
+players, it is possible to show that there exists ˜ci : ΦCCE → [−1, 1]m such that the following holds for every z ∈ ∆A:
+˜ci(φi) := ci(φi ⋄ z).
+Indeed, for every φi ∈ ΦCCE, by definition of ΦCCE there exists a probability distribution h ∈ ∆Ai : φi[b, a] = h[a]
+for all b, a ∈ Ai. Then, for every ai ∈ Ai and a−i ∈ A−i, we can write:
+(φi ⋄ z)[ai, a−i] =
+�
+b∈Ai
+φi[b, ai]z[b, a−i]
+=
+�
+b∈Ai
+h[ai]z[b, a−i]
+= h[ai]
+�
+b∈Ai
+z[b, a−i].
+Moreover, it holds:
+ci(φi ⋄ z)[ai, a−i] =
+�
+a∈A
+ci(a)(φi ⋄ z)[ai, a−i]
+=
+�
+a∈A
+ci(a)h[ai]
+�
+b∈Ai
+z[b, a−i]
+=
+�
+ai∈Ai
+ci(ai, ·)h[ai]
+�
+a−i∈A−i
+�
+b∈Ai
+z[b, a−i]
+=
+�
+ai∈Ai
+ci(ai, ·)h[ai],
+which only depends on φi, as desired. Notice that, in the equations above, for every a ∈ Ai we let ci(a, ·) be the
+(unique) value of ci(a) for all a ∈ A : ai = a.
+23
+

1tE2T4oBgHgl3EQfNQaE/content/tmp_files/2301.03735v1.pdf.txt

@@ -0,0 +1,2223 @@
+arXiv:2301.03735v1  [math.RA]  10 Jan 2023
+Multipliers and weak multipliers of algebras
+Yuji Kobayashi and Sin-Ei Takahasi
+Laboratory of Mathematics and Games
+(https://math-game-labo.com)
+2020 MSC Numbers: Primary 43A22; Secondary 17A99, 46J10.
+Keywords: (weak) multiplier, (non)associative algebra, Jordan algebra, zeropotent algebra,
+annihilator, nihil decomposition, matrix representation.
+Abstract
+We study general properties of multipliers and weak multipliers of
+algebras.
+We apply the results to determine the (weak) multipliers of
+associative algebras and zeropotent algebras of dimension 3 over an alge-
+braically closed field.
+1
+Introduction
+Multipliers of algebras, in particular, multipliers of Banach algebras, have been
+discussed in analysis. In this paper we will discuss them in a purely algebraic
+manner.
+Let B be a Banach algebra. A mapping T : B → B is called a multiplier of
+B, if it satisfies the condition (I) xT (y) = T (xy) = T (x)y (x, y ∈ B). Let M(B)
+denote the collection of all multipliers of B, and let B(B) be the collection of all
+bounded linear operators on B. Then M(B) forms an algebra and B(B) forms
+a Banach algebra. B is called without order if it has no nonzero left or right
+annihilator.
+If B is without order, then M(B) forms a commutative closed
+subalgebra of B(B) (see [2], Proposition 1.4.11). In 1953, Wendel [7] proved
+an important result that the multiplier algebra of L1(G) on a locally compact
+abelian group G is isometrically isomorphic to the measure algebra on G. The
+general theory of multipliers of Banach algebras has been developed by Johnson
+[1]. A good reference to the theory of multipliers of Banach algebra is given in
+Larsen [5].
+When B is without order, T is a multiplier if it satisfies the condition (II)
+xT (y) = T (x)y (x, y ∈ B). Many researchers had been unaware of difference
+between conditions (I) and (II) until Zivari-Kazempour [8] (see also [9]) recently
+clearly stated the difference. We call a mapping T satisfying (II) a weak mul-
+tiplier and denote the set of weak multipliers of B by M ′(B). Then, M(B)
+is in general a proper subset of M ′(B). Furthermore, (weak) multipliers can
+1
+
+be defined for an algebra A not necessarily associative, and they are not lin-
+ear mappings in general. We denote the spaces of linear multipliers and linear
+weak multipliers of A by LM(A) and LM ′(A) respectively. M(A) and LM(A)
+are subalgebras of the algebra AA consisting of all mappings from A to itself.
+Meanwhile, M ′(A) and LM ′(A) are closed under the operation ◦ defined by
+T ◦ S = T S + ST , and they form a Jordan algebra.
+In Sections 2 - 5 we study general properties of (weak) multipliers. In par-
+ticular, in sections 3 and 4 we give a decomposition theorem (Theorem 3.1), and
+a matrix equation (Theorem 4.2) for (weak) multipliers. They play an essential
+role to analyze (weak) multipliers.
+Complete classifications of associative algebras and zeropotent algebras of
+dimension 3 over an algebraically closed field of characteristic not equal to 2
+were given in Kobayashi et al, [3] and [4]. In Sections 6 and 7 we completely
+determine the (linear) (weak) multipliers of those algebras.
+2
+Multipliers and weak multipliers
+Let K be a field and A be a (not necessarily associative) algebra over K. The
+set AA of all mappings from A to A forms an associative algebra over K in the
+usual manner. Let L(A) denotes the subalgebra of AA of all linear mappings
+from A to A.
+A mapping T : A → A is a weak multiplier of A, if
+xT (y) = T (x)y
+(1)
+holds for any x, y ∈ A, and T is a multiplier, if
+xT (y) = T (xy) = T (x)y
+(2)
+for any x, y ∈ A. Let M(A) (resp. M ′(A)) denote the set of all multipliers
+(resp. weak multipliers) of A. Define
+LM(A)
+def
+= M(A) ∩ L(A) and LM ′(A)
+def
+= M ′(A) ∩ L(A).
+Proposition 2.1. M(A) (resp. LM(A)) is a unital subalgebra of AA (resp.
+L(A)), and M ′(A) (resp. LM ′(A)) is a Jordan subalgebra of AA (resp. L(A)).
+Proof. First, the zero mapping is a multiplier because all the three terms in (2)
+are zero. Secondly, the identity mapping is also a multiplier because the three
+terms in (2) are xy. Let T, U ∈ M(A). Then we have
+x(T +U)(y) = xT (y)+xU(y) = T (xy)+U(xy) = T (x)y+U(x)y = (T +U)(x)y
+(3)
+and
+x(T U)(y) = xT (U(y)) = T (xU(y)) = T U(xy) = T (U(x)y) = (T U)(x)y
+(4)
+2
+
+for any x, y ∈ A. Hence, T + U, T U ∈ M(A). Finally let k ∈ K, then
+x(kT )(y) = kxT (y) = kT (xy) = kT (x)y = (kT )(x)y,
+(5)
+and so kT ∈ M(A).
+Therefore, M(A) is a unital subalgebra of AA, and
+LM(A) = M(A) ∩ L(A) is a unital subalgebra of L(A).
+Next, let T, U ∈ M ′(A). Then, the equalities in (3) and (5) hold removing
+the center terms T (xy) + U(xy) and kT (xy) respectively. Hence, M ′(A) is a
+subspace of AA. Moreover, we have
+x(T U)(y) = xT (U(y)) = T (x)U(y) = U(T (x))y = UT (x)y
+and similarly x(UT )(y) = T U(x)y for any x, y ∈ A. Hence,
+x(T U + UT )(y) = (T U + UT )(x)y.
+It follows that T U + UT ∈ M ′(A).1
+The opposite Aop of A is the algebra with the same elements and the addition
+as A, but the multiplication ∗ in it is reversed, that is, x∗y = yx for all x, y ∈ A.
+Proposition 2.2. A and Aop have the same multipliers and weak multiplies,
+that is,
+M(Aop) = M(A) and M ′(Aop) = M ′(A).
+Proof. Let T ∈ AA. Then, T ∈ M ′(A), if and only if
+x ∗ T (y) = T (y)x = yT (x) = T (x) ∗ y
+for any x, y ∈ A, if and only if T ∈ M ′(Aop). Further, T ∈ M(A), if and only if
+x ∗ T (y) = T (y)x = T (yx) = T (x ∗ y) = yT (x) = T (x) ∗ y
+for any x, y ∈ A, if and only if T ∈ M(Aop).
+Let Annl(A) (resp. Annr(A)) be the left (resp. right) annihilator of A and
+let A0 be their intersection, that is,
+Annl(A) = {a ∈ A | ax = 0 for all x ∈ A},
+Annr(A) = {a ∈ A | xa = 0 for all x ∈ A}
+and
+A0 = Annl(A) ∩ Annr(A).
+They are all subspaces of A, and when A is an associative algebra, they are
+two-sided ideals. For a subset X of A, ⟨X⟩ denotes the subspace of A generated
+by X.
+1In general, for an associative algebra A over a field K of characteristic ̸= 2, the Jordan
+product ◦ on A is defined by x ◦ y = (xy + yx)/2 for x, y ∈ A.
+3
+
+Proposition 2.3. A weak multiplier T of A such that ⟨T (A)⟩ ∩ A0 = {0} is a
+linear mapping.
+Proof. Let x, y, z ∈ A and a, b ∈ K, and let T be a weak multiplier. We have
+T (ax + by)z = (ax + by)T (z) = axT (z) + byT (z) = aT (x)z + bT (y)z
+= (aT (x) + bT (y))z.
+Because z is arbitrary, we have w = T (ax + by) − aT (x) − bT (y) ∈ Annl(A).
+Similarly, we can show w ∈ Annr(A), and so w ∈ A0. Hence, if ⟨T (A)⟩ ∩ A0 =
+{0}, then w = 0 because w ∈ ⟨T (A)⟩. Since a, b, x, y are arbitrary, T is a linear
+mapping.
+Corollary 2.4. If A0 = {0}, then any weak multiplier is a linear mapping over
+K, that is, M ′(A) = LM ′(A) and M(A) = LM(A).
+Proposition 2.5. If T is a weak multiplier, then T (Annl(A)) ⊆ Annl(A),
+T (Annr(A)) ⊆ Annr(A) and T (A0) ⊆ A0 .
+Proof. Let x ∈ Annl(A), then for any y ∈ A we have
+0 = xT (y) = T (x)y.
+Hence, T (x) ∈ Annl(A). The other cases are similar.
+In this paper we denote the subset {xy | x, y ∈ A} of A by A2, though usually
+A2 denotes the subspace of A generated by this set.
+Proposition 2.6. Any mapping T : A → A such that T (A) ⊆ A0 is a weak
+multiplier. Such a mapping T is a multiplier if and only if T (A2) = {0}. In
+particular, if A is the zero algebra, every mapping T is a weak multiplier, and
+it is a multiplier if only if T (0) = 0.
+Proof. If T (A) ⊆ A0, the both sides are 0 in (1) and T is a weak multiplier.
+This T is a multiplier, if only if the term T (xy) in the middle of (2) is 0 for all
+x, y ∈ A, that is, T (A2) = {0}. If A is the zero algebra, then A = A0 and A2
+= {0}. Hence, any T is a weak multiplier and it is a multiplier if and only if
+T (0) = 0.
+3
+Nihil decomposition
+Let A1 be a subspace of A such that
+A = A1 ⊕ A0.
+(6)
+Here, A1 is not unique, but choosing an appropriate A1 will become important
+later. When A1 is fixed, any mapping T ∈ AA is uniquely decomposed as
+T = T1 + T0
+(7)
+4
+
+with T1(A) ⊆ A1 and T0(A) ⊆ A0. We call (6) and (7) a nihil decompositions
+of A and T respectively. Let π : A → A1 be the projection and µ : A1 → A be
+the embedding, that is, π(x1 + x0) = µ(x1) = x1 for x1 ∈ A1 and x0 ∈ A0.
+Let M1(A) (resp.
+M0(A)) denote the set of all multipliers T of A with
+T (A) ⊆ A1 (resp. T (A) ⊆ A0). Similarly, the sets M ′
+1(A) and M ′
+0(A) of weak
+multipliers of A are defined. Also, set
+LMi(A) = Mi(A) ∩ L(A) and LM ′
+i(A) = M ′
+i(A) ∩ L(A)
+for i = 0, 1. By Proposition 2.3 we see
+M ′
+1(A) = LM ′
+1(A) and M1(A) = LM1(A),
+and by Proposition 2.6 we have
+M ′
+0(A) = AA
+0 , M0(A) = {T ∈ AA
+0 | T (A2) = {0}}.
+(8)
+Theorem 3.1. Let A = A1 ⊕ A0 and T = T1 + T0 be nihil decompositions of A
+and T ∈ AA respectively.
+(i) T is a weak multiplier, if and only if T1 is a weak multiplier. If T is a
+weak multiplier, T1 is a linear mapping satisfying T1(A0) = {0}.
+(ii) If T1 is a multiplier and T0(A2) = {0}, then T is a multiplier. If A1 is
+a subalgebra of A, the converse is also true.
+Suppose that A1 is a subalgebra of A, and let Φ be a mapping sending R ∈
+(A1)A1 to µ ◦ R ◦ π ∈ AA. Then,
+(iii) Φ gives an algebra isomorphism from M(A1) onto M1(A) and a Jordan
+isomorphism from M ′(A1) onto M ′
+1(A).
+Proof. Let x, y ∈ A.
+(i) If T is a weak multiplier, then
+xT1(y) = x(T (y) − T0(y)) = xT (y) = T (x)y = T1(x)y.
+Thus, T1 is also a weak multiplier. Moreover, T1 is a linear mapping by Propo-
+sition 2.3 and T1(A0) ⊆ A1 ∩ A0 = {0} by Proposition 2.5. Conversely, if T1 is
+a weak multiplier, then
+xT (y) = xT1(y) = T1(x)y = T (x)y,
+and so T is a weak multiplier.
+(ii) If T1 is a multiplier and T0(A2) = 0, then T is a multiplier because
+xT (y) = xT1(y) = T1(xy) = T (xy) − T0(xy) = T (xy) = T1(x)y = T (x)y.
+Next suppose that A1 is a subalgebra. If T is a multiplier, then for any
+x, y ∈ A we have
+T1(xy) + T0(xy) = T (xy) = xT (y) = x1T1(y),
+(9)
+5
+
+where x = x1 + x0 with x1 ∈ A1 and x0 ∈ A0. Here, x1T1(y) ∈ A1 because A1
+is a subalgebra, and thus, we have T0(xy) = x1T1(y) − T1(xy) ∈ A0 ∩ A1 = {0}.
+Since x, y are arbitrary, we get T0(A2) = {0}. Moreover, T1 is a multiplier
+because T1(xy) = x1T1(y) = xT1(y) by (9) and similarly T1(xy) = T1(x)y. The
+converse is already proved above.
+(iii) Let S ∈ (A1)A1 and x = x1 + x0, y = y1 + y0 ∈ A with x1, y1 ∈ A1 and
+x0, y0 ∈ A0. Then, π(x) = µ(x1) = x1, π(y) = µ(y1) = y1 and
+Φ(S)(x) = µ(S(π(x))) = µ(S(x1)) = S(x1).
+Thus, if S ∈ M ′(A1), we have
+xΦ(S)(y) = xS(y1) = x1S(y1) = S(x1)y1 = Φ(S)(x)y1 = Φ(S)(x)y.
+Hence, Φ(S) ∈ M ′
+1(A). Moreover, if S ∈ M(A1), then because π is a homomor-
+phism, we have
+Φ(S)(xy) = S(π(xy)) = S(x1y1) = x1S(y1) = xΦ(S)(y),
+and hence Φ(S) ∈ M1(A). Conversely, let T ∈ M ′
+1(A), then because T is a
+linear mapping satisfying T (A0) = {0}, there is a linear mapping S ∈ L(A1) on
+A1 such that Φ(S) = T , that is, S(x1) = T (x) = T (x1). We have
+x1S(y1) = x1T (y1) = T (x1)y1 = S(x1)y1,
+and hence S ∈ M ′(A1). Therefore, Φ is a linear isomorphism from M ′(A1)
+to M ′
+1(A).
+Similarly, Φ gives a linear isomorphism from M(A1) to M1(A).
+Moreover, for T, U ∈ M ′(A1), we have
+Φ(T U) = µ ◦ T ◦ U ◦ π = µ ◦ T ◦ π ◦ µ ◦ U ◦ π = Φ(T )Φ(U).
+Thus, Φ gives an isomorphism of algebras from M(A1) to M1(A) and a Jordan
+isomorphism from M ′(A1) to M ′
+1(A).
+Theorem 3.1 implies
+M ′(A) = M ′
+1(A) ⊕ M ′
+0(A), M1(A) ⊕ M0(A) ⊆ M(A),
+where M ′
+0(A) and M0(A) are given as (8). Moreover, if A1 is a subalgebra, we
+have
+M ′(A) ∼= M ′(A1)⊕(A0)A, M(A) ∼= M(A1)⊕{T ∈ (A0)A | T (A2) = {0}}. (10)
+Corollary 3.2. Any weak multiplier T is written as
+T = T1 + R
+(11)
+with T1 ∈ LM ′
+1(A) and R ∈ (A0)A, and it is a multiplier if and only if
+R(x1y1) = x1T1(y1) − T1(x1y1)
+(12)
+for any x1, y1 ∈ A1.
+6
+
+Proof. As stated above T is written as (11). Let x = x1 + x0, y = y1 + y0 ∈ A
+with x1, y1 ∈ A1 and x0, y0 ∈ A0 be arbitrary, then we have
+xT (y) = x1(T1(y) + R(y)) = x1T1(y) = x1T1(y1)
+(13)
+because R(A) ⊆ A0 and T1(A0) = {0}. The last term in (13) is also equal
+to T1(x1)y1 = T (x)y. Hence, T is a multiplier, if and only if it is equal to
+T (xy) = T (x1y1) = T1(x1y1) + R(x1y1), if and only if (12) holds.
+4
+Linear multipliers and matrix equation
+In this section, A is a finite dimensional algebra over K. We drive a matrix
+equation for a linear mapping on A to be a (weak) multiplier. Suppose that A
+is n-dimensional with basis E = {e1, e2, . . . , en}.
+Lemma 4.1. A linear mapping T : A → A is a weak multiplier if and only if
+eiT (ej) = T (ei)ej,
+(14)
+and it is a multiplier if and only if
+T (eiej) = eiT (ej) = T (ei)ej,
+(15)
+for all ei, ej ∈ E.
+Proof. The necessity of the conditions (14) and (15) is obvious. Let x = x1e1 +
+x2e2+· · ·+xnen, y = y1e1+y2e2+· · ·+ynen ∈ A with x1, x2, . . . , xn, y1, y2, . . . , yn ∈
+K. If T satisfies (14), then we have
+xT (y)
+=
+(
+�
+i
+xiei)T (
+�
+j
+yjej) = (
+�
+i
+xiei)(
+�
+j
+yjT (ej)) =
+�
+i,j
+xiyjeiT (ej)
+=
+�
+i,j
+xiyjT (ei)ej = (
+�
+i
+xiT (ei))(
+�
+j
+yjej) = T (x)y.
+Hence, T is a weak multiplier. Moreover, if T satisfies (15), it is a multiplier in
+a similar way
+Let A (we use the bold character) be the multiplication table of A on E. A
+is a matrix whose elements are from A defined by
+A = EtE,
+(16)
+where E = (e1, e2, . . . , en) (we again use the bold face E) is the row vector
+consisting the basis elements. For a linear mapping T on A and a matrix B
+over A, T (B) denotes the matrix obtained by applying T component-wise, that
+is, the (i, j)-element of T (B) is T (bij) for the (i, j)-element bij of B.2 We use
+the same character T for the representation matrix of T on E, that is,
+T (E) = ET.
+(17)
+.
+2This is called a broadcasting (cf. [6]).
+7
+
+Theorem 4.2. A linear mapping T is a weak multiplier of A if and only if
+AT = T tA,
+(18)
+and T is a multiplier if and only if
+T (A) = AT = T tA.
+(19)
+Proof. By (16) and (17) we have
+(e1, e2, . . . , en)t(T (e1), T (e2), . . . , T (en)) = EtT (E) = EtET = AT
+(20)
+and
+(T (e1), T (e2), . . . , T (e2))t(e1, e2, . . . , en) = T (E)tE = T tEtE = T tA.
+(21)
+By Lemma 4.1, T is a weak multiplier, if and only if (20) and (21) are equal,
+if and only if (18) holds. Moreover, T is multiplier if and only if, the leftmost
+sides of (20) and (21) are equal to (T (eiej))i,j=1,2,...,n = T (A), if and only if
+(19) holds.
+The multiplication table of the opposite algebra Aop of A is At. So, the alge-
+bras with multiplication tables transposed to each other have the same (weak)
+multipliers.
+5
+Associative algebras
+In this section A is an associative algebra over K.
+Proposition 5.1. If A0 = {0}, then we have
+M(A) = M ′(A) = LM(A) = LM ′(A).
+Proof. Let T ∈ M ′(A), then we have
+T (xy)z = xyT (z) = xT (y)z and zT (xy) = T (z)xy = zT (x)y
+for any x, y, z ∈ A. It follows that
+T (xy) − xT (y) ∈ Annl(A) ∩ Annr(A) = A0 = {0}.
+Hence, T (xy) = xT (y) and we see T ∈ M(A).
+Moreover, T ∈ LM(A) by
+Proposition 2.3.
+Let a ∈ A. If xay = axy (resp. xay = xya) for any x, y ∈ A, a is called a
+left (resp. right) central element, and a is called a central element if ax = xa
+for any x ∈ A. Let Zl(A), (resp. Zr(A), Z(A)) denotes the set of all left central
+(resp. right central, central) elements.
+8
+
+Lemma 5.2. Zl(A) (resp. Zr(A), Z(A)) are subalgebra of A containing Annl(A)
+(resp. Annr(A), A0).
+Proof. Straightforward.
+For a ∈ A, la (resp. ra) denotes the left (resp. right) multiplication by a,
+that is,
+la(x) = ax,
+ra(x) = xa
+for x ∈ A. They are linear mappings.
+Lemma 5.3. For a ∈ A the following statements are equivalent.
+(i) la (resp. ra) is a multiplier,
+(ii) la (resp. ra) is a weak multiplier,
+(iii) a is left (resp. right) central.
+Proof. If la is a weak multiplier, then
+xay = xla(y) = la(x)y = axy
+for any x, y ∈ A. Hence, a is left central. Because la(x)y = axy = la(xy) and
+xla(y) = xay for any x, y ∈ A, la is a multiplier if a is left central. The other
+case is similar, and we see that the three statements are equivalent.
+Lemma 5.4. Suppose that A has a right (resp. left) identity. Then, a left (resp.
+right) central element is central, and Annl(A) = {0} (resp. Annr(A) = {0}).
+Proof. Easy.
+Because Annl(A) ⊆ Zl and Annr(A) ⊆ Zr by Lemma 5.2, we can make the
+quotient algebras ¯Zl(A) = Zl(A)/Annl(A) and ¯Zr(A) = Zr(A)/Annr(A).
+Theorem 5.5. Suppose that A has a left (resp. right) identity e. Then, any
+multiplier is a left (resp. right) multiplication by a left (resp. right) central
+element and is a linear multiplier. The mapping φ : Zl(A) → M(A) sending
+a ∈ Zl(A) to la induces an isomorphism ¯φ : Zl(A) → M(A) of algebras. In
+particular, if A is unital, M(A) is isomorphic to Z(A).
+Proof. Suppose that A has a left identity e. Let T ∈ M ′(A) and set a = T (e).
+Then we have
+T (x) = eT (x) = T (e)x = ax
+for any x ∈ A. Hence, T = la, where a ∈ Zl(A) and T is a linear multiplier
+by Lemma 5.3. Therefore, M ′(A) = LM(A) and φ is surjective. Moreover,
+for a ∈ Zl(A), φ(a) = 0, if and only if ax = 0 for any x ∈ A, if and only
+if a ∈ Annl(A). Thus we have Ker(φ) = Annl(A), and φ induces the desired
+isomorphism. Similarly, if A has a right identity, M(A) is isomorphic to Zr(A).
+Finally, if A has the identity, then Annl(A) = Annr(A) = {0} and hence M(A)
+is isomorphic to Z(A).
+9
+
+6
+3-dimensional associative algebras
+Over an algebraically closed field K of characteristic not equal to 2, we have,
+up to isomorphism, 24 families of associative algebras of dimension 3. They are
+5 unital algebras U0, U1, U2, U3, U4 defined on basis E = {e, f, g} by
+�e
+f
+g
+f
+0
+0
+g
+0
+0
+�
+,
+�e
+f
+g
+f
+0
+f
+g
+−f
+e
+�
+,
+�e
+0
+0
+0
+f
+0
+0
+0
+g
+�
+,
+�e
+0
+0
+0
+f
+g
+0
+g
+0
+�
+,
+�e
+f
+g
+f
+g
+0
+g
+0
+0
+�
+,
+5 curled algebras C0, C1, C2, C3, C4 defined by
+�0
+0
+0
+0
+0
+0
+0
+0
+0
+�
+,
+�0
+0
+0
+0
+0
+e
+0
+−e
+0
+�
+,
+�0
+0
+0
+e
+f
+0
+0
+g
+0
+�
+,
+�0
+0
+0
+0
+0
+0
+e
+f
+g
+�
+,
+�0
+0
+e
+0
+0
+f
+0
+0
+g
+�
+,
+non-unital 4 straight algebras S1, S2, S3, S4 defined by
+�f
+g
+0
+g
+0
+0
+0
+0
+0
+�
+,
+�e
+0
+0
+0
+g
+0
+0
+0
+0
+�
+,
+�e
+0
+0
+0
+f
+0
+0
+0
+0
+�
+,
+�e
+f
+0
+f
+0
+0
+0
+0
+0
+�
+,
+and non-unital 10 families of waved algebras W1, W2, W4, W5, W6, W7, W8,
+W9, W10 and
+�
+W3(k)
+�
+k∈H defined by
+�0
+0
+0
+0
+0
+0
+0
+0
+e
+�
+,
+�0
+0
+0
+0
+0
+0
+0
+e
+0
+�
+,
+�e
+0
+0
+0
+0
+0
+0
+0
+0
+�
+,
+�0
+0
+0
+0
+0
+0
+0
+f
+g
+�
+,
+�0
+0
+0
+0
+0
+f
+0
+0
+g
+�
+,
+�e
+0
+0
+0
+0
+0
+0
+f
+g
+�
+,
+�e
+0
+0
+0
+0
+f
+0
+0
+g
+�
+,
+�0
+e
+0
+e
+f
+0
+0
+g
+0
+�
+,
+�0
+e
+0
+e
+f
+g
+0
+0
+0
+�
+and
+�0
+0
+0
+0
+e
+0
+0
+ke
+e
+�
+,
+respectively, where H is a subset of K such that K = H∪−H and H∩−H = {0}
+(see [3] for details). We determine the (weak) multipliers of them below.
+(0) A = C0 is the zero algebra, so by Proposition 2.6, we have
+M ′(A) = AA, M(A) = {T ∈ AA | T (0) = 0}
+and
+LM(A) = LM ′(A) = L(A).
+(i) The unital algebras U0, U2, U3, U4 are commutative, so for such A we have
+M(A) = LM(A) = M ′(A) = LM ′(A) = {lx|x ∈ A} ∼= A
+by Theorem 5.5. For A = U1, we have
+M(A) = LM(A) = M ′(A) = LM ′(A) ∼= Z(A) = Ke.
+(ii) For A = C1, We have A0 = Annl(A) = Annr(A) = Ke, and a nihil decompo-
+sition A = A1 ⊕ A0 with A1 = Kf + Kg. Let T1 ∈ M ′
+1(A), then by Theorem 3.1, T1
+is a linear mapping such that T1(Ke) = {0}. Let
+T1 =
+
+
+0
+0
+0
+0
+q
+r
+0
+t
+u
+
+
+(22)
+10
+
+with q, r, t, u ∈ K be the representation matrix of T1 on E. By Theorem 4.2, T1 is a
+weak multiplier, if and only if
+
+
+0
+0
+0
+0
+te
+ue
+0
+−qe
+−re
+
+ = AT1 = T t
+1A =
+
+
+0
+0
+0
+0
+−te
+qe
+0
+−ue
+re
+
+ ,
+if and only if r = t = 0 and q = u. Hence, M ′
+1(A) = {Tq
+�� q ∈ K}, where Tq =
+
+
+0
+0
+0
+0
+q
+0
+0
+0
+q
+
+. By Theorem 3.1 we see
+M ′(A) = {Tq
+�� q ∈ K} ⊕ (Ke)A,
+and
+LM ′(A) =
+
+
+
+
+
+a
+b
+c
+0
+q
+0
+0
+0
+q
+
+
+��� a, b, c, q ∈ K
+
+
+ .
+By the multiplication table of A, we have αβ = (xv − yz)e
+for α = xf + yg, β =
+zf + vg ∈ A1 with x, y, z, v ∈ K. By Corollary 3.2, T ∈ M ′(A) is given by T = Tq + R
+with R ∈ (Ke)A and this T is a multiplier, if and only if
+R((xv − yz)e)
+=
+R(αβ) = αTq(β) − Tq(αβ)
+=
+α(qβ) − Tq((xv − yz)e) = q(xv − yz)e
+for any α and β, if and only if R(xe) = qxe for all x ∈ K. Let Sq =
+
+
+q
+0
+0
+0
+q
+0
+0
+0
+q
+
+ be
+the scalar multiplication by q ∈ K. Then, we see (T − Sq)(A) ⊆ A0 = Ke and
+(T − Sq)(xe) = Tq(xe) + R(xe) − Sq(xe) = 0 + qxe − qxe = 0,
+for any x ∈ K, that is, (T − Sq)(Ke) = {0}. Thus, we have
+M(A) = {Sq
+�� q ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}},
+and
+LM(A) =
+
+
+
+
+
+a
+b
+c
+0
+a
+0
+0
+0
+a
+
+
+��� a, b, c ∈ K
+
+
+ .
+(iii) A = C2: Because Annl(A) = Ke and Annr(A) = Kg, we see A0 = {0}.
+Hence, any weak multiplier T is a linear multiplier by Proposition 5.1. By Theorem
+4.2,
+T =
+
+
+a
+b
+c
+p
+q
+r
+s
+t
+u
+
+
+(23)
+is a (weak) multiplier, if and only if
+
+
+0
+0
+0
+ae + pf
+be + qf
+ce + rf
+pg
+qg
+rg
+
+ = AT = T tA =
+
+
+pe
+pf + sg
+0
+qe
+qf + tg
+0
+re
+rf + ug
+0
+
+ ,
+11
+
+if and only if b = c = p = r = s = t = 0 and a = q = u, that is, T is the scalar
+multiplication Sa by a. Consequently,
+M(A) = M ′(A) = LM(A) = LM ′(A) = {Sa
+�� a ∈ K} ∼= K.
+(iv) C3 and C4 are opposed to each other, and have the same (weak) multipliers
+by Proposition 2.2.
+Let A = C3, then, A has a left identity g, Zl(A) = A and
+Annl(A) = Ke + Kf. Hence, by Theorem 5.4,
+M(A) = M ′(A) = LM(A) = LM ′(A) = A/(Ke + Kg) = {Sa
+�� a ∈ K}.
+(v) A = S1: We have A0 = Annl(A) = Annr(A) = Kg, and A = A1 ⊕ A0 with
+A1 = Ke + Kf. Then, T1 ∈ M ′
+1(A) is a linear mapping with T (Kg) = {0}. Let
+T1 =
+
+
+a
+b
+0
+p
+q
+0
+0
+0
+0
+
+
+(24)
+be its representation on E. T1 is a weak multiplier, if and only if
+
+
+af + pg
+bf + qg
+0
+ag
+bg
+0
+0
+0
+0
+
+ = AT1 = T t
+1A =
+
+
+af + pg
+ag
+0
+bf + qg
+bg
+0
+0
+0
+0
+
+ ,
+if and only if b = 0 and a = q. Hence,
+M ′(A) = {T a,p
+1
+| a, p ∈ K} ⊕ (Kg)A,
+where T a,p
+1
+=
+
+
+a
+0
+0
+p
+a
+0
+0
+0
+0
+
+. So, T ∈ M ′(A) is written as T = T a,p
+1
++R with R ∈ (Kg)A,
+and this T is multiplier, if and only if
+R(xzf + (xv + yz)g)
+=
+R(αβ) = αT a,p
+1
+(β) − T a,p
+1
+(αβ)
+=
+α(aze + (pz + av)f) − T a,p
+1
+(xzf + (xv + yz)g)
+=
+axzf + (pxz + axv + ayz)g − axzf
+=
+(pxz + a(xv + yz))g
+for any α = xe + yf, β = ze + vf ∈ A1 with x, y, z, v ∈ K, if and only if R(xf + yg) =
+(px + ay)g for all x, y ∈ K. Let T a,p =
+
+
+a
+0
+0
+p
+a
+0
+0
+p
+a
+
+, then (T − T a,p)(A) ⊆ Kg, and
+(T − T a,p)(xf + yg)
+=
+(T a,p
+1
++ R − T a,p) (xf + yg)
+=
+axf + (px + ay)g − (axf + pxg + ayg) = 0.
+for any x, y ∈ K. Thus, (T − T a,p)(Kf + Kg) = {0}, and hence
+M(A) = {T a,p | a, p ∈ K} ⊕ {R ∈ (Kg)A | R(Kf + Kg) = {0}}.
+Taking the intersections of M ′(A) and M(A) with L(A), we obtain
+LM ′(A) =
+
+
+
+
+
+a
+0
+0
+p
+a
+0
+s
+t
+u
+
+
+��� a, p, s, t, u ∈ K
+
+
+
+12
+
+and
+LM(A) =
+
+
+
+
+
+a
+0
+0
+p
+a
+0
+s
+p
+a
+
+
+��� a, p, s ∈ K
+
+
+ .
+(vi) A = S2: We have A0 = Annl(A) = Annr(A) = Kg, and A = A1 ⊕ A0 with
+A1 = Ke + Kf. Let a linear mapping T1 ∈ M ′
+1(A) be represented as (24), then T1 is
+a weak multiplier, if and only if
+
+
+ae
+be
+0
+pg
+qg
+0
+0
+0
+0
+
+ = AT = T tA =
+
+
+ae
+pg
+0
+be
+qg
+0
+0
+0
+0
+
+ ,
+if and only if b = p = 0. Hence,
+M ′(A) = {T a,q
+1
+�� a, q ∈ K} ⊕ (Kg)A,
+where T a,q
+1
+=
+
+
+a
+0
+0
+0
+q
+0
+0
+0
+0
+
+, and
+LM ′(A) =
+
+
+
+
+
+a
+0
+0
+0
+q
+0
+s
+t
+u
+
+
+��� a, q, s, t, u ∈ K
+
+
+ .
+By Corollary 3.2, a weak multiplier T written as T = T a,q
+1
++ R for a, q ∈ K and
+R ∈ (Kg)A is multiplier, if and only if
+R(xze + yvg)
+=
+R(αβ) = αT a,q
+1
+(β) − T a,q
+1
+(xze + yvg)
+=
+(xe + yf)(aze + qvf) − axze
+=
+yqvg,
+for any α = xe + yf, β = ze + vf ∈ A1 with x, y, z, v ∈ K, if only if R(xe + yg) =
+qyg for all x, y ∈ K.
+Let T a,q =
+
+
+a
+0
+0
+0
+q
+0
+0
+0
+q
+
+, then T a,q ∈ M(A) and we have
+(T − T a,q)(xe + yg) = 0 for any x, y ∈ K in the same way as (v) above.
+Hence,
+(T − T a,q)(Ke + Kg) = {0}, and we have
+M(A) = {T a,p | a, p ∈ K} ⊕ {R ∈ (Kg)A | R(Ke + Kg) = {0}}
+and
+LM(A) =
+
+
+
+
+
+a
+0
+0
+0
+p
+0
+0
+t
+0
+
+
+��� a, p, t ∈ K
+
+
+ .
+(vii) A = S3: We have A0 = Kg and A = A1 ⊕ A0 with A1 = Ke + Kf. Since
+A1 is a subalgebra of A, by Theorem 3.1 we obtain the equalities (10) in Section 3.
+Because A1 is a commutative unital algebra,
+M(A1) = M ′(A1) = A1 =
+��
+a
+0
+0
+b
+� ��� a, b ∈ K
+�
+by Theorem 5.5. Hence,
+M ′(A) = A1 ⊕ (Kg)A and M(A) = A1 ⊕ {T ∈ (Kg)A | T (Ke + Kf) = 0}.
+13
+
+Intersecting with L(A) we have
+LM ′(A) =
+
+
+
+
+
+a
+0
+0
+0
+b
+0
+s
+t
+u
+
+
+��� a, b, s, t, u ∈ K
+
+
+ and LM(A) =
+
+
+
+
+
+a
+0
+0
+0
+b
+0
+0
+0
+u
+
+
+��� a, b, u ∈ K
+
+
+ .
+(viii) A = S4: We have A = A1 ⊕ A0 with A0 = Kg and A1 = Ke + Kf. Because
+A1 is a commutative unital subalgebra of A, similarly to above we have
+M ′(A) = A1 ⊕ (Kg)A =
+��a
+0
+b
+a
+� ��� a, b ∈ K
+�
+⊕ (Kg)A,
+M(A)
+=
+A1 ⊕
+�
+T ∈ (Kg)A | T (A2) = 0
+�
+=
+��a
+0
+b
+a
+� ��� a, b ∈ K
+�
+⊕ {T ∈ (Kg)A | T (Ke + Kf) = 0},
+LM ′(A) =
+
+
+
+
+
+a
+0
+0
+b
+a
+0
+s
+t
+u
+
+
+��� a, b, s, t, u ∈ K
+
+
+ and LM(A) =
+
+
+
+
+
+a
+0
+0
+b
+a
+0
+0
+0
+u
+
+
+��� a, b, u ∈ K
+
+
+ .
+(ix) A = W1 : We have A = A1 ⊕ A0 with A0 = Ke + Kf and A1 = Kg. Let
+T1 ∈ M ′
+1(A), then T1 is a linear mapping with T1(A0) = {0}. So T1 is determined by
+T1(g) = ag with a ∈ K. Let denote this T1 by T a
+1 . We have
+M ′(A) = {T a
+1 | a ∈ K} ⊕ (Ke + Kf)A.
+A weak multiplier T = T a
+1 + R with R ∈ (Ke + Kf)A is a multiplier, if and only if
+R(xye) = R((xg)(yg)) = xgT a
+1 (yg) − T a
+1 (xye) = axye
+for all x, y ∈ K, if and only if R(xe) = axe for any x ∈ K. Let Ta =
+
+
+a
+0
+0
+0
+0
+0
+0
+0
+a
+
+.
+Then, (T − Ta)(Ke) = {0} and it follows that
+M(A) = {Ta
+�� a ∈ K} ⊕ {R ∈ (Ke + Kf)A �� R(Ke) = {0}}.
+Also we have
+LM ′(A) =
+
+
+
+
+
+a
+b
+c
+p
+q
+r
+0
+0
+u
+
+
+��� a, b, c, p, q, r, u ∈ K
+
+
+
+and
+LM(A) =
+
+
+
+
+
+a
+b
+c
+0
+q
+r
+0
+0
+a
+
+
+��� a, b, c, q, r ∈ K
+
+
+ .
+(x) A = W2
+3: We have A = A1 ⊕ A0 with A0 = Ke and A1 = Kf + Kg.
+T ∈ M ′
+1(A) is a linear mapping with T (Ke) = {0}. Let T be represented as (22), then
+T is a weak multiplier if and only if
+
+
+0
+0
+0
+0
+0
+0
+0
+qe
+re
+
+ = AT = T tA =
+
+
+0
+0
+0
+0
+te
+0
+0
+ue
+0
+
+ ,
+3This is the algebra taken up in [8]
+14
+
+if and only if r = t = 0, q = u. Hence,
+M ′(A) = {Tq | q ∈ K} ⊕ (Ke)A,
+where Tq =
+
+
+0
+0
+0
+0
+q
+0
+0
+0
+q
+
+. So,
+LM ′(A) =
+
+
+
+
+
+a
+b
+c
+0
+q
+0
+0
+0
+q
+
+
+��� a, b, c, q ∈ K
+
+
+ .
+A weak multiplier T = Tq + R with R ∈ (Ke)A is a multiplier, if and only if
+R(yze) = R(αβ) = αTq(β) − Tq(yze) = α(qβ) = qyze
+for any α = xf +yg, β = zf +vg ∈ A1 with x, y, z, v ∈ K, if and only if R(xe) = qxe for
+all x ∈ K. Let Sa be the scalar multiplication by a ∈ K. Then, (T − Sa)(Ke) = {0},
+and hence,
+M(A) = {Sa | a ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}}
+and
+LM(A) =
+
+
+
+
+
+a
+b
+c
+0
+a
+0
+0
+0
+a
+
+
+��� a, b, c ∈ K
+
+
+ .
+(xi) A = W4: We have A = A1 ⊕ A0 with A0 = Kf + Kg and A1 = Ke. Because
+A1 is a subalgbra isomorphic to the base field K, for T a
+1 ∈ M(A1) with a ∈ K given
+by T a
+1 (e) = ae, we see
+M ′(A) = {T a
+1 | a ∈ K} ⊕ (fK + gK)A
+and
+M(A) = {T a
+1 | a ∈ K} ⊕ {R ∈ (fK + gK)A | R(Ke) = 0}
+by Theorem 3.1. Taking the intersection with L(A) we have
+LM ′(A) =
+
+
+
+
+
+a
+0
+0
+p
+q
+r
+s
+t
+u
+
+
+��� a, p, q, r, s, t, u ∈ K
+
+
+
+and
+LM(A) =
+
+
+
+
+
+a
+0
+0
+0
+q
+r
+0
+t
+u
+
+
+��� a, q, r, t, u ∈ K
+
+
+ .
+(xii) W5 and W6 are opposed. Let A = W5, then A = A1 ⊕ A0 with A0 = Ke and
+A1 = Kf + Kg. Since A1 is a subalgebra of A, we have the equalities (10). Because
+A1 has a left identity g, we have
+M(A1) = LM(A1) = M ′(A1) = LM ′(A1) ∼= (A1)/Kf ∼= Kg
+by Theorem 5.5. So, any element in M(A1) is a scalar multiplication Sq
+1 in A1 by
+q ∈ K. By Theorem 3.1 we have
+M ′(A) = {Sq
+1 | q ∈ K} ⊕ (Ke)A,
+15
+
+M(A) = {Sq
+1 | q ∈ K} ⊕ {R ∈ (Ke)A | R(Kf + Kg) = 0},
+LM ′(A) =
+
+
+
+
+
+a
+b
+c
+0
+q
+0
+0
+0
+q
+
+
+��� a, b, c, q ∈ K
+
+
+ and LM(A) =
+
+
+
+
+
+a
+0
+0
+0
+q
+0
+0
+0
+q
+
+
+��� a, q ∈ K
+
+
+ .
+(xiii) W7 and W8 are opposed. Let A = W7. We see A0 = Annr(A) = {0}. Hence,
+any weak multiplier is a linear multiplier by Proposition 5.1, and T represented as (23)
+is a weak multiplier, if and if
+
+
+ae
+be
+ce
+0
+0
+0
+pf + sg
+qf + tg
+rf + ug
+
+ = AT = T tA =
+
+
+ae
+sf
+sg
+be
+tf
+tg
+ce
+uf
+ug
+
+ ,
+if and only if b = c = p = r = s = t = 0, q = u. Therefore,
+M(A) = LM(A) = M ′(A) = LM ′(A) = LM ′(A) =
+
+
+
+
+
+a
+0
+0
+0
+q
+0
+0
+0
+q
+
+
+��� a, q ∈ K
+
+
+ .
+(xiv) W9 and W10 are opposed. Let A = W9. Then, because A0 = Annl(A) = {0},
+any weak multiplier is a linear multiplier and a linear mapping T represented as (23)
+is a weak multiplier if and only if
+
+
+pe
+qe
+re
+ae + pf
+be + qf
+ce + rf
+pg
+qg
+rg
+
+ = AT = T tA =
+
+
+pe
+ae + pf + sg
+0
+qe
+be + qf + tg
+0
+re
+ce + rf + ug
+0
+
+
+c = p = r = s = t = 0, a = q = u. Therefore,
+LM(A) = M(A) = LM ′(A) = M ′(A) =
+
+
+
+
+
+a
+b
+0
+0
+a
+0
+0
+0
+a
+
+
+��� a, b ∈ K
+
+
+ .
+(xv) A = W3(k). We have A = A1 ⊕ A0 with A0 = Ke and A1 = Kf + Kg.
+T ∈ M ′
+1(A) is a linear mapping with T (Ke) = {0}. Let T be represented as (22), then
+T is a weak multiplier if and only if
+
+
+0
+0
+0
+0
+qe
+re
+0
+(kq + t)e
+(kr + u)e
+
+ = AT = T tA =
+
+
+0
+0
+0
+0
+(q + kt)e
+te
+0
+(r + ku)e
+ue
+
+ .
+If k = 0, then the above holds if and only if r = t, and otherwise it holds if and only
+if r = t = 0, q = u. Thus,
+M ′(A) = {T q,r,u
+1
+�� q, r, u ∈ K}⊕(Ke)A, LM ′(A) =
+
+
+
+
+
+a
+b
+c
+0
+q
+r
+0
+r
+u
+
+
+��� a, b, c, q, r, u ∈ K
+
+
+
+if k = 0, and
+M ′(A) = {T q
+1
+�� q ∈ K} ⊕ (Ke)A, LM ′(A) =
+
+
+
+
+
+a
+b
+c
+0
+q
+0
+0
+0
+q
+
+
+��� a, b, c, q ∈ K
+
+
+
+16
+
+if k ̸= 0, where
+T q,r,u
+1
+=
+
+
+0
+0
+0
+0
+q
+r
+0
+r
+u
+
+ and T q
+1 =
+
+
+0
+0
+0
+0
+q
+0
+0
+0
+q
+
+.
+When k = 0, T = T q,r,u
+1
++ R with R ∈ (Ke)A is multiplier, if and only if
+R((xz + yv)e)
+=
+R(αβ) = αT q,r,u
+1
+(β) − T q,r,u
+1
+((xz + yv)e)
+=
+α((qz + rv)f + (rz + uv)g) = (qxz + r(xv + yz) + uyv)e
+for any α = xf + yg, β = zf + vg ∈ A1 with x, y, z, v ∈ K, if and only if q = u, r = 0
+and R(xe) = qxe for all x ∈ K. While, when k ̸= 0, T = T q
+1 + R with R ∈ (Ke)A is a
+multiplier, if and only if
+R((xz + y(kz + v))e)
+=
+R(αβ) = αT q
+1 (β) − T q
+1 (xue + y(kz + v)e)
+=
+α(qβ) = q(xz + y(kz + v))e
+for any α, β, if and only if R(xe) = qxe for any x ∈ K.
+In the both cases, with
+the scalar multiplication Sa by a ∈ K, we have (T − Sa)(Ke) = {0}. Therefore, for
+arbitrary k (either k is zero or nonzero) we see
+M(A) = {Sa
+�� a ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}}
+and
+LM(A) =
+
+
+
+
+
+a
+b
+c
+0
+a
+0
+0
+0
+a
+
+
+��� a, b, c ∈ K
+
+
+ .
+7
+3-dimensional zeropotent algebras
+A is a zeropotent algebra if x2 = 0 for all x ∈ A. The zeropotent algebra A over K is
+anti-commutative, that is, xy = −yx for all x, y ∈ A. Thus we see
+A0 = Annl(A) = Annr(A).
+Let A be a zeropotent algebras of dimension 3 over K with char(K) ̸= 2. Let
+E = {e, f, g} be a basis of A. Because A is anti-commutative, the multiplication table
+A of A on E is given as
+A =
+
+
+0
+α
+−β
+−α
+0
+γ
+β
+−γ
+0
+
+ ,
+where
+
+
+
+
+
+γ = fg = a11e + a12f + a13g
+β = ge = a21e + a22f + a23g
+α = ef = a31e + a32f + a33g
+for aij ∈ K. We call A =
+
+
+a11
+a12
+a13
+a21
+a22
+a23
+a31
+a32
+a33.
+
+ the structural matrix of A (we use the
+same symbol A for the algebra and its structural matrix).
+Lemma 7.1. If rank(A) ≥ 2, then A0 = {0}.
+17
+
+Proof. If rank(A) ≥ 2, at least two of fg, ge, ef are linearly independent. Suppose
+that α = ef and β = ge are linearly independent (the other cases are similar). If
+x = ae + bf + cg with a, b.c ∈ K is in Annl(A), then xe = −bα + cβ, xf = aα − cγ
+and xg = −aβ + bγ are all zero. It follows that a = b = c = 0. Hence, we have
+Annl(A) = {0} and A0 = Annl(A) = {0}.
+Theorem 7.2. Let A be a zeropotent algebra A of dimension 3 with rank(A) ≥ 2 over
+K. Then, any weak multiplier of A is the scalar multiplication Sa for some a ∈ K,
+that is,
+M(A) = M ′(A) = LM(A) = LM ′(A) = {Sa | a ∈ K}.
+Proof. By Lemma 7.1 and Corollary 2.4, any weak multiplier T is a linear mapping.
+Let T ∈ L(A) be represented as (23). By Theorem 4.2, T is a weak multiplier, if and
+only if AT = T tA, if and only if
+
+
+pα − sβ
+qα − tβ
+rα − uβ
+−aα + sγ
+−bα + tγ
+−cα + uγ
+aβ − pγ
+bβ − qγ
+cβ − rγ
+
+ =
+
+
+−pα + sβ
+aα − sγ
+−aβ + pγ
+−qα + tβ
+bα − tγ
+−bβ + qγ
+−rα + uβ
+cα − uγ
+−cβ + rγ
+
+
+(25)
+holds. Suppose that α = ef, β = ge are linearly independent (the other cases are
+similar). Then, because pα − sβ = −pα + sβ by comparing the (1,1)-elements of two
+matrices in (25), we have p = s = 0. Comparing (1,2)-elements and (1,3)-elements,
+we have qα − tβ = aα − sγ = aα and rα − uβ = −aβ + pγ = −aβ respectively. It
+follows that a = q = u and r = t = 0. Comparing (2,2)-elements and (3,3)-elements,
+we see b = c = 0. Consequently, (23) holds if and only if
+b = c = p = r = s = t =
+0 and a = q = u, that is, T = Sa.
+In [4] we classify the zeropotent algebras of dimension 3 over an algebraically
+field K of characteristic not equal to 2. Up to isomorphism, we have 10 families of
+zeropotent algebras. They are
+Z0, Z1, Z2, Z3, {Z4(a)}a∈H, Z5, Z6, {Z7(a)}a∈H, Z8 and Z9
+defined by the structural matrices
+
+
+0
+0
+0
+0
+0
+0
+0
+0
+0
+
+ ,
+
+
+0
+0
+0
+0
+0
+0
+0
+0
+1
+
+ ,
+
+
+0
+0
+1
+0
+0
+0
+0
+0
+1
+
+ ,
+
+
+0
+1
+0
+−1
+0
+0
+0
+0
+0
+
+ ,
+
+
+0
+0
+0
+0
+1
+a
+0
+0
+1
+
+ ,
+
+
+0
+1
+0
+0
+0
+0
+0
+0
+1
+
+ ,
+
+
+0
+1
+1
+0
+0
+1
+0
+0
+1
+
+ ,
+
+
+1
+a
+0
+0
+1
+0
+0
+0
+1
+
+ ,
+
+
+1
+2
+2
+0
+1
+2
+0
+0
+1
+
+ and
+
+
+1
+3
+3
+0
+1
+3
+0
+0
+1
+
+ ,
+respectively.
+Z0 is the zero algebra, and Z1 is isomorphic to the 3-dimensional associative algebra
+C1, and their (weak) multipliers are already determined in Section 6. The algebras Z3
+to Z9 have rank greater or equal to 2, which are covered by Theorem 7.2.
+Thus, only A = Z2 is left to be analyzed. The multiplication table A of A is
+
+
+0
+g
+0
+−g
+0
+g
+0
+−g
+0
+
+. We see A0 = Annl(A) = Annr(A) = K(e + g), and we have the nihil
+decomposition A = A0 ⊕ A1 with A1 = Ke + Kf. A weak multiplier T ∈ M ′
+1(A) is a
+linear mapping represented by
+
+
+a
+b
+c
+p
+q
+r
+0
+0
+0
+
+ satisfying
+18
+
+
+
+pg
+qg
+rg
+−ag
+−bg
+−cg
+−pg
+−qg
+−rg
+
+ = AT = T tA =
+
+
+−pg
+ag
+pg
+−qg
+bg
+qg
+−rg
+cg
+rg
+
+ .
+Hence, a = −c = q and b = p = r = 0. Let Ta be this linear mapping, then by
+Theorem 3.1 we have
+M ′(A) = {Ta | a ∈ K} ⊕ (K(f + g))A
+and
+LM ′(A) =
+
+
+
+
+
+a + s
+t
+−a + u
+0
+a
+0
+s
+t
+u
+
+
+��� a, s, t, u ∈ K
+
+
+ .
+By Corollary 3.2, a weak multiplier T = Ta+R with R ∈ (K(e+g))A is a multiplier
+if and only if for any ζ = xe + yf and η = ze + vf with x, y, z, v ∈ K,
+R((xv − yz)g)
+=
+R(ζη) = ζTa(η) − Ta((xv − yz)g)
+=
+ζ(aη) + a(xv − yz)e = a(xv − yz)(e + g)
+holds. It follows that R(xg) = ax(e + g) for all x ∈ K. Let Sa be the scalar multi-
+plication by a, then (T − Sa)(A) ⊆ K(e + g) and (T − Sa)(Kg) = {0}. Hence, we
+obtain
+M(A) = {Sa | a ∈ K} ⊕ {R ∈ (K(e + g))A | R(Kg) = 0},
+and
+LM(A) =
+
+
+
+
+
+a + s
+t
+0
+0
+a
+0
+s
+t
+a
+
+
+��� a, s, t ∈ K
+
+
+ =
+
+
+
+
+
+a
+b
+0
+0
+c
+0
+a − c
+b
+c
+
+
+��� a, b, c ∈ K
+
+
+ .
+References
+[1] B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math.
+Soc., 14 (1964), 299–320.
+[2] E. Kaniuth, A Course in commutative Banach algebras, Springer, 2008.
+[3] Y. Kobayashi, K. Shirayanagi, M. Tsukada and S.-E. Takahasi, A complete clas-
+sification of three-dimensional algebras over R and C , Asian-European J. Math.,
+14 (2021) 2150131.
+[4] Y. Kobayashi, K. Shirayanagi, M. Tsukada and S.-E. Takahasi, Classification of
+three dimensional zeropotent algebras over an algebraically closed field, Commu-
+nication in Algebra, vol. 45, 2017, 5037—5052.
+[5] R. Larsen, An introduction to the theory of multipliers, Berlin, New York,
+Springer–Verlag, 1971.
+[6] T. Tsukada and et al,, Linear algebra with Python, Theory and Applications, to
+be published in Springer.
+[7] J. G. Wendel, Left Centralizers and Isomorphisms on group algebras, Pacific J.
+Math., 2 (1952), 251–261.
+[8] A. Zivari-Kazempour, Almost multipliers of Frechet algebras, The J. Anal., 28(4)
+(2020), 1075-1084
+[9] A. Zivari-Kazempour, Approximate θ-multipliers on Banach algebras, Surv. Math.
+Appl., 77 (2022), 79–88.
+19
+

69E1T4oBgHgl3EQfnASE/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e3994f04f9eba4ba72e6ef5c0d9f4e4a2ccda13c7b7e2da07ff103ba5f96f350
+size 3211309

6NFKT4oBgHgl3EQf_C4s/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:30658742d8b6d16bc3311299727d777fb4bbba8007252f784a62a51ee0a9fc95
+size 2031661

7tE1T4oBgHgl3EQf7QXS/content/tmp_files/2301.03533v1.pdf.txt

@@ -0,0 +1,889 @@
+arXiv:2301.03533v1  [hep-th]  9 Jan 2023
+FIAN/TD/16/2022
+Unfolded Point Particle as a Field in Minkowski Space
+A.A. Tarusov1,2 and M.A. Vasiliev1,2
+1 I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,
+Leninsky prospect 53, 119991, Moscow, Russia
+2 Moscow Institute of Physics and Technology, Institutsky pereulok 9, 141701,
+Dolgoprudny, Moscow region, Russia
+Abstract
+Point-particle dynamics is reformulated as a field theory. This is achieved by using
+the unfolded dynamics approach that makes it possible to give dynamical interpretation
+to the concept of physical dimension which is 1 for a point particle in the d-dimensional
+space-time. The main idea for the description of a k-dimensional on-shell system in
+the d-dimensional space is to keep the evolution along d − k dimensions off-shell or,
+alternatively, restrict it in a specific way respecting the compatibility conditions of the
+resulting unfolded system. The developed approach gives some hints how a non-linear
+realization of the symmetry G of a larger-dimensional space in a lower-dimensional
+system can emerge from a geometrical realization on the fields in an appropriate G-
+invariant space.
+For the example of a relativistic point particle considered in this
+paper, G is the Poincar´e group. The proposed general scheme is illustrated by simple
+examples that reproduce conventional results.
+1
+
+Table of contents
+1
+Introduction
+3
+2
+Unfolding
+3
+3
+Free particle
+6
+4
+Off-shell system
+8
+4.1
+General setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+4.2
+Covariant constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+5
+Examples of on-shell systems
+11
+5.1
+Lorentz force in a constant field . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+5.2
+Gravitational interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+5.3
+Interaction with higher spins . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+6
+Conclusion
+14
+2
+
+1
+Introduction
+Among different approaches to relativistic theories, one can distinguish between the
+world-line particle approach and the field-theoretic one. In this article, we unify these ap-
+proaches within the unfolded formulation of dynamical systems [1, 2], which will allow us to
+ascribe a dynamical sense to physical dimensions of a system [3, 4].
+An example of this phenomenon was given in [5], where, following Fronsdal’s prescription
+[6], an infinite system of massless fields of all spins in four-dimensional space has been
+described by a single field in the ten-dimensional space (Analogous results were achieved in
+the particle approach somewhat earlier in [7]).
+In this approach space-time geometry in which the dynamical equations are formulated is
+determined by the symmetry acting on the space, while the physical dimension is associated
+with the set of initial data, that determine the evolution of the system [3]. In the papers
+[5, 3, 7] (see also [8]) this idea was realized for the symmetry acting linearly on the fields in
+both the four- and ten-dimensional space.
+The description of the dynamics of point particles elaborated in this article assumes a
+non-linear realization of the Lorentz symmetry on the dynamic variables of the point particle
+as a consequence of the Einstein constraint on the velocity four-vector
+unun = 1 .
+(By selecting any un, that obeys this constraint, the Lorentz symmetry gets spontaneously
+broken.) The application of the unfolded formalism in this case differs significantly from the
+linear case. It requires the introduction of additional fields which encode unconstrained or
+specific evolution along ”extra” dimensions of space-time as is explained in this paper.
+The paper is organised as follows: Section 2 recalls the unfolded formalism used in the
+paper, illustrated by the well-known scalar field example. Section 3 details the application
+of that formalism to the case of a free point particle. In Section 4 an off-shell formulation
+of the unfolded particle dynamics is presented both in terms of component fields and in
+terms of generating functions.
+In Section 5 it is explained how an external force to the
+equations of motion can be introduced and a number of simple examples of the on-shell
+systems is considered.
+Section 6 contains brief conclusions with some emphasize on the
+further applications and open problems.
+2
+Unfolding
+The possibility of decreasing the order of a differential equation by introducing new
+variables and transitioning to an equivalent system of differential equations is well known.
+Extension of this approach to partial differential equations is based on the jet formalism [9].
+Also it was elaborated in the framework of BV-BRST formalism e.g. in [10, 11, 12, 13].
+Unfolded dynamics approach [1, 2] (see also [14]), that is most appropriate for the gauge
+theories in the framework of gravity, is a generalization of the first-order formulation of a
+system via replacing a partial derivative by de Rham derivative d := ξn
+∂
+∂xn and dynamical
+3
+
+variables by space-time differential forms W(x), which allows one to rewrite the system of
+equations in the form
+dW Ω(ξn, x) = GΩ(W(ξn, x)) ,
+(2.1)
+where ξn is the anticommuting differential used as a placeholder for dxn, and GΩ(W(ξ, x))
+is some function of W containing only exterior products of the differential forms W(ξ, x) at
+the same x (no space-time derivatives in GΩ(W(ξ, x)); wedge products are implicit). The
+functions GΩ(W(ξ, x)) cannot be arbitrary, as the de Rham derivative is nilpotent and thus
+the compatibility condition dGΩ = 0 must hold. This demands
+GΛ(W)∂GΩ(W)
+∂W Λ
+= 0 .
+(2.2)
+This constraint allows one to show that system (2.1) is manifestly invariant under the fol-
+lowing gauge transformations:
+δgaugeW Ω(ξn, x) = dǫΩ(ξn, x) + ǫΛ(ξn, x)∂GΩ(W(ξn, x))
+∂W Λ(ξn, x)
+,
+(2.3)
+where
+deg ǫΛ(ξn, x) = deg W Λ(ξn, x) − 1 ,
+with deg ω being a differential form degree of ω.
+Generally speaking, gauge invariance only takes place in so-called universal systems [15,
+16], in which the compatibility conditions hold as a consequence of the system itself without
+taking into account the number of space-time dimensions, i.e. the fact that d+1-forms vanish
+in d-dimensional space. Indeed, for non-universal systems partial derivative ∂GΩ
+∂W Λ might have
+no sense, leading to a non-zero derivative of zero represented by a d + 1-form.
+In other terms, the fact that GΛ(ξn, x) is a function of W Ω(ξn, x) can be written as
+GΛ =
+∞
+�
+n=1
+f Λ
+Ω1,...ΩnW Ω1...W Ωn ,
+f Λ
+Ω1,...Ωk,Ωk+1,...Ωn = (−1)degΩkdegΩk+1f Λ
+Ω1,...Ωk+1,Ωk,...Ωn .
+(2.4)
+(From now on we omit arguments of GΛ and W Ω if it does not lead to misunderstandings.)
+Compatibility condition (2.2) then yields generalized Jacobi identities on the structure con-
+stants f
+m
+�
+n=0
+(n + 1)f Λ
+[Ω1,...Ωm−nf Φ
+Λ,Ωm−n+1,...Ωm} = 0 ,
+(2.5)
+where [} indicates an appropriate (anti)symmetrisation of indices. From this point of view,
+the universality of a system means that the generalized Jacobi identities hold true regardless
+of the dimension d. The underlying mathematical structure is called the strong homotopy
+L∞ algebra [17].
+For universal systems it is also possible to introduce a Q-differential (homological vector
+field) of the form [16]
+Q = GΩ
+∂
+∂W Ω ,
+(2.6)
+4
+
+which turns out to be nilpotent,
+Q2 = 0 ,
+as a consequence of compatibility conditions (2.2). In these terms, any universal unfolded
+system can be rewritten as
+dF(W) = QF(W) ,
+where F(W) is an arbitrary function.
+This way of describing the system as a so-called
+Q-manifold relates the de Rham derivation on the world-sheet with coordinates xn to the
+derivation Q on the target space with coordinates W Λ.
+Unfolded formalism allows for a natural way of description of background geometry via
+Maurer–Cartan equations which have the unfolded form. Indeed, let g be a Lie algebra.
+Setting W = w and G = 1
+2[w , w] for a one-form w ∈ g, one observes that equation (2.1)
+yields the zero-curvature condition
+dw + 1
+2[w , w] = 0 .
+(2.7)
+For g being Poincar´e algebra with the one-form gauge fields (connection) en and wnm, the
+unfolded system (2.7) yields the coordinate-independent description of Minkowski space in
+the form
+Rn = 0 ,
+Rnm = 0 .
+(2.8)
+Consider a system of equations
+DCA(x) = 0 ,
+where the fields CA(x) in Minkowski space are valued in a Poincar´e-module V while D is
+the exteriour covariant derivative in V with the flat connection w = enPn +ωnmLnm obeying
+(2.7): D = d + w. (For instance, Lorentz transformations act on the tensor indices.) This
+system is invariant under the following gauge transformations:
+δCA = −εA
+BCB,
+(2.9)
+δw(x) = Dε(x).
+(2.10)
+From here it follows that for a fixed w = w0 the gauge symmetry parameters are restricted
+by the condition
+D0ε(x) = 0 ,
+D0 = d + w0.
+(2.11)
+Since D2
+0 = 0, in the topologically trivial case the zero-form parameter εBA, can be recon-
+structed from any point hence generating global symmetries of the system.
+Minkowski space in Cartesian coordinates is described by the connection
+en = dxn ,
+ωnm = 0 ,
+(2.12)
+in which case (2.11) yields
+∂nεn − εn
+men
+m = 0 ,
+∂nεnm = 0 ,
+(2.13)
+which can be solved as εnm = −εmn = const, εn = εnmxm + ǫn, where ǫn is x-independent.
+This obviously forms the Poincar´e transformations.
+5
+
+3
+Free particle
+The classical point particle is conventionally described by generalized coordinates qi(s)
+depending on the evolution parameter s. In this paper, we also use the generalized coor-
+dinates qi = qi(x) which, however, will be treated as space-time fields, i.e., functions of all
+space-time coordinates xn. Let us, for simplicity, work with Cartesian coordinates xn of a
+flat Minkowski space, which will allow us to omit the Lorentz connection dependent terms.
+We start with a first-step equation
+Dqi(x) = ejqi
+j(x) ,
+(3.1)
+where qij(x) is an arbitrary (for now) matrix and D is the Lorentz covariant derivative. In
+Cartesian coordinates this yields
+dqi(x) = ejqi
+j(x).
+(3.2)
+At j = 0 one arrives at a differential equation with respect to time t = x0 with an
+arbitrary right hand side.
+However, in contrast with classical dynamics, other values of
+j also produce non-trivial equations, which means that the equations of motion involve
+all space-time variables. This unusual modification makes sense when treating generalized
+coordinates as embedding functions from our laboratory system to some other. In this case
+the space sector of the right hand side is just a Jacobian of that transformation.
+It is
+convenient to assume that the dimensions of the original and target spaces are the same,
+thus the space sector of our matrix has to be non-degenerate. Apart from the dependence
+on ”extra” variables, this is similar to the description of a classical particle in terms of the
+transformation from a chosen reference frame to the one in which the particle is at rest.
+The analysis does not stop here though, since qij must satisfy the compatibility condition
+ejDqi
+j(x) = 0 .
+(3.3)
+The general solution to this condition has the form of a one-form with tensor coefficients that
+are symmetric in lower indices, which solves (3.3) due to anticommutativity of the one-forms
+en, eiej = −ejei,
+Dqi
+j(x) = ekqi
+jk(x) ,
+qi
+jk(x) = qi
+kj(x) .
+(3.4)
+Equation (3.4) also produces the compatibility condition which has the analogous form
+Dqi
+jk(x) = elqi
+(jkl)(x) .
+(3.5)
+The process can be continued resulting in the infinite set of equations
+Dqi
+(j1...jn)(x) = ekqi
+(j1...jnk)(x) .
+(3.6)
+This system provides an example of an off-shell unfolded system that does not describe
+any non-trivial equations of motion. It is fully analogous to that described in [18] for the
+scalar field case. (Supersymmetric extensions of the off-shell unfolded systems were recently
+6
+
+considered in [19].) Every equation expresses the compatibility of the previous one, but
+involves a new object that requires its own compatibility condition. To get nontrivial dy-
+namics, however, one has to introduce additional conditions on the coefficients qi(j1...jnk)
+describing higher derivatives of the field qi. Imposing constraints on the fields qi(j1...jnk) is
+equivalent to imposing some differential equations on the fields qi.
+The equations (3.6) admit a more compact form using auxiliary variables yi, so that the
+right hand side of the equations results from differentiation of the generating functions with
+respect to yi,
+Dqi
+(j1...jn)(x, y) = ek d
+dykqi
+(j1...jn)(x, y) ,
+(3.7)
+where qi(x, y) is the generating function
+qi(x, y) =
+∞
+�
+n=0
+1
+n!qi
+j1,...jn(x)yj1...yjn ,
+(3.8)
+with the original field qi(x) recovered at y = 0.
+To describe a free relativistic particle this way we introduce a time-like “velocity 4-vector”
+V i(x) as a new variable, imposing the equations
+Dqi(x) = ejVj(x)V i(x),
+(3.9)
+DV i(x) = 0,
+(3.10)
+V i(x)Vi(x) = 1.
+(3.11)
+Let us show that this system indeed describes a free relativistic particle. In Cartesian
+coordinates the system takes the form
+ej ∂
+∂xj qi(x) = ejVj(x)V i(x),
+(3.12)
+ej ∂
+∂xj V i(x) = 0.
+(3.13)
+Here the second equation implies that V i is a constant while the first one contains a one-form
+κ = eiVi ,
+(3.14)
+which, in a sense, serves as a projector on the world line of the particle.
+There is some freedom in the parametrization of the world line. For example, to take
+time x0 as the evolution parameter, one has to reduce dxn to dx0 (equivalently, en → dxnδ0
+n).
+This reproduces the familiar equations of motion. Indeed, after such a reduction, the velocity
+vector produces a factor of V0, which is just a relativistic gamma-factor γ = (1 − v2
+c2 )−1/2.
+This is not surprising, since the differentiation on the left hand side is over laboratory time,
+˙qi(x) = γV i(x),
+(3.15)
+˙V i(x) = 0.
+(3.16)
+7
+
+Thus, equations (3.9)-(3.11) indeed describe propagation of a free point particle with the
+4-velocity V i(x).
+Since our unfolded system can be easily extended to include the Lorentz connection by
+appending (2.7), it inherits the full Poincar´e symmetries as outlined at the end of Section
+2.
+The unfolded formalism allows us to straightforwardly derive the symmetries of the
+system. In this case (2.3) generates the background Poincar´e transformation as well as a
+transformation of qi
+δqi = 0ǫj ∂ekVkV i
+∂ej
+= 0ǫjVjV i .
+(3.17)
+Note that in this formalism nontrivial particle dynamics is only along the direction asso-
+ciated with κ. In other (transversal) directions the dynamics is trivial with no dependence
+on the other coordinates.
+4
+Off-shell system
+4.1
+General setup
+The world-line one-form κ (3.14) makes it possible to formulate the off-shell unfolded
+system of a specific form distinguishing between the directions along κ and transversal ones.
+The evolution of the system in the transversal directions is necessary for consistency. Indeed,
+the naive system
+Dqi(x) = κV i(x) ,
+DV i(x) = κF i(x) ,
+(4.1)
+is inconsistent for arbitrary F i because now κ is not closed
+Dκ = −eiDVi = −eiκFi .
+(4.2)
+While the classical behavior of the system is defined by the terms aligned with V i, to
+achieve compatibility in all directions one has to adjust the evolution along the transversal
+directions appropriately.
+To achieve this it is convenient to introduce the transversal one-forms
+ηi := ei − κV i
+V 2 ,
+V iηi = 0 ,
+(4.3)
+where an additional normalization is introduced, since the condition (3.11) is relaxed, as it
+is not necessarily true off-shell. (Still we assume that V 2(x) := V i(x)Vi(x) ̸= 0.) The system
+then takes the form
+Dqi = κV i(x) + ηjHi
+j(x),
+(4.4)
+DV i = κF i(x) + ηjGi
+j(x).
+(4.5)
+Since ηi is V i-transversal, this system is invariant under the “gauge” transformations
+H
+′i
+j(x) = Hi
+j(x) + φi(x)Vj(x),
+(4.6)
+G
+′i
+j(x) = Gi
+j(x) + ψi(x)Vj(x) ,
+(4.7)
+8
+
+with arbitrary functions φi(x), ψi(x), that can be gauge fixed by demanding
+Hi
+jV j = 0 ,
+Gi
+jV j = 0 .
+(4.8)
+Firstly, let us note that the system is indeed off-shell as long as the condition V iVi = 1
+is not enforced. Indeed, the left hand sides contain d2 components of first derivatives of
+qi(x) (or V i(x) for the second equation). On the right hand side, the Hij (Gij) contain
+d(d−1) components due to the transversality condition while the V i (F i) span the d leftover
+components.
+To check the compatibility conditions of this system, one has to act by D on the both
+sides of the equations then solving them with respect to Gij and Hij. We analyze the system
+in an arbitrary torsion free geometry with Dei = 0, which yields
+Dκ = −eiκFi − eiηjGij,
+(4.9)
+Dηi = 1
+V 4
+�
+(ekκFkV i + ekηjGkjV i + κηjGi
+j)V 2 − 2κV iVkηjGk
+j
+�
+.
+(4.10)
+The compatibility of (4.4) yields using DDAi = RikAk = elejRik,ljAk .
+DDqi = (Dκ)V i(x) − κDV i(x) + D(ηj)Hi
+j(x) − ηjDHi
+j(x) =
+= −V iejκFj − V iekηjGkj − κηjGi
+j + 1
+V 2κηlGj
+lHi
+j − ηjDHi
+j = elejRi
+k,ljqk .
+(4.11)
+Expanding the last equation in the basis two-forms κηi and ηiηj, we obtain
+(4.12)
+ηjDHi
+j = −V iηjκFj − 1
+V 2V iκV kηjGkj − V iηkηjGkj − κηjGi
+j
++ 1
+V 2κηlGj
+lHi
+j − 2
+V 2κV lηjRi
+k,ljqk − ηlηjRi
+k,ljqk ,
+which is equivalent to
+(4.13)
+DHi
+j = κ(−FjV i + Gi
+j + 1
+V 2V iV kGkj − 1
+V 2Gk
+jHi
+k + 2
+V 2V lRi
+k,ljqk)
++ ηkGkjV i + ηlRi
+k,ljqk + ηkAi
+jk + κVjBi + VjηkCi
+k,
+where the last three terms with arbitrary Bi, Cik and symmetric Aijk = Aikj parameterize
+the general solution of the homogeneous equation ηjDHij = 0. Just as for Hij itself, the
+transversality condition can be imposed on Aijk and Cik,
+Ai
+jkV k = 0 ,
+Ci
+kV k = 0 .
+(4.14)
+Analogously for equation (4.5), with the only difference that we now impose the unfolded
+equations on the field F i,
+DF i = κJi + ηjKi
+j
+(4.15)
+9
+
+again demanding KijV j = 0. Then the compatibility condition for V i yields
+DDV i = (Dκ)F i(x) − κ(DF i) + (Dηj)Gi
+j − ηj(DGi
+j) =
+= −ejκFjF i − elηjGljF i − κηjKi
+j + 1
+V 2κηkGj
+kGi
+j − ηjDGi
+j = elejRi
+k,ljV k ,
+(4.16)
+or, equivalently,
+(4.17)
+ηjDGi
+j = ηjκ(−FjF i + Ki
+j + 1
+V 2F iV kGkj − 1
+V 2Gk
+jGi
+k + 2
+V 2V lRi
+k,ljV k)
++ ηjηk(GkjF i + Ri
+l,kjV l).
+The solution again consists of the inhomogeneous part and the terms parameterizing a
+general solution of the homogeneous equation,
+(4.18)
+DGi
+j = κ(−FjF i + Ki
+j + 1
+V 2F iV kGkj − 1
+V 2Gk
+jGi
+k + 2
+V 2V lRi
+k,ljV k)
++ ηk(GkjF i + Ri
+l,kjV l) + ηlMi
+jl + κVjNi + VjηkLi
+k
+with Mijl = Milj. Once again, Mijl and Lik obey the transversality conditions
+Mi
+jkV k = 0 ,
+Li
+kV k = 0 .
+(4.19)
+In its turn, consistency of equations (4.13) and (4.18) imposes differential constraints
+on the yet unconstrained coefficients A, B, C and M, N, L in terms of new unconstrained
+variables. This process continues indefinitely leading eventually to a totally consistent infinite
+set of equations on the infinite set of variables. Since the analysis of all these conditions in
+terms of component fields like H, G, A, B, C, M, N, L quickly gets complicated we now revisit
+them in a more compact form of generating functions.
+4.2
+Covariant constraints
+Though the description of a point particle considered in Section 4.1 is clear in principle
+it is algebraically involved and not instructive. It can be simplified at least in Minkowski
+background by imposing appropriate constraints in terms of generating functions of Section
+3. To this end, we introduce auxiliary variables yi as in (3.7), rewriting the system (4.4),
+(4.5) as
+Dqi(x, y) = ej d
+dyj qi(x, y),
+(4.20)
+DV i(x, y) = ej d
+dyj V i(x, y).
+(4.21)
+10
+
+These equations are clearly consistent, as derivatives commute while the vielbein one-forms
+anticommute. They do not describe any dynamics, imposing no conditions on qi(x, 0) and
+V i(x, 0). The results of Section 4.1 can be reproduced by imposing the following conditions:
+V i(x, y) d
+dyiqj(x, y) = V 2(x, y)V j(x, y) .
+(4.22)
+One has to check that this constraint is compatible with (4.20), (4.21), i.e. its differenti-
+ation does not produce new constraints, giving zero by virtue of (4.22). Indeed,
+D
+�
+V i(x, y) d
+dyiqj(x, y)−V 2(x, y)V j(x, y)
+�
+= ek d
+dyk
+�
+V i(x, y) d
+dyiqj(x, y)−V 2(x, y)V j(x, y)
+�
+= 0,
+(4.23)
+(In the sequel, the arguments of the generating functions qi(x, y), V i(x, y) are implicit.)
+Let us now show that supplemented with constraint (4.22) equations (4.20), (4.21) repro-
+duce the equations from the previous section. By virtue of (4.22), and since ei = κV i
+V 2 + ηi,
+Eq. (4.20) yields
+Dqi = κV i + ηj d
+dyj qi .
+(4.24)
+Equation (4.4) is reproduced with ηj d
+dyj qi|y=0= ηjHij. To fix an obvious freedom up to a
+function φiVj in a way preserving trasversality one can set
+Hi
+j =
+� d
+dyj qi − 1
+V 2VjV k d
+dykqi����
+y=0 .
+Analogously, (4.21) yields equation (4.5) with
+V j d
+dyj V i���
+y=0 = F i ,
+(4.25)
+� d
+dyj V i − 1
+V 2VjV k d
+dyk V i����
+y=0 = Gi
+j .
+(4.26)
+Let us note that the second derivative of the generating function has d2(d+1)
+2
+independent
+components, of which, keeping in mind the transversality conditions, d2(d+1)
+2
+−d2 are encoded
+in Aijk, d2 − d in Cik and d more in Bi. That means that the system (4.20)-(4.22) indeed
+concisely reproduces the off-shell formulation of Section 4.1 in all orders.
+5
+Examples of on-shell systems
+To put the system on-shell one has to set the field F i, that determines the evolution of
+V i along itself, to some function F i(q, V ). Restriction of some combination of derivatives
+parameterized by F i then would impose some partial differential equations on qi giving rise
+to the equations of motion. Generally, a non-zero force F i(q, V ) would demand some higher
+11
+
+components of additional fields associated with the higher components in yj of qi(x, y) and
+V i(x, y) (descendants) to be nonzero. There are two somewhat opposite options.
+One is that all these descendants are kept non-zero and arbitrary in the sense that they
+parameterize a general solution to the compatibility conditions. Another one is that these
+descendants give as simple as possible specific solution to the compatibility conditions. In the
+former case the system turns out to be off-shell in all directions transversal to the trajectory.
+In the latter, the evolution along transversal directions has a specific form compatible with
+F i(q, V ) ̸= 0 in the full unfolded system. Postponing a general analysis of this issue for the
+future publication here we consider a few simple examples of the second kind.
+5.1
+Lorentz force in a constant field
+A particular choice of F i(q, V ) linear in V , F i(q, V ) = F ij(q)V j, Fij = −Fji, replicates
+the Lorentz force.
+As a toy example consider a particular solution to the compatibility
+conditions of (4.4), (4.5) in flat space, that easily puts the system on-shell. Namely, let
+F ij(q) be a constant field, i.e. dF ij = 0. Antisymmetry of Fij allows us to impose condition
+(3.11) and write down the following on-shell system:
+dqi(x) = κV i + ηi = ei,
+(5.1)
+dV i(x) = κV jF i
+j + ηjF i
+j = ejF i
+j,
+(5.2)
+which is obviously consistent without introducing higher components in yj of qi(x, y) and
+V i(x, y).
+Note that the free particle case considered in Section 3 is reproduced at F i = 0 and
+also corresponds to the specific (trivial) choice of the descendants associated with higher
+components in yj of qi(x, y) and V i(x, y).
+5.2
+Gravitational interaction
+Within the exterior algebra formalism underlying the unfolded dynamics approach, the
+gravitational background is naturally taken into account by using appropriate covariant
+derivatives of the Cartan formulation of gravity. To introduce it in the metric formalism, i.e.
+with Christoffel symbols, one has to distinguish between laboratory Lorentz indices denoted
+by Latin letters and the underlined world sheet indices. For instance, V i = eiiV i, where the
+vielbein eii relates laboratory and world indices. Let us start with the Cartan formulation.
+The on-shell covariant condition (4.5) for V i with zero force reads as
+∂kV i = −ωk
+i
+jV j + ηk
+jGi
+j .
+(5.3)
+On-shell, it is possible to impose the condition (3.11), compatibility with which then implies
+the anticipated antisymmetry of ω,
+ωk
+i
+j = −ωk
+j
+i .
+(5.4)
+12
+
+Here the higher-order compatibility with Gij = 0 is easily achieved for the case of flat
+(zero-curvature) gravitational fields while the general case demands some Gij ̸= 0.
+It is not difficult to see that the condition (5.3), after applying the frame postulate
+∂kel
+i − Γi
+klei
+i + ωk
+i
+jel
+j = 0 ,
+(5.5)
+can be equivalently rewritten, leaving out the derivatives of the vielbein, i.e only in terms of
+V i and the Christoffel symbols.
+∂kV i = (∂kei
+i)V i + ei
+i∂kV i = −ωk
+i
+jei
+jV i + ηk
+jGi
+j =⇒ ∂kV i = Γi
+klV l + ηk
+jGi
+j .
+(5.6)
+From here it is possible to use the metric formalism in the equations for V i
+dV i = κV k∂kV i + ηk∂kV i = κΓi
+klV kV l + ηjGi
+j ,
+(5.7)
+where
+∂k = ek
+k∂k .
+(5.8)
+After the vielbein reduction on V : P(ei) = κV i, one gets the expected geodesic equation.
+The example above of the link between spin-connection and Christoffel symbols realizes
+the transition between worldsheet and fiber indices. This highlights the difference between
+the usual dynamics formulated in terms of xi, ˙xk and our unfolded system formulated in
+terms of qi, V i.
+The relation between the two formalisms can be uplifted to the action level. As noted in
+Section 3, after an appropriate reduction of the vielbein, Dqi becomes dqi
+dτ ((3.15), (3.16)),
+where τ is the natural evolution parameter. Using that qi(x) are embedding functions for
+the coordinates xi, let us write an action, quadratic in dqi
+dτ using the metric gij in the target
+space (not necessarily corresponding to Minkowski’s space) and adding the gauge parameters
+α for reparametrization invariance by α′dτ ′ = αdτ,
+S = 1
+2
+�
+dτα
+� 1
+α2gij
+dqi
+dτ
+dqj
+dτ + m2�
+.
+(5.9)
+In terms of xi, we obtain the regular action
+S = 1
+2
+�
+dτα
+� 1
+α2gij
+dqi
+dxk
+dqj
+dxl
+dxk
+dτ
+dxl
+dτ + m2�
+= 1
+2
+�
+dτα
+� 1
+α2 ˜gkl
+dxk
+dτ
+dxl
+dτ + m2�
+.
+(5.10)
+Here ˜gkl is the induced metric from the target space. As usual, Euler-Lagrange equations
+for α are algebraic
+α = 1
+m
+�
+˜gkl
+dxk
+dτ
+dxl
+dτ ,
+(5.11)
+which allows us to substitute them back into the action to arrive at the conventional result
+S = m
+�
+dτ
+�
+˜gkl
+dxk
+dτ
+dxl
+dτ .
+(5.12)
+13
+
+5.3
+Interaction with higher spins
+Note that the condition (5.3) allows a direct generalization onto higher-spin interactions
+via introducing an appropriate higher-spin connection ωn1,...ns−1,m = dxkωkn1,...ns−1,m [20],
+dV i = −ωn1,...ns−1,
+iV n1...V ns−1 + ηjGi
+j .
+(5.13)
+In this case, the compatibility with the constraint (3.11) demands ω(n1,...ns−1,m) = 0, which
+means that, in agreement with the general higher-spin theory [20], ωn1,...ns−1,m is described
+by the Young diagram n1 ... ns−1
+m
+.
+Note that the force associated with the field of an arbitrary spin can also be written in
+terms of generalized Christoffel symbols [21] (see also [22, 23]).
+6
+Conclusion
+In this paper, we suggest an approach to the description of a relativistic classical point
+particle as a field on which relativistic symmetries act geometrically. This is achieved by
+rewriting equations in the unfolded formalism that supports manifest invariance under dif-
+feomorphisms and the Lorentz group. The point particle is represented as a field obeying
+unfolded equations.
+A mechanism of projectors specifying the evolution parameter in a
+covariant way is introduced.
+The proposed approach can be useful for different types of theories including the double
+field theory, [24, 25] where, as we hope, it can be used to provide an alternative way of
+enforcing the section constraint (see, e.g., [26].) More generally, the description of lower-
+dimensional objects within a proper extension of the proposed approach is of great interest,
+in particular, for the description of branes in superstring theory as well as string theory
+itself: it would be interesting to reformulate the string theory as a 2d theory described from
+the start in terms of fields in the target space. It would also be interesting to analyze the
+relation of the suggested mechanism with the models with non-linearly realized symmetries
+[27, 28].
+It should be stressed that in the presence of extra dimensions the unfolded dynamics
+solely along the parameters associated with the particle trajectory does not allow for com-
+patible unfolded equations demanding an evolution along the transverse directions respecting
+appropriate compatibility conditions. This raises a number of questions for the future study
+such as, for instance, whether any solutions of the compatibility conditions can be associated
+with an evolution along a one-dimensional trajectory and, if not, what are the sufficient con-
+ditions for this to be true? A related interesting problem is to obtain the on-shell conditions
+from the variational principle along the lines of [29].
+14
+
+Acknowledgement
+We are grateful to Ruslan Metsaev for the correspondence. This work was supported by
+the Russian Basic Research Foundation Grant No 20-02-00208.
+References
+[1] M. A. Vasiliev, Phys. Lett. B 209 (1988) 491.
+[2] M. A. Vasiliev, Annals Phys. 190 (1989) 59.
+[3] M. Vasiliev, [arXiv:hep-th/0111119 [hep-th]].
+[4] M. Vasiliev, Lect. Notes Phys. 892 (2015), 227-264 [arXiv:1404.1948 [hep-th]].
+[5] M. Vasiliev, Phys. Rev. D 66 (2002), 066006 [arXiv:hep-th/0106149 [hep-th]].
+[6] C. Fronsdal, “Massless Particles, Ortosymplectic Symmetry and Another Type of
+Kaluza-Klein Theory”, Preprint UCLA/85/TEP/10, in Essays on Supersymmetry, Rei-
+del, 1986 (Mathematical Physics Studies, v.8).
+[7] I.
+Bandos,
+J.
+Lukierski
+and
+D.
+Sorokin,
+Phys.Rev.
+D61
+(2000)
+045002,
+[hep-th/9904109].
+[8] I. Bandos, X. Bekaert, J. de Azcarraga, D. Sorokin and M. Tsulaia, JHEP 05 (2005),
+031 [arXiv:hep-th/0501113 [hep-th]].
+[9] Vinogradov, A.M. Geometry of nonlinear differential equations. J Math Sci 17,
+1624–1649 (1981). https://doi.org/10.1007/BF01084594 .
+[10] I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B 69 (1977), 309-312.
+[11] G. Barnich, M. Grigoriev, A. Semikhatov and I. Tipunin, Commun. Math. Phys. 260
+(2005), 147-181 [arXiv:hep-th/0406192 [hep-th]].
+[12] G. Barnich and M. Grigoriev, JHEP 01 (2011), 122 [arXiv:1009.0190 [hep-th]].
+[13] M. Grigoriev, JHEP 12 (2012), 048 [arXiv:1204.1793 [hep-th]].
+[14] N. Boulanger, C. Iazeolla and P. Sundell, JHEP 07 (2009), 013 [arXiv:0812.3615 [hep-
+th]].
+[15] X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, hep-th/0503128.
+[16] M. A. Vasiliev, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 37 [hep-th/0504090].
+[17] T.
+Lada
+and
+J.
+Stasheff,
+Int.
+J.
+Theor.
+Phys.
+32
+(1993),
+1087-1104
+[arXiv:hep-th/9209099 [hep-th]].
+[18] O. Shaynkman and M. A. Vasiliev,
+Theor. Math. Phys. 123 (2000),
+683-700
+[arXiv:hep-th/0003123 [hep-th]].
+[19] N. G. Misuna, [arXiv:2201.01674 [hep-th]].
+[20] M. A. Vasiliev, Yad. Fiz. 32 (1980) 855 [Sov. J. Nucl. Phys. 32 (1980) 439].
+15
+
+[21] B. de Wit and D. Z. Freedman, Phys. Rev. D 21, 358 (1980).
+[22] A. Y. Segal, [arXiv:hep-th/0008105 [hep-th]].
+[23] I. Bars and C. Deliduman, Phys. Rev. D 64 (2001), 045004 [arXiv:hep-th/0103042
+[hep-th]].
+[24] W. Siegel, Phys. Rev. D 48, 2826-2837 (1993).
+[25] W. Siegel, Phys. Rev. D 47, 5453-5459 (1993).
+[26] O. Hohm, E. T. Musaev and H. Samtleben, JHEP 10 (2017), 086 [arXiv:1707.06693
+[hep-th]].
+[27] E. Ivanov and V. Ogievetsky, Teor. Mat. Fiz. 25 (1975), 164-177.
+[28] E. Ivanov and A. Kapustnikov, J. Phys. A 11 (1978), 2375-2384.
+[29] A. A. Tarusov and M. A. Vasiliev, Phys. Lett. B 825 (2022), 136882 [arXiv:2111.12691
+[hep-th]].
+16
+

89E5T4oBgHgl3EQfQw7M/content/tmp_files/2301.05516v1.pdf.txt

@@ -0,0 +1,4257 @@
+arXiv:2301.05516v1  [math.PR]  13 Jan 2023
+CLT for real β-Ensembles at High Temperature∗
+Charlie Dworaczek Guera†, Ronan Memin‡
+Abstract
+We establish a central limit theorem for the fluctuations of the empirical measure in the beta-
+ensemble of dimension N at a temperature proportional to N and with convex, smooth potential.
+The space of test functions for which the CLT holds includes C1, vanishing functions at infinity. It is
+obtained by the inversion of an operator which is a pertubation of a Sturm-Liouville operator. The
+method that we use is based on a change of variables introduced in [BFG15] and in [Shc14].
+Contents
+1
+Introduction and main result
+1
+2
+Regularity of the equilibrium measure and Hilbert transform
+6
+3
+Concentration inequality, proof of Theorem 1.5
+9
+4
+Localization of the edge of a configuration
+14
+5
+Laplace transform for smooth test functions, proof of Theorem 1.3
+17
+6
+Inversion of L
+22
+7
+Regularity of the inverse of L and completion of the proof of Theorem 1.3
+26
+A Appendix: proof of Theorem 6.2
+29
+1
+Introduction and main result
+The beta-ensemble of dimension N ⩾ 1 with parameter β > 0 and potential V is the probability measure
+on RN given by
+dPβ,V
+N (x1, . . . , xN) =
+1
+ZN(V, β)
+�
+i<j
+|xi − xj|βe−�N
+i=1 V (xi)dx1 . . . dxN .
+(1)
+The potential V has to be chosen so that the partition function
+ZN(V, β) =
+ˆ
+RN
+�
+i<j
+|xi − xj|βe−�N
+i=1 V (xi)dx1 . . . dxN
+∗This project has received funding from the European Research Council (ERC) under the European Union Horizon 2020
+research and innovation program (grant agreement No. 884584).
+†Université de Lyon, ENSL, CNRS, France
+email: charlie.dworaczek@ens-lyon.fr
+‡Université de Lyon, ENSL, CNRS, France
+email: ronan.memin@ens-lyon.fr
+1
+
+is finite. This is the case for example if for some β′ > max(1, β),
+lim inf
+|x|→∞
+V (x)
+Nβ′ ln |x| > 1 ,
+(2)
+see [AGZ10, equation (2.6.2)]. The parameter β, which is allowed to depend on N, is the so-called inverse
+temperature.
+Under the special choice V (x) = x2
+2 , the measure (1) can be seen as the joint law of the (unordered)
+eigenvalues of certain matrix models:
+• For β = 1 (resp. β = 2), it is the law of the eigenvalues of the Gaussian Orthogonal Ensemble
+(resp. Gaussian Unitary Ensemble), [AGZ10][Theorem 2.5.2].
+• For general β > 0, potentially depending on N, it is the law of the spectrum of certain tri-diagonal
+random matrices as shown by Dumitriu and Edelman in [DE02].
+We consider here the high temperature regime where β scales as 1/N, and write β = 2P
+N for some
+P > 0. The corresponding measure is therefore
+dPV,P
+N (x1, . . . , xN) =
+1
+ZV,P
+N
+�
+i<j
+|xi − xj|
+2P
+N e−�N
+i=1 V (xi)dx1 . . . dxN ,
+(3)
+with partition function
+ZV,P
+N
+=
+ˆ
+RN
+�
+i<j
+|xi − xj|
+2P
+N e−�N
+i=1 V (xi)dx1 . . . dxN .
+(4)
+It was shown in [GZ19] that under PV,P
+N , the sequence of empirical measures
+ˆµN = 1
+N
+N
+�
+i=1
+δxi
+satisfies a large deviation principle at speed N with strictly convex, good rate function. As a consequence,
+ˆµN converges almost surely in distribution towards a deterministic measure µV
+P as N goes to infinity,
+meaning that almost surely, for every bounded continuous f : R → R,
+ˆ
+R
+fdˆµN −→
+N→∞
+ˆ
+R
+fdµV
+P .
+The limiting measure µV
+P can be seen to have a density ρV
+P which satisfies for almost every x ∈ R
+V (x) − 2P
+ˆ
+R
+ln |x − y|ρV
+P (y)dy + ln ρV
+P (x) = λV
+P ,
+(5)
+where λV
+P is constant (see [GM22, Lemma 3.2] for example).
+The β-ensemble in the regime βN −→
+N→∞ 2P > 0 has drawn a lot of attention from the random matrix
+and statistical physics communities lately. The limiting density was first described in the case of the
+quadratic potential in [ABG12], as a crossover between the Wigner semicircle law (fixed β > 0 case) and
+the Gaussian density (case β = 0). The fluctuations of the eigenvalues in the bulk and at the edge of a
+configuration were studied for example in [BGP15],[NT18],[NT20],[Pak18], [Lam21]. These fluctuations
+were shown to be described by Poisson statistics in this regime. Recently, Spohn uncovered in [Spo20]
+a link between the study of the Classical Toda chain and the β-ensemble in this regime, the result was
+later extended to more general potentials in [GM22]. It is explained in [Spo21][Section 6] how the central
+2
+
+limit theorem for the empirical measure in the high temperature β ensemble is linked with the currents
+of the Toda chain.
+The Central Limit Theorem for the fluctuations of the linear statistics of beta-ensembles was first
+established by [Joh98] for β = 2 polynomial potential, then generalized and further developed in the
+regime where β is fixed in [Shc13], [BG13a], [BG13b], [BLS18]. Also an optimal local law was found
+in this regime in [BMP22]. The CLT was obtained in the high-temperature regime βN → 2P > 0 by
+Nakano and Trinh in [NT18, Theorem 4.9] for quadratic V , relying on the tridiagonal representation for
+the beta-ensemble with quadratic potential in [DE02]. In [HL21], the authors prove the CLT in the case
+of the circular beta-ensemble at high temperature with general potential, using a normal approximation
+method involving the spectral analysis of an operator associated to the limiting covariance structure.
+Their method allowed them to derive a Berry-Esseen bound, i.e. a speed of convergence of the fluctuations
+towards a Gaussian variable.
+In this paper, we adapt part of the arguments of [HL21] to our setup. More precisely, we show that
+for a class of regular, convex potentials V satisfying a growth condition of the type
+lim
+|x|→∞
+V ′′(x)
+V ′(x)2 = 0 ,
+denoting νN = ˆµN − µV
+P and considering test functions f belonging to the range of a certain integro-
+differential operator, the scaled fluctuations of ˆµN, defined by
+√
+NνN(f) :=
+√
+N
+�ˆ
+R
+fdµN −
+ˆ
+R
+fdµV
+P
+�
+,
+converge in law towards centered Gaussian law with variance depending on f.
+When considering the fixed temperature regime, i.e. β fixed, one has to renormalize the xi’s by
+√
+N.
+It is shown in [AGZ10][Theorem 2.6.1] that the measure
+1
+N
+N
+�
+i=1
+δxi/
+√
+N
+satisfies a large deviation principle, and the limiting measure is characterized in [AGZ10][Lemma 2.6.2]
+by an equation similar to (5). In fact, the term ln ρV
+P in the left-hand side of (5) is the only difference in
+the equation characterizing the limiting measure in the fixed β case. We point out the very similar char-
+acterization of the equilibrium measure corresponding to the minimization problem arising in [BGK16].
+There again, the limiting measure is compactly supported. The term ln ρV
+P is of prime importance because
+its presence implies that the support of ρV
+P is the whole real line. It leads to technicalities to deal with
+the behavior at infinity of most of the associated objects, namely dealing with weighted Lebesgue spaces
+L2(ρV
+P ) and the corresponding Sobolev spaces Hk(ρV
+P ).
+Our strategy is based on a change of variables in the partition function ZV,P
+N
+(4), used for the beta-
+ensemble at fixed temperature introduced in [BFG15] and [Shc14], and used in [Gui19] and in [BGK16]
+to derive the loop equations and in [BLS18] to derive a CLT in the β-ensemble with β fixed. The outline
+of the argument goes as follows: Take φ : R → R smooth, vanishing fast enough at infinity, and do the
+change of variables in ZV,P
+N
+, xi = yi +
+t
+√
+N φ(yi), 1 ⩽ i ⩽ N, to get
+ZP,V
+N
+=
+ˆ
+RN
+�
+i<j
+����yi − yj +
+t
+√
+N
+(φ(yi) − φ(yj))
+����
+2P/N
+e−�N
+i=1 V �
+yi+
+t
+√
+N φ(yi)� N
+�
+i=1
+�
+1 +
+t
+√
+N
+φ′(yi)
+�
+dNy .
+Expanding the different terms in this integral, one gets
+ZP,V
+N
+=
+ˆ
+RN
+�
+i<j
+|yi − yj|
+2P
+N e−�N
+i=1 V (yi)e
+t
+√
+N
+�
+2P
+N
+�
+i<j
+φ(yi)−φ(yj )
+yi−yj
++�N
+i=1(φ′(yi)−V ′(yi)φ(yi))
+�
+e
+t2
+2 σ2
+N (φ)dNy ,
+3
+
+where the term σ2
+N(φ) converges towards a limiting variance σ2(φ) depending on φ, P and V . After
+dividing both parts of the equation by ZP,V
+N
+, and because of equation (5) characterizing µV
+P , one can
+deduce from the last equation the convergence of the Laplace transform
+E
+�
+et
+√
+N(νN (Ξφ)+error term)�
+−→
+N→∞ e
+t2
+2 σ2(φ) ,
+(6)
+where Ξ is a linear operator acting on test functions and defined by
+(Ξφ)(x) = 2P
+ˆ
+R
+φ(x) − φ(y)
+x − y
+dµV
+P (y) + φ′(x) − V ′(x)φ(x) .
+(7)
+Once the error term is taken care of, (6) shows the central limit theorem for test functions of the form
+Ξφ. Following [HL21], the operator L given by
+Lφ = Ξφ′
+(8)
+can be analyzed using Hilbert space techniques. In particular, the operator L, seen as an unbounded
+operator of the Hilbert space
+H =
+�
+u ∈ L2(ρV
+P )
+��� u′ ∈ L2(ρV
+P ),
+ˆ
+R
+uρV
+P dx = 0
+�
+,
+⟨u, v⟩H = ⟨u′, v′⟩L2(ρV
+P ) ,
+can be decomposed as
+−L = A + 2PW ,
+where A is a positive Sturm-Liouville operator and W is positive and self-adjoint. Such a writing allows
+us to show that −L is invertible, see Theorem 6.7.
+We now state the assumptions we make on the potential V .
+Assumptions 1.1. The potential V satisfies:
+i) V ∈ C3(R), goes to infinity at infinity and is convex.
+ii) For all polynomial Q ∈ R[X] and α > 0, Q
+�
+V ′(x)
+�
+e−V (x) =
+o
+|x|→∞(x−α) .
+iii) |V ′(x)|
+−→
+|x|→+∞ +∞ and x �→
+1
+V ′(x)2 is integrable at infinity. Furthermore, for any sequence xN
+such that |xN| goes to infinity, and for all real a < b, we have, as N goes to infinity,
+1
+V ′(xN)2
+sup
+a⩽x⩽b
+|V ′′(xN + x)| −→
+N→∞ 0 .
+iv) V ′′(x)
+V ′(x) =
+O
+|x|→∞(1) and V (3)(x)
+V ′(x)
+=
+O
+|x|→∞(1).
+The convexity assumption in i) will guarantee the Poincaré inequality, stated in Proposition 2.4 for the
+equilibrium measure µV
+P . Because i) implies that V goes to infinity faster than linearly, we will see that
+it also ensures exponential decay at infinity of ρV
+P . Recalling the sufficient condition for PV,P
+N
+of equation
+(2) to be defined, this first assumption implies that there exists α > 0 such that lim inf|x|→∞
+V (x)
+|x|
+> α.
+This guarantees in particular that the beta-ensemble (3) is well-defined for all N ⩾ 1 and P ⩾ 0.
+The second assumption ensures that any power of V ′ and V ′′ is in L2(ρV
+P ) and ρV
+P , which behaves
+like e−V up to a sub-exponential factor, belongs to the Sobolev space H2(R) ⊂ C1(R). Indeed, for k ⩽ 2,
+using iv), ρV
+P
+(k) behaves at infinity like (V ′)kρV
+P as shown in (2.2) which is in L2(R) by assumption ii).
+Assumption iii) will be used to localize the minimum/maximum point of a typical configuration
+(x1, . . . , xN) following the law PV,P
+N : this will be done in Corollary 4.2, which comes as a consequence of
+4
+
+[Lam21][Theorem 3.4]. More precisely, Corollary 4.2 establishes that for sequences (α+
+N)N, (α−
+N)N going
+to infinity, the random variables
+α+
+N
+�
+max
+1⩽j⩽N xj − E+
+N
+�
+and
+α−
+N
+�
+max
+1⩽j⩽N xj − E−
+N
+�
+converge in distribution. For large N, the scalars E+
+N and E−
+N can thus be seen as the edges of a typical
+configuration. Furthermore,
+V (E±
+N) ∼ ln N .
+(9)
+We refer to Section 4 for detailed statements. We will need another technical assumption to ensure that
+Taylor remainders arising in the proof of Theorem 5.2 are negligible. This final step in the proof of
+Theorem 1.3 consists in lifting the result of Proposition 5.1 from compactly supported functions to more
+general functions. We use Assumption iv) to ensure that L−1 is regular enough ie that for sufficiently
+smooth functions f,
+�
+L−1f
+�′
+∈ H2(R).
+Assumption 1.2. With the notations of Theorem 4.1, we have
+sup
+d(x,IN)⩽1
+���V (3)(x)
+��� = o(N 1/2) ,
+where IN =
+�
+E−
+N − 2; E+
+N + 2
+�
+.
+In view of the asymptotics (9), Assumption 1.2 is reasonable, at least in the cases where V is of the
+form |x|a + R(x) for a = 2 or a > 3 (we choose a so that V is three times differentiable), and with
+R ∈ C3(R), convex, small enough (see the comment following Theorem 1.3) or V = cosh, for example.
+On the other hand a scaled potential like ex2 doesn’t satisfy assumptions iii) ans iv)..
+We are now able to state the main result, ie the central limit theorem for functions belonging to the
+image of the operator L introduced in (8).
+Theorem 1.3. Assume that V satisfies Assumptions 1.1 and Assumption 1.2. Then for φ verifying the
+following conditions:
+• φ ∈ C1(R)
+• φ(x) = O(1/x) and φ′(x) = O(1/x2) at infinity
+•
+ˆ
+R
+φ(x)dµV
+P (x) = 0
+we have the convergence in law
+√
+NνN(φ) → N
+�
+0, (σV
+P )2(φ)
+�
+(10)
+where the limiting variance (σV
+P )2(φ) is given by
+(σV
+P )2(φ) =
+ˆ
+R
+�
+�
+L−1φ
+�′′(x)2 + V ′′(x)
+�
+L−1φ
+�′(x)2
+�
+dµV
+P (x)
++ P
+¨
+R2
+��
+L−1φ
+�′(x) −
+�
+L−1φ
+�′(y)
+x − y
+�2
+dµV
+P (x)dµV
+P (y) .
+(11)
+5
+
+Remark 1.4. Since νN(φ + c) = νN(φ) for all constant c ∈ R, the assumption
+ˆ
+R
+φ(x)dµV
+P = 0 can be
+dropped by replacing φ by φ −
+ˆ
+R
+φ(x)dµV
+P in the expression of the limiting variance.
+As a tool to deal with the error term of equation (6), we establish a concentration inequality for the
+empirical measure. This inequality is stated in terms of the following distance over the set of probability
+distributions P(R).
+For µ, µ′ ∈ P(R) we define the distance
+d(µ, µ′) =
+sup
+∥f∥Lip⩽1
+∥f∥1/2⩽1
+�����
+ˆ
+fdµ −
+ˆ
+fdµ′
+����
+�
+,
+(12)
+where ∥f∥Lip denotes the Lipschitz constant of f, and ∥f∥2
+1/2 =
+ˆ
+R
+|t| |F[f](t)|2 dt, where F denotes the
+Fourier transform on L2(R) which takes the following expression F[f](t) =
+ˆ
+R
+f(x)e−itxdx for
+f ∈ L1(R) ∩ L2(R).
+We then have
+Theorem 1.5. There exists K ∈ R (depending on P and on V ), such that for any N ⩾ 1 and r > 0,
+PV,P
+N
+�
+d(ˆµN, µV
+P ) > r
+�
+⩽ e−Nr2 P π2
+2
++5P ln N+K .
+(13)
+This result is the analog of [HL21, Theorem 1.4].
+The paper is organized as follows. In Section 2 we discuss the regularity of the equilibrium
+density ρV
+P under Assumption 1.1. In Section 3 we prove Theorem 1.5. Section 4 is dedicated to the
+localization of the edge of a typical configuration, mentioned in the discussion preceding the statement of
+Assumption 1.2. We next prove in Section 5 the convergence of the Laplace transform of
+√
+NνN(Lφ) for
+general functions φ which establishes Theorem 1.3 for functions of the form Lφ. Section 6 is dedicated to
+the diagonalization and inversion of L given by (8). In Section 7, we show regularity properties of L−1 to
+establish Theorem 1.3. We detail in Appendix A elements of proof for the spectral theory of Schrödinger
+operators, used in Section 6.
+Acknowledgments The authors wish to thank Alice Guionnet and Karol Kozlowski for their helpful
+suggestions. We also thank Arnaud Debussche for pointing out the link with Schrödinger operators theory
+and Gautier Lambert for pointing out [Lam21]. We would also like to thank Jeanne Boursier, Corentin
+Le Bihan and Jules Pitcho for their intuition about the regularity of the inverse operator.
+2
+Regularity of the equilibrium measure and Hilbert transform
+In this section, we discuss the regularity properties of the equilibrium density ρV
+P , namely its decay at
+infinity and its smoothness, and give formulas for its two first derivatives.
+The Hilbert transform, whose definition we recall, plays a central role in the analysis of the equilibrium
+measure. It is first defined on the Schwartz class through ∀φ ∈ S(R), ∀x ∈ R,
+H[φ](x) :=
+ 
+R
+φ(t)
+t − xdt = lim
+ε↓0
+ˆ
+|t−x|>ε
+φ(t)
+t − xdt =
+ˆ +∞
+0
+φ(x + t) − φ(x − t)
+t
+dt,
+(14)
+where
+ 
+denotes the Cauchy principal value integral, and then extended to L2(R) thanks to property ii)
+of Lemma 2.1: ∥f∥L2(dx) = 1
+π ∥H[f]∥L2(dx). The last expression in (14) is a definition where the integral
+converges in the classical sense.
+6
+
+We also recall the definition of the logarithmic potential U f of a density of probability f : R → R,
+given for x ∈ R by
+U f(x) = −
+ˆ
+R
+ln |x − y|f(y)dy .
+(15)
+Because we assume f ∈ L1(R) to be nonnegative, U f takes values in [−∞, +∞). If f integrates the
+function ln, i.e
+´
+R ln |x|f(x)dx < +∞, then U f takes real values.
+Additionally, one can check that the logarithmic potential and the Hilbert transform of f are linked
+through the distributional identity
+�
+U f�′ = H[f].
+We recall in the next lemma some properties of the Hilbert transform that we will use in the rest of
+the paper.
+Lemma 2.1 (Properties of the Hilbert transform).
+i) Fourier transform: For all φ ∈ L2(R), F
+�
+H[φ]
+�
+(ω) = iπsgn(ω)F[φ](ω) for all ω ∈ R.
+ii) As a consequence, 1
+π H is an isometry of L2(R), and H satisfies on L2(R) the identity H2 = −π2I.
+iii) Derivative: For any f ∈ H1(R), H[f] is also H1(R) and H[f]′ = H[f ′].
+iv) For all p > 1, the Hilbert transform can be extended as a bounded operator H : Lp(R) → Lp(R).
+v) Skew-self adjointness: For any f, g ∈ L2(R), ⟨H[f], g⟩L2(R) = −⟨f, H[g]⟩L2(R).
+Proof. We refer to [Kin09] for the proofs of these properties.
+As a consequence of [GZ19], ˆµN converges almost surely under PV,P
+N
+towards the unique minimizer of
+the energy-functional EV
+P , defined for µ ∈ P(R) by
+EV
+P (µ) =
+
+
+
+ˆ
+R
+�
+V + ln
+�dµ
+dx
+��
+dµ − P
+¨
+R2 ln
+��x − y
+��dµ(x)dµ(y) if µ ≪ dx
++∞ otherwise
+.
+(16)
+(Here we wrote µ ≪ dx for "µ is absolutely continuous with respect to Lebesgue measure")
+Consequently, following [GM22, Lemma 3.2], the density ρV
+P of µV
+P satisfies equation (5), which we
+rewrite here for convenience.
+V (x) − 2P
+ˆ
+R
+ln |x − y|ρV
+P (y)dy + ln ρV
+P (x) = λV
+P ,
+(17)
+where λV
+P is a constant (depending on V and P). Using this equation, we show in the next lemma that
+ρV
+P decays exponentially and is twice continuously differentiable.
+We now drop the superscript of ρV
+P and µV
+P and denote it ρP and µP for convenience.
+Lemma 2.2. Under Assumption 1.1,
+• The support of µP is R and there exists a constant CV
+P such that for all x ∈ R,
+ρP (x) ⩽ CV
+P (1 + |x|)2P e−V (x) .
+• The density ρP is in C2(R) and we have
+ρ′
+P = −
+�
+V ′ + 2PH[ρP ]
+�
+ρP
+(18)
+and
+ρ′′
+P =
+�
+− 2PH[ρP ]′ − V ′′ + V ′2 + 4P 2H[ρP ]2 + 4PV ′H[ρP ]
+�
+ρP .
+(19)
+7
+
+Proof. For the first point, [GM22, Lemma 3.2] establishes that the support of µP is the whole real axis,
+and that under the first condition of 1.1, we have the bound, valid for all x ∈ R
+ρP (x) ⩽
+KV
+P
+(1 + |x|)2 ,
+(20)
+with KV
+P a positive constant. Using (17) and the fact that
+ln |x − y| ⩽ ln
+�
+1 + |x|
+�
++ ln
+�
+1 + |y|
+�
+,
+we see that for all x ∈ R,
+ρP (x) ⩽ CV
+P exp
+�
+− V (x) + 2P ln(1 + |x|)
+�
+,
+(21)
+with
+CV
+P = exp
+�
+2P
+ˆ
+R
+ln(1 + |y|)ρP (y)dy + λV
+P
+�
+which is indeed finite by (20).
+For the second point, we use that
+�
+U ρP �′ = H[ρP ] weakly and equation (17) to conclude on the
+distributional identity
+ρ′
+P =
+�
+− V ′ − 2PH[ρP ]
+�
+ρP .
+By the second point of Assumption 1.1, V ′(x)e−V (x)+2P ln(1+|x|) = o(x−1) as |x| → ∞, thus by (21),
+V ′ρP ∈ L2(R). Also since ρP is L2(R) and bounded, we deduce, by using that H
+�
+L2(R)
+�
+= L2(R), that
+H[ρP ]ρP ∈ L2(R). Adding up these terms we get ρP ∈ H1(R). Because H[ρP ]′ = H[ρ′
+P ] in a weak sense
+by Lemma 2.1, H[ρP ] ∈ H1(R). By the classical fact that H1(R) is contained in the set of 1/2-Hölder
+functions C1/2(R), we have H[ρP ] ∈ C1/2(R) and so U ρP ∈ C1,1/2(R), the set of functions in C1(R) with
+derivative of class 1/2-Hölder.
+Using the fact that V is continuously differentiable, the previous equation for the weak derivative of ρP
+then ensures that ρP ∈ C1(R) and equation (18) holds in the strong sense.
+Differentiating (in a weak sense) equation (18) we obtain
+ρ′′
+P =
+�
+− 2PH[ρP ]′ − V ′′ + V ′2 + 4P 2H[ρP ]2 + 4PV ′H[ρP ]
+�
+ρP .
+The three first terms belong to L2(R) for the same reasons as before. By equation (21), ρP ∈ L4(R) and
+by lemma 1.1, so is H[ρP ], thus using the boundedness of ρP we see that ρP H[ρP ]2 ∈ L2(R). For the last
+term, we use that V ′ρP and H[ρP ] belong to L4(R) to ensure by Cauchy-Schwarz inequality that it is in
+L2(R). Finally, we can conclude that ρP ∈ H2(R) and so that H[ρP ] ∈ H2(R) with H[ρP ]′′ = H[ρ′′
+P ] (in a
+weak sense). As before, we conclude that ρP ∈ C2(R) and that equation (19) holds in a strong sense.
+We next show that the Hilbert transform of ρP is continuous and decays at infinity.
+Lemma 2.3. Let u ∈ L2(R) such that
+´
+R u(t)dt exists and f : t �→ tu(t) ∈ H1(R) then
+H[u](x)
+∼
+|x|→∞
+−
+´
+R u(t)dt
+x
+.
+Moreover if
+ˆ
+R
+u(t)dt = 0,
+´
+R f(t)dt exists and g : t �→ t2u(t) ∈ H1(R), then
+H[u](x)
+∼
+|x|→∞
+−
+´
+R tu(t)dt
+x2
+.
+As a consequence, we obtain that H[ρP ](x)
+∼
+|x|→∞ x−1 and the logarithmic potential U ρP is Lipschitz
+bounded, with bounded derivative H[ρP ].
+8
+
+Proof. Let u ∈ L2(R), such that
+´
+R u(t)dt exists and f : t �→ tu(t) ∈ H1(R). Then
+xH[u](x) +
+ˆ
+R
+u(t)dt =
+ˆ
+R
+�xu(x + t) − xu(x − t)
+2t
++ u(x + t)
+2
++ u(x − t)
+2
+�
+dt = H[f](x).
+Since f ∈ H1(R), so is H[f], proving that it goes to zero at infinity. Hence
+H[u](x)
+∼
+|x|→∞
+−
+´
+R u(t)dt
+x
+Moreover if
+ˆ
+R
+u(t)dt = 0,
+´
+R f(t)dt exists and g : t �→ t2u(t) ∈ H1(R), then by the same argument:
+x2H[u](x) = xH[f](x) = H[g](x) −
+ˆ
+R
+f(t)dt
+where g(t) = t2u(t). We deduce that H[u](x)
+∼
+|x|→∞
+−
+´
+R tu(t)dt
+x2
+since H[g] goes to zero at infinity.
+We conclude this section by stating the Poincaré inequality for the measure µP under the assumption
+that V is convex.
+Proposition 2.4. The measure µP satisfies the following Poincaré inequality: There exists a constant
+C such that for all f ∈ C∞
+c (R),
+VarµP (f) ⩽ C
+ˆ
+R
+|f ′|2dµP .
+(22)
+This fact comes as a direct consequence of [BBCG08][Corollary 1.9], which states that if the probability
+measure µ has a log-concave density on R, then it satisfies (22) for f smooth enough (actually the result
+is true in Rd, replacing f ′ by ∇f). Indeed, by convexity of V and concavity of x �→ ´
+R ln |x − y|ρP (y)dy,
+equation (17) implies that ln ρP is concave.
+Remark 2.5. We will apply later inequality (22) to more general functions than C∞
+c (R), namely functions
+of the weighted Sobolev space H1(ρP ), defined in Section 6; which can be seen as the completion of C∞
+c (R)
+with respect to the norm ∥u∥L2(ρP ) + ∥u′∥L2(ρP ).
+3
+Concentration inequality, proof of Theorem 1.5
+We prove in this section the concentration Theorem 1.5. Its proof is a direct adaptation of Theorem 1.4
+of [HL21], which shows the analogous estimate in the circular setup. It is inspired by [MMS14] and based
+on a comparison between a configuration xN = (x1, . . . , xN) sampled with PV,P
+N
+and a regularized version
+yN = (y1, . . . , yN), which we describe here.
+Definition 3.1. Let xN = (x1, . . . , xN) ∈ RN, and suppose (up to reordering) that x1 ⩽ x2 . . . ⩽ xN.
+We define yN ∈ RN by:
+y1 := x1, and for 0 ⩽ k ⩽ N − 1, yk+1 := yk + max{xk+1 − xk, N −3}.
+Note that the configuration yN given by the previous definition satisfies yk+1 − yk ⩾ N −3, and yN is
+close to xN in the sense that
+N
+�
+k=1
+|xk − yk| ⩽
+1
+2N .
+(23)
+Indeed, by construction we have |xk − yk| = yk − xk ⩽ (k − 1)N −3, and we get the bound by summing
+these inequalities.
+9
+
+The key point of the proof of Theorem 1.5 is comparing the empirical measure ˆµN = 1
+N
+�N
+i=1 δxi, where
+xN follows PV,P
+N , to the regularized measure
+�µN := λN −5 ∗ 1
+N
+N
+�
+i=1
+δyi,
+(24)
+ie the convolution of λN −5 and the empirical measure, where λN −5 is the uniform measure on [0, N −5].
+The interest of introducing the measure �µN is that it is close to ˆµN, while having a finite energy EV
+P (�µN),
+given by (16). Finally, notice that the empirical measure doesn’t change when reordering x1, . . . , xN, and
+thus we do not lose in generality for our purposes in assuming that x1 ⩽ . . . ⩽ xN in definition 3.1.
+We now introduce a distance on P(R) which is well-suited to our context.
+Definition 3.2. For µ, µ′ ∈ P(R) we define the distance (possibly infinite) D(µ, µ′) by
+D(µ, µ′) :=
+�
+−
+ˆ
+ln |x − y|d(µ − µ′)(x)d(µ − µ′)(y)
+�1/2
+(25)
+=
+�ˆ +∞
+0
+1
+t
+��F[µ − µ′]
+��2dt
+�1/2
+.
+(26)
+where the Fourier transform of a signed measure ν is defined by F[ν](x) :=
+´
+e−itxd(µ − µ′)(x)
+Let f : R ��� R with finite 1/2 norm ∥f∥1/2 :=
+�´
+R |t| |F[f](t)|2 dt
+�1/2
+. By Plancherel theorem and
+Hölder inequality, for any µ, µ′ ∈ P(R), setting ν = µ − µ′,
+����
+ˆ
+R
+fdµ −
+ˆ
+R
+fdµ′
+����
+2
+=
+�����
+1
+2π
+ˆ
+R
+|t|1/2F[f](t)F[ν](t)
+|t|1/2 dt
+�����
+2
+⩽
+1
+2π2 ∥f∥2
+1/2D2(µ, µ′).
+Therefore the metric d defined in (12) is dominated by D:
+d(µ, µ′) ⩽
+1
+√
+2π D(µ, µ′).
+(27)
+The following lemma shows how the distance D is related to the energy-functional EV
+P defined in (16),
+we will write EP for simplicity.
+Lemma 3.3. We have for any absolutely continuous µ ∈ P(R) with finite energy EV
+P (µ),
+EP (µ) − EP (µP ) = PD2(µ, µP ) +
+ˆ
+ln
+� dµ
+dµP
+�
+dµ .
+(28)
+Proof of Lemma 3.3. Subtracting EP (µ) − EP (µP ) we find
+EP (µ) − EP (µP ) =
+ˆ
+V d(µ − µP ) +
+ˆ
+ln dµ
+dx dµ −
+ˆ
+ln ρP dµP − P
+¨
+ln |x − y|dµ(x)dµ(y)
++ P
+¨
+ln |x − y|dµP (x)dµP (y) .
+(29)
+Now, if ν is a signed measure of mass zero, integrating (17) we get
+ˆ
+V (x)dν(x) − 2P
+¨
+ln |x − y|dν(x)dµP (y) +
+ˆ
+ln(ρP )(x)dν(x) = 0 .
+10
+
+We take ν = µ − µP , and get
+ˆ
+V (x)d(µ − µP )(x) = 2P
+¨
+ln |x − y|dµ(x)dµP (y) − 2P
+¨
+ln |x − y|dµP (x)dµP (y)
+−
+ˆ
+ln(ρP )(x)dµ(x) +
+ˆ
+ln(ρP )(x)dµP (x) .
+Plugging this last identity in (29), we find
+EP (µ) − EP (µP ) = −P
+¨
+ln |x − y|dν(x)dν(y) +
+ˆ
+ln
+� dµ
+dµP
+�
+(x)dµ(x)
+which establishes the result.
+Proof of Theorem 1.5. We first give a lower bound for the partition function ZV,P
+N
+(4) of PV,P
+N . We rewrite
+it as
+ZV,P
+N
+=
+ˆ
+RN exp
+�
+2P
+N
+�
+i<j
+ln |xi − xj| −
+N
+�
+i=1
+�
+V (xi) + ln ρP (xi)
+��
+dρP (x1) . . . dρP (xN) ,
+and apply Jensen inequality to obtain:
+ln ZV,P
+N
+⩾
+ˆ
+RN
+�
+2P
+N
+�
+i<j
+ln |xi − xj| −
+N
+�
+i=1
+�
+V (xi) + ln ρP (xi)
+��
+dρP (x1) . . . dρP (xN)
+⩾ P(N − 1)
+¨
+ln |x − y|dρP (x)dρP (y) − N
+ˆ
+R
+�
+V + ln ρP
+�
+dρP
+⩾ −NEV
+P
+�
+µP
+�
+− P
+¨
+ln |x − y|dρP (x)dρP (y).
+Using this estimate and the fact that for 1 ⩽ i, j ⩽ N we have |xi −xj| ⩽ |yi−yj|, with yN = (y1, . . . , yN)
+of definition 3.1, we deduce the bound on the density of probability
+dPV,P
+N
+dx
+(x1, . . . , xN) ⩽ e
+NEP (µP )+P
+˜
+ln |x−y|dµP (x)dµP (y)+ P
+N
+�
+i̸=j ln |yi−yj|−�N
+i=1 V (xi) .
+(30)
+Recalling (24), we now show the following estimate
+�
+i̸=j
+ln |yi − yj| ⩽ 2N + N 2
+¨
+ln |x − y|d�µN(x)d�µN(y) + 3N ln N + 3
+2N .
+(31)
+Let i ̸= j and u, v ∈ [0, N −5]. Since for x ̸= 0 and |h| ⩽ |x|
+2 , we have
+�� ln |x + h| − ln |x|
+�� ⩽ 2|h|
+|x| , we deduce
+�� ln |yi − yj + u − v| − ln |yi − yj|
+�� ⩽ 2|u − v|
+|yi − yj| ⩽ 2N −5
+N −3 =
+2
+N 2 .
+Thus, summing over i ̸= j and integrating with respect to u and v, we get
+�
+i̸=j
+ln |yi − yj| ⩽ 2 +
+�
+i̸=j
+¨
+ln |yi − yj + u − v|dλN −5(u)dλN −5(v)
+= 2 + N 2
+¨
+ln |x − y|d�µN(x)d�µN(y) − N
+¨
+ln |u − v|dλN −5(u)dλN −5(v) .
+11
+
+The last integral is equal to − 3
+2 − 5 ln N, so we deduce (31). We now combine (30) and (31). Recall (16)
+and set
+cN = P
+�¨
+ln |x − y|dµP (x)dµP (y) + 3/2 + 2/N
+�
+.
+We get
+dPV,P
+N
+dx
+(x1, . . . , xN) ⩽ ecN+5P ln N exp
+�
+N
+�
+EP (µP ) − EP (�µN) +
+ˆ �
+V + ln d�µN
+dx
+�
+d�µN
+�
+−
+N
+�
+i=1
+V (xi)
+�
+= ecN+5P ln N exp
+�
+−NPD2(�µN, µP ) + N
+ˆ
+(V + ln ρP ) d�µN −
+N
+�
+i=1
+V (xi)
+�
+where we used equation (28) in the last equality. Using again equation (17) we then see that the density
+dPV,P
+N
+dx (x1, . . . , xN) is bounded by
+ecN+5P ln N exp
+�
+−NPD2(�µN, µP ) + 2PN
+¨
+ln |x − y|d(�µN − ˆµN)(x)dµP (y)
+� N
+�
+i=1
+ρP (xi) .
+Recalling (15),
+˜
+ln |x − y|d(�µN − ˆµN)(x)dµP (y) = −
+´
+U ρP d(�µN − ˆµN). As a consequence of the bound
+on the density
+dPV,P
+N
+dx (x1, . . . , xN) we established, we have for all r > 0
+PV,P
+N
+�
+D2(�µN, µP ) > r
+�
+⩽ e−NP r+cN+5P ln N
+ˆ
+RN exp
+�
+−2PN
+ˆ
+U ρP d(�µN − ˆµN)
+� N
+�
+i=1
+ρP (xi)dxi . (32)
+Next, we show that −N ´ U ρP d(�µN − ˆµN) is bounded.
+By Lemma 2.3, U ρP is differentiable with bounded derivative H[ρP ] on R. As a consequence,
+����N
+ˆ
+U ρP d(�µN − ˆµN)
+���� ⩽
+N
+�
+i=1
+ˆ
+|U ρP (yi + u) − U ρP (xi)| dλN −5(u)
+⩽ ∥H[ρP ]∥∞
+� N
+�
+i=1
+|yi − xi| + N
+ˆ
+udλN −5(u)
+�
+⩽ ∥H[ρP ]∥∞
+� 1
+2N + N −4/2
+�
+,
+where we used (23) in the last inequality. Therefore, we deduce from (32)
+PV,P
+N
+�
+D2(�µN, µP ) > r
+�
+⩽ e−NP r+cN+5P ln N+2P ∥H[ρP ]∥∞ = e−NP r+5P ln N+KN
+(33)
+with KN := cN + 2P∥H [ρP ] ∥∞. Since cN is bounded, so is KN.
+Finally, let f be a Lipschitz bounded function with ∥f∥Lip ⩽ 1, then, we have (as we did for U ρP )
+����
+ˆ
+fdˆµN −
+ˆ
+fd�µN
+���� ⩽ N −2 .
+Thus,
+d(ˆµN, µP ) ⩽ d(ˆµN, �µN) + d(�µN, µP ) ⩽ N −2 +
+1
+√
+2π D(�µN, µP ) ,
+and for any N such that r − N −2 ⩾ r/2 (in particular r − N −2 > 0) we get
+PV,P
+N
+(d(ˆµN, µP ) > r) ⩽ PV,P
+N
+� 1
+2π2 D2(�µN, µP ) > (r − N −2)2
+�
+⩽ PV,P
+N
+� 1
+2π2 D2(�µN, µP ) > r2/4
+�
+,
+and the last term is bounded by e−Nr2 P π2
+2
++5P ln N+K for some K large enough, which concludes the
+proof.
+12
+
+As a consequence of Theorem 1.5, we are able to control the quantities
+ζN(φ) :=
+¨
+R2
+φ(x) − φ(y)
+x − y
+d(ˆµN − µP )(x)d(ˆµN − µP )(y)
+(34)
+for a certain class of test functions φ.
+Corollary 3.4. There exists C, K > 0 such that for all φ ∈ C2(R)∩H2(R) with bounded second derivative,
+we have for ε > 0 and N large enough,
+PV,P
+N
+�√
+N|ζN(φ)| ⩽ N −1/2+ε�
+⩾ 1 − exp
+�
+−
+PN ε
+2CN2(φ) + 5P ln N + K
+�
+with N2(φ) = ∥φ′∥L2(dx) + ∥φ′′∥L2(dx).
+Proof. We follow the proof given in [Gui19][Cor. 4.16] and adapt it to our setting. Let us denote by
+�
+ζN(φ) the quantity
+¨
+R2
+φ(x) − φ(y)
+x − y
+d(�µN − µP )(x)d(�µN − µP )(y) .
+We have the almost sure inequality, by a Taylor estimate
+|ζN(φ) − �
+ζN(φ)| ⩽ 2N −2∥φ′′∥∞ .
+(35)
+Thus, for any δ > 0,
+PV,P
+N
+(|ζN(φ)| > δ) ⩽ PV,P
+N
+�
+|ζN(φ) − �
+ζN(φ)| > δ/2
+�
++ PV,P
+N
+�
+|�
+ζN(φ)| > δ/2
+�
+⩽ PV,P
+N
+�
+2N −2∥φ′′∥∞ > δ/2
+�
++ PV,P
+N
+�
+|�
+ζN(φ)| > δ/2
+�
+,
+where the first term of the right-hand side is either 0 or 1. With δ = N −1+ε, ε > 0, it is zero for N large
+enough. For such a choice of δ, and for N large enough,
+PV,P
+N
+�
+|ζN(φ)| > N −1+ε�
+⩽ PV,P
+N
+�
+|�
+ζN(φ)| > 1
+2N −1+ε
+�
+.
+We next show that, for some C > 0 independent of φ, we have
+|�
+ζN(φ)| ⩽ CD2(�µN, µP )N2(φ) .
+(36)
+We begin by showing this inequality for ψ ∈ S(R). By using the inverse Fourier transform we have
+�ζN(ψ) = 1
+2π
+¨ ´
+dtF[ψ](t)e−itx −
+´
+dtF[ψ](t)e−ity
+x − y
+d
+�
+�µN − µP
+�
+(x)d
+�
+�µN − µP
+�
+(y)
+= −1
+2π
+ˆ
+dtitF[ψ](t)
+¨
+e−ity e−it(x−y) − 1
+−it(x − y) d
+�
+�µN − µP
+�
+(x)d
+�
+�µN − µP
+�
+(y)
+= −1
+2π
+ˆ
+dtitF[ψ](t)
+¨
+e−ity
+ˆ 1
+0
+dαe−iαt(x−y)d
+�
+�µN − µP
+�
+(x)d
+�
+�µN − µP
+�
+(y)
+= −1
+2π
+ˆ
+dtitF[ψ](t)
+ˆ 1
+0
+dα
+ˆ
+e−iαtxd
+�
+�µN − µP
+�
+(x)
+ˆ
+e−i(1−α)tyd
+�
+�µN − µP
+�
+(y)
+13
+
+We then apply in order the triangular inequality, Cauchy-Schwarz inequality, a change of variable and
+the fact that |F [�µN − µP ]|2 is an even function.
+|�ζN(ψ)| ⩽ 1
+2π
+ˆ
+R
+dt |tF[ψ](t)|
+ˆ 1
+0
+dα |F [�µN − µP ] (αt)| .
+��F [�µN − µP ]
+�
+(1 − α)t
+���
+⩽ 1
+2π
+ˆ
+R
+dt |tF[ψ](t)|
+� ˆ 1
+0
+dα |F [�µN − µP ] (αt)|2 � 1
+2 � ˆ 1
+0
+dα
+��F [�µN − µP ]
+�
+(1 − α)t
+���2 � 1
+2
+⩽ 1
+2π
+ˆ
+R
+dt |tF[ψ](t)|
+ˆ 1
+0
+dα |F [�µN − µP ] (αt)|2
+⩽ 1
+2π
+ˆ +∞
+0
+dt |tF[ψ](t)|
+ˆ 1
+0
+tdα
+tα |F [�µN − µP ] (αt)|2 + 1
+2π
+ˆ 0
+−∞
+dt |tF[φ](t)|
+ˆ 1
+0
+−tdα
+−tα |F [�µN − µP ] (αt)|2
+⩽ 1
+2π
+ˆ
+R
+dt |tF[ψ](t)| D2(�µN, µP )
+⩽ 1
+2π
+� ˆ
+R
+dt |tF[ψ](t)|2 (1 + t2)
+� 1
+2 � ˆ
+R
+dt
+1 + t2
+� 1
+2 D2(�µN, µP )
+⩽
+1
+2√π D2(�µN, µP )N2(ψ)
+By density of S(R) in L2(R), and since �ζN :
+�
+H2(R), N2
+�
+→ R is continuous, the inequality still holds
+for φ. Thus, using equation (33),
+PV,P
+N
+�
+|�
+ζN(φ)| > 1
+2N −1+ε
+�
+⩽ PV,P
+N
+�
+D2(�µN, µP ) >
+N −1+ε
+2CN2(φ)
+�
+⩽ exp
+�
+−P
+N ε
+2CN2(φ) + 5P ln N + K
+�
+,
+which concludes the proof.
+4
+Localization of the edge of a configuration
+In [Lam21][Theorem 1.8, Theorem 3.4], Lambert was able to control the edge (i.e the minimum and the
+maximum) of a typical configuration (x1, . . . , xN) distributed according to PV,P
+N , by showing that the
+random measure
+ΞN :=
+N
+�
+j=1
+δϕ−1
+N (xj)
+converges in distribution towards a Poisson point process for a function ϕN which takes the form
+ϕN(x) := EN + α−1
+N x .
+Before being more precise on the construction of (EN)N and (αN)N, we explain, following [Lam21], how
+one can use this convergence to localize the edge of a typical configuration (x1, . . . , xN). Let us assume for
+a moment that ΞN converges towards a Poisson point process with intensity θ(x) = e−x, with EN → +∞.
+In particular, the random variable
+ΞN(t, +∞)
+converges in distribution towards a Poisson random variable with mean
+´ +∞
+t
+e−xdx. Combined with the
+equalities
+PV,P
+N
+�
+ΞN(t, +∞) = 0
+�
+= PV,P
+N
+�
+∀ 1 ⩽ j ⩽ N, ϕ−1
+N (xj) = αN(xj − EN) ⩽ t
+�
+= PV,P
+N
+�
+αN
+�
+max
+1⩽j⩽N xj − EN
+�
+⩽ t
+�
+,
+14
+
+we deduce that for all t ∈ R
+PV,P
+N
+�
+αN
+�
+max
+1⩽j⩽N xj − EN
+�
+⩽ t
+�
+−→
+N→∞ exp(−e−t) .
+Therefore, the random variable
+αN
+�
+max
+1⩽j⩽N xj − EN
+�
+converges in distribution to the Gumbel law, showing that the maximum of a configuration is of order
+EN. Furthermore, as will be clear from the construction of αN and EN, αN is positive, and goes to
+infinity as N goes to infinity.
+Replacing in the previous analysis θ(x) = ex and EN → −∞, we would have deduced in the same
+fashion that
+αN
+�
+min
+1⩽j⩽N xj − EN
+�
+converges in law.
+With the above notations, we can apply [Lam21][Theorem 3.4] to our context.
+Theorem 4.1. Let v = ±. There exists (Ev
+N)N, (αv
+N)N sequences of real numbers with |Ev
+N| → +∞,
+αv
+N > 0 for large enough N, satisfying V ′(Ev
+N) = αv
+Nv, such that:
+a) Ne−V (Ev
+N)+2P ln |Ev
+N|+λP
+V
+αv
+N
+−→
+N→∞ 1 (recall λV
+P is defined through equation (5)),
+b)
+ln(αv
+N)
+N
+−→
+N→∞ 0 and αv
+N|Ev
+N| −→
+N→∞ +∞ ,
+c) For all compact K ⊂ R,
+(αv
+N)−2 sup
+x∈K
+|V ′′(ϕN(x))| −→
+N→∞ 0 .
+As a consequence, the random measure ΞN converges in distribution as N → ∞ to a Poisson point process
+with intensity θ(x) = e−vx.
+Proof. We prove it in the case v = +, the case where v = − being similar. We show that there exists a
+sequence (E+
+N)N going to +∞ satisfying f(E+
+N) = − ln N, where we defined the function f by
+f(x) = −V (x) + 2P ln |x| + λV
+P − ln |V ′(x)| .
+(we will then have α+
+N = V ′(E+
+N) > 0. In the case v = −1 we would have looked for a sequence E−
+N going
+to −∞ and α−
+N = −V ′(E−
+N))
+As a consequence of Assumptions 1.1,ii), one shows that ln |V ′| is negligible with respect to V at infinity.
+Therefore, because ln |x|
+V (x)
+−→
+|x|→∞ 0,
+f(x) = −V (x) + o(V (x))
+at infinity (in particular, f(x)
+−→
+x→+∞ −∞). We deduce that for 0 < ε < 1 fixed, there exists A > 0 such
+that for all x > A,
+−(1 + ε)V (x) < f(x) < −(1 − ε)V (x) ,
+(37)
+and because f(x)
+−→
+x→+∞ −∞ there exists (E+
+N)N going to infinity such that for all N ⩾ 1, f(E+
+N) = − ln N.
+Setting x = E+
+N in (37), we obtain that −V (E+
+N) ∼ f(E+
+N) = − ln N. By convexity of V and the fact that
+it goes to infinity at infinity, V is increasing on some [M, +∞[, where M ⩾ 0. Thus −(1±ε)V (x) = − ln N
+15
+
+iff x = V −1
+� ln N
+1 ± ε
+�
+, where V −1 denotes
+�
+V|[M,+∞[
+�−1. We conclude by (37) that such an E+
+N must
+satisfy
+V −1
+� ln N
+1 + ε
+�
+⩽ E+
+N ⩽ V −1
+� ln N
+1 − ε
+�
+.
+(38)
+By convexity of V and the fact that it goes to infinity at infinity, (α+
+N)N is non-decreasing and goes to
+infinity. It is thus positive for N large enough, ensuring that αN|E+
+N| −→
+N→∞ +∞. Property c) of the
+theorem follows from Assumptions 1.1, point ii). It remains to show that ln(α+
+N)
+N
+= ln |V ′(E+
+N)|
+N
+−→
+N→∞ 0.
+By construction, we have
+ln |V ′(E+
+N)|
+N
+=
+ln
+�
+Ne−V (E+
+N)+2P ln N+λP
+�
+N
+= −V (E+
+N)
+N
++ o(1) .
+Using that V (E+
+N) ∼ ln N, we can conclude that ln |V ′(E+
+N)| = o(N) which concludes the proof.
+By the discussion preceding Theorem 4.1, we deduce
+Corollary 4.2 (Edge of a configuration). Let E±
+N, α±
+N := |V ′(E±
+N)| be the sequences of Theorem 4.1
+associated with v = ±1. Then, both random variables
+α+
+N
+�
+max
+1⩽j⩽N xj − E+
+N
+�
+and
+α−
+N
+�
+min
+1⩽j⩽N xj − E−
+N
+�
+converge to a Gumbel law, whose distribution function is given for t ⩾ 0 by G([0, t]) = exp(e−t). Further-
+more, V (E±
+N) ∼ ln N and α±
+N −→
+N→∞ ±∞.
+Remark 4.3. Note that[Lam21][Theorem 3.4] applies for V of class C2 outside of a compact set, allowing
+to take V (x) = |x|a for a ⩾ 1. Furthermore, if V (x) = |x|a + R(x) for a ⩾ 1, R ∈ C2(R) and convex
+and where R(x), R′(x) and R′′(x) are negligible respectively with respect to xa, xa−1 and xa−2, we find
+E±
+N ∼ ±(ln N)1/a.
+If V (x) = cosh(x), we find E+
+N ∼ −E−
+N ∼ arg cosh(ln N) ∼ ln ln N.
+The next lemma will be convenient in the proof of Theorem 5.2 when dealing with error terms.
+Lemma 4.4. With the notations of Corollary 4.2, we have
+µP ([E−
+N, E+
+N]c) = o(N −1/2) .
+Proof. Let 0 < δ < 1, to be specified later. We have
+ˆ +∞
+E+
+N
+ρP dx =
+ˆ +∞
+E+
+N
+(ρP )δ(ρP )1−δdx ⩽
+ˆ
+R
+(ρP )δdx
+sup
+[E+
+N,+∞[
+(ρP )1−δ .
+By the first inequality of Lemma 2.2, the integral is finite. Also from the same inequality, we have for
+some constant C′ and x big enough ρP (x) ⩽ C′e− 3
+4 V (x). Because V is increasing in a neighborhood of
++∞, we get for N large enough
+sup
+[E+
+N,+∞[
+(ρP )1−δ ⩽ C′1−δe−(1−δ) 3
+4 V (E+
+N) .
+16
+
+Taking δ > 0 such that 1
+2 − (1 − δ) 3
+4 =: −γ < 0 and using that V (E+
+N) = ln N + o(ln N) (established in
+the proof of Theorem 4.1),
+√
+N
+ˆ +∞
+E+
+N
+ρP dx ⩽ Ke−γ ln N+(1−δ) 3
+4 o(ln N) ,
+and the right-hand side goes to zero as N goes to infinity. We deal with the integral
+´ E−
+N
+−∞ ρP dx in the
+same way.
+Remark 4.5. We could improve the proof to show that µP ([E−
+N, E+
+N]c) ∼ 1
+N but showing that it is o(N
+1
+2 )
+is sufficient for what we need and requires less carefulness.
+5
+Laplace transform for smooth test functions, proof of Theo-
+rem 1.3
+Section 3 allows us to justify in Proposition 5.1 the heuristics we gave in equation (6) for φ having
+compact support. We will then extend in Theorem 5.2 this result to a more general set of functions, by
+an approximation by compactly supported functions, using Corollary 4.2.
+Proposition 5.1. For φ ∈ C1(R, R) with compact support, we have for any real t, as N goes to infinity,
+EV,P
+N
+�
+et
+√
+NνN (Ξφ)�
+→ exp
+�t2
+2 qP (φ)
+�
+,
+(39)
+where Ξφ is given by equation (7), and qP (φ) is given by
+qP (φ) :=
+ˆ
+R
+�
+φ′(x)2 + V ′′(x)φ(x)2
+�
+dµP (x) + P
+¨
+R2
+�φ(x) − φ(y)
+x − y
+�2
+dµP (x)dµP (y) .
+(40)
+Proof. Let φ ∈ C1
+c(R, R), and let t ∈ R. We perform in equation (4) the change of variables
+xi = yi +
+t
+√
+N φ(yi), 1 ⩽ i ⩽ N, which is a diffeomorphism for N big enough. We thus have
+ZV,P
+N
+=
+ˆ
+�
+1⩽i<j⩽N
+����yi − yj +
+t
+√
+N
+�
+φ(yi) − φ(yj)
+�����
+2P/N
+.e−�N
+i=1 V �
+yi+
+t
+√
+N φ(yi)�
+.
+N
+�
+i=1
+�
+1 +
+t
+√
+N
+φ′(yi)
+�
+dNy,
+(41)
+and we develop separately the different terms of this integral. The first term can be written as:
+�
+i<j
+|yi − yj|2P/N �
+i<j
+����1 +
+t
+√
+N
+φ(yi) − φ(yj)
+yi − yj
+����
+2P/N
+,
+The second product above, setting ∆φi,j := φ(yi)−φ(yj)
+yi−yj
+and using Taylor-Lagrange theorem, equals
+exp
+�2P
+N
+�
+i<j
+ln
+����1 +
+t
+√
+N
+φ(yi) − φ(yj)
+yi − yj
+����
+�
+= exp
+�2P
+N
+�
+i<j
+�
+t
+√
+N
+∆φi,j − t2
+2N (∆φi,j)2 + RN,1(i, j)
+� �
+,
+where we noticed that 1 +
+t
+√
+N ∆φi,j ⩾ 1 −
+t
+√
+N ∥φ′∥∞ > 0 if N is big enough, and where
+|RN,1(i, j)| ⩽
+|t|3
+3N 3/2 ∥φ′∥3
+∞.
+17
+
+Again by Taylor-Lagrange theorem, the second term in (41) equals
+exp
+�
+−
+N
+�
+i=1
+�
+V (yi) +
+t
+√
+N
+V ′(yi)φ(yi) + t2
+2N V ′′(yi)φ(yi)2 + RN,2(i)
+� �
+where RN,2(i) =
+t3
+6N 3/2 V (3) �
+yi + tθi
+√
+N φ(yi)
+�
+φ(yi)3 for some θi ∈ [0, 1], thus for N large enough
+|RN,2(i)| ⩽
+|t|3
+6N 3/2 ∥φ∥3
+∞
+sup
+d(x,supp φ)⩽1
+|V (3)(x)|.
+The last term reads
+N
+�
+i=1
+�
+1 +
+t
+√
+N
+φ′(yi)
+�
+= exp
+� N
+�
+i=1
+�
+t
+√
+N
+φ′(yi) − t2
+2N φ′(yi)2 + RN,3(i)
+� �
+,
+with |RN,3(i)| ⩽
+t3
+3N 3/2 ∥φ′∥3
+∞. Dividing both sides of equation (41) by ZV,P
+N
+we get
+EV,P
+N
+�
+exp
+�
+t
+√
+N
+�
+P
+¨
+R2
+φ(x) − φ(y)
+x − y
+dˆµN(x)dˆµN(y) +
+ˆ
+R
+(φ′ − V ′φ)dˆµN
+��
+× exp {KN(t, φ)}
+× exp
+�
+t2
+2
+�
+−P
+¨
+R2
+�φ(x) − φ(y)
+x − y
+�2
+dˆµN(x)dˆµN(y) −
+ˆ
+R
+(V ′′φ2 + φ′2)dˆµN
+�� �
+= 1,
+with |KN(t, φ)| ⩽ c(t,φ)
+√
+N
+where c(t, φ) ⩾ 0 is independent of N. This bound shows that taking the limit
+N → ∞ we can get rid of KN:
+lim
+N→∞ EV,P
+N
+�
+exp
+�
+t
+√
+N
+�
+P
+¨
+R2
+φ(x) − φ(y)
+x − y
+dˆµN(x)dˆµN(y) +
+ˆ
+R
+(φ′ − V ′φ)dˆµN
+��
+× exp
+�
+t2
+2
+�
+−P
+¨
+R2
+�φ(x) − φ(y)
+x − y
+�2
+dˆµN(x)dˆµN(y) −
+ˆ
+R
+(V ′′φ2 + φ′2)dˆµN
+�� �
+= 1.
+Using Fubini’s theorem (the function (x, y) �→ φ(x)−φ(y)
+x−y
+being bounded continuous on R2), the first line
+in the expectation value can be rewritten as et
+√
+NΛN with
+ΛN := 2P
+¨
+R2
+φ(x) − φ(y)
+x − y
+dµP (x)d(ˆµN − µP )(y) +
+ˆ
+R
+(φ′ − V ′φ)d(ˆµN − µP ) + PζN(φ)
+(42)
+where we used equation (5) and ζN(φ) is given by (34). Let F : P(R) → R be defined by
+F(µ) = −P
+¨
+R2
+�φ(x) − φ(y)
+x − y
+�2
+dµ(x)dµ(y) −
+ˆ
+R
+(V ′′φ2 + φ′2)dµ .
+(43)
+It is continuous for the topology of weak convergence since all the functions in the integrals are bounded
+continuous. So far we have established that
+lim
+N→∞ EV,P
+N
+�
+et
+√
+NΛN+ t2
+2 F (ˆµN )�
+= 1,
+with ΛN given by (42). We now replace in the latter equation the term F(ˆµN) by its limiting expression,
+F(µP ). Fix a metric that is compatible with the weak convergence of probability measures on R. For
+example,
+dLip(µ, ν) = sup
+����
+ˆ
+fdµ −
+ˆ
+fdν
+���� ,
+(44)
+18
+
+where the supremum runs over f : R → R bounded and Lipschitz with ∥f∥∞ ⩽ 1 and Lipschitz constant
+|f|Lip ⩽ 1. By the large deviations principle for (ˆµN)N under the probability (3) established by [GZ19,
+Theorem 1.1], for all δ > 0 the event {dLip(ˆµN, µP ) > δ} has (for N big enough) probability smaller than
+e−Ncδ where cδ > 0. Hence,
+lim
+N→∞ EV,P
+N
+�
+et
+√
+NΛN + t2
+2 F (ˆµN)�
+= lim
+N→∞ EV,P
+N
+�
+1{dLip(ˆµN ,µP )⩽δ}et
+√
+NΛN + t2
+2 F (ˆµN)�
+.
+By continuity of F there is some ε(δ) which goes to 0 as δ → 0 such that, for dLip(ν, µP ) ⩽ δ, we have
+|F(ν) − F(µP )| ⩽ ε(δ). Taking the (decreasing) limit as δ goes to zero we deduce
+lim
+N→∞ EV,P
+N
+�
+et
+√
+NΛN + t2
+2 F (ˆµN )�
+= lim
+δ→0 lim
+N→∞ EV,P
+N
+�
+1{dLip(ˆµN ,µP )⩽δ}et
+√
+NΛN �
+e
+t2
+2 F (µP ).
+But the same large deviations argument shows that
+lim
+δ→0 lim
+N→∞ EV,P
+N
+�
+1{dLip(ˆµN ,µP )⩽δ}et
+√
+NΛN �
+= lim
+N→∞ EV,P
+N
+�
+et
+√
+NΛN �
+.
+Thus, we have shown that
+lim
+N→∞ EV,P
+N
+�
+et
+√
+N�
+2P
+˜
+R2
+φ(x)−φ(y)
+x−y
+dµP (x)d(ˆµN−µP )(y)+
+´
+R(φ′−V ′φ)d(ˆµN−µP )+P ζN (φ)��
+= e− t2
+2 F (µP ) ,
+(45)
+Which establishes that
+√
+NΛN =
+√
+N
+�
+νN(Ξφ) + PζN(φ)
+�
+converges in law towards a centered Gaussian
+random variable with announced variance. We finally get rid of the remaining term ζN(φ), using Corollary
+3.4: taking ε = 1/4 for example, we see in particular that
+√
+NζN(φ) converges in probability towards
+zero. The conclusion follows from Slutsky’s lemma.
+We now extend the result of Proposition 5.1 to a more general set of functions. With the notations
+of Proposition 5.1, we have
+Theorem 5.2. Let φ ∈ H2(R) ∩ C2(R) such that φ′′ is bounded. Additionally, suppose that V (3)φ2,
+V ′′φφ′, V ′′φ2 and V ′φ are bounded. Then, recalling (40) we have the convergence in distribution as N
+goes to infinity
+√
+NνN(Ξφ) → N(0, qP (φ)) .
+Proof. For N ⩾ 1, let E−
+N, E+
+N be given by Corollary 4.2. Let χN : R → [0, 1] be C2 with compact support
+such that
+χN(x) = 1 for x ∈ [E−
+N − 1, E+
+N + 1] and χN(x) = 0 for x ∈ [E−
+N − 2, E+
+N + 2]c
+and such that, denoting φN = φχN, supN ∥φ′
+N∥∞ + ∥φ′
+N∥L2(R), supN ∥φ′′
+N∥∞ + ∥φ′′
+N∥L2(R) < +∞ (we
+assumed φ ∈ H2(R), in particular φ′ is bounded and such a χN exists). The point of cutting φ outside
+the set [E−
+N −1, E+
+N +1] is that with high probability, the empirical measure ˆµN doesn’t see the difference
+between φ and φN.
+The support of φN is then contained in [E−
+N −2, E+
+N+2], and we now argue that the proof of Proposition
+5.1 can be adapted so that
+√
+NνN(ΞφN) → N(0, qP (φ)) .
+(46)
+Similarly as in Proposition 5.1, we perform in ZV,P
+N
+the change of variables xi = yi +
+t
+√
+N φN(yi),
+1 ⩽ i ⩽ N, which is the same as before, but with φ replaced by φN. First, with IN := [E−
+N − 2, E+
+N + 2],
+the error term
+KN(t, φN) ⩽ 2
+t3
+3N 1/2 ∥φ′
+N∥3
+∞ +
+t3
+6N 1/2 ∥φN∥∞
+sup
+d(x,IN)⩽1
+|V (3)(x)|
+19
+
+of the proof of Proposition 5.1 is still going to zero, because of our choice of χN and Assumption 1.2. As
+previously, we then have
+lim
+N→∞ EV,P
+N
+�
+et
+√
+NΛN (φN)+ t2
+2 FN(ˆµN )�
+= 1
+(47)
+with
+ΛN(φN) := 2P
+¨
+R2
+φN(x) − φN(y)
+x − y
+dµP (x)d(ˆµN − µP )(y) +
+ˆ
+R
+(φ′
+N − V ′φN)d(ˆµN − µP ) + PζN(φN) ,
+where ζN is given by (34), and
+FN(ˆµN) = −P
+¨
+R2
+�φN(x) − φN(y)
+x − y
+�2
+dˆµN(x)dˆµN(y) −
+ˆ
+R
+(V ′′φ2
+N + φ′2
+N)dˆµN .
+Taking again the distance dLip defined in (44), one can check that for µ, ν probability measures over R,
+|FN(µ) − FN(ν)| ⩽ CNdLip(µ, ν) ,
+where CN is a term depending on the norms ∥φ′
+N∥∞, ∥φ′′
+N∥∞, ∥V ′′φ2
+N∥∞ and ∥(V ′′φ2
+N)′∥∞. The choice
+of χN and the fact that φ is chosen so that V (3)φ2 and V ′′φφ′ are bounded guarantee that ∥(V ′′φ2
+N)′∥∞
+is bounded in N. The other norms are easily bounded by hypothesis. Therefore CN can be seen to be
+uniformly bounded in N, and we find some C ⩾ 0 independent of N such that
+|FN(µ) − FN(ν)| ⩽ CdLip(µ, ν) .
+As in proposition 5.1, we use the large deviation principle for (ˆµN) to deduce
+lim
+N→+∞ EV,P
+N
+�
+et
+√
+NΛN (φN)+ t2
+2 FN(ˆµN )�
+=
+lim
+N→+∞ EV,P
+N
+�
+et
+√
+NΛN (φN)�
+e
+t2
+2 FN (µP ) .
+By dominated convergence, FN(µP ) converges to F(µP ), the function F being given by (43). This shows
+the convergence as N goes to infinity
+lim
+N→+∞ EV,P
+N
+�
+et
+√
+NΛN (φN)�
+= e− t2
+2 F (µP ) ,
+and
+√
+N
+�
+νN(ΞφN)+PζN(φN)
+�
+converges towards a centered Gaussian variable with variance −F(µP ) =
+qP (φ). Because supN ∥φ′
+N∥L2(dx) + ∥φ′′
+N∥L2(dx) is finite, we can apply again Corollary 3.4 to deduce the
+convergence in law (46).
+We now have the ingredients to conclude, by showing that the characteristic function
+EV,P
+N
+�
+eit
+√
+NνN (Ξφ)�
+= EV,P
+N
+�
+eit
+√
+N
+´
+ΞφdˆµN�
+e−it
+√
+N
+´
+ΞφdµP
+converges to the characteristic of a Gaussian variable with appropriate variance. By Corollary 4.2, the
+probability under PV,P
+N
+of the event EN =
+�
+x1, . . . , xN ∈ [E−
+N − 1, E+
+N + 1]
+�
+converges to 1. Along with
+the convergence (46), we deduce
+e− t2
+2 qP (φ) = lim
+N EV,P
+N
+�
+eit
+√
+N
+´
+ΞφNdˆµN �
+e−it
+√
+N
+´
+ΞφNdµP = lim
+N EV,P
+N
+�
+1ENeit
+√
+N
+´
+ΞφNdˆµN�
+e−it
+√
+N
+´
+ΞφNdµP ,
+Where we used
+���EV,P
+N
+�
+1Ec
+Neit
+√
+N
+´
+ΞφNdˆµN �
+e−it
+√
+N
+´
+ΞφNdµP
+��� ⩽ PV,P
+N (Ec
+N) −−−−−→
+N→+∞ 0 .
+20
+
+Using that φN = φ on JN = [E−
+N − 1, E+
+N + 1],
+ˆ
+ΞφNdµP = 2P
+¨ φN(x) − φN(y)
+x − y
+dµP (x)dµP (y) +
+ˆ
+(φ′
+N − V ′φN)dµP
+= 2P
+¨
+J2
+N
+φ(x) − φ(y)
+x − y
+dµP (x)dµP (y) + 2P
+¨
+(J2
+N)c
+φN(x) − φN(y)
+x − y
+dµP (x)dµP (y)
++
+ˆ
+JN
+(φ′ − V ′φ)dµP +
+ˆ
+Jc
+N
+(φχ′
+N + φ′χN − V ′φχN)dµP .
+By boundedness of (∥φ′
+N∥∞)N, the second term is bounded by
+CP
+¨
+(J2
+N)c dµP dµP ⩽ 2CP µP (Jc
+N) = o(N −1/2) ,
+where we used the union bound and Lemma 4.4. By the same estimate and the fact that χN can be
+chosen so that (∥χ′
+N∥∞)N is bounded, and because φ′, V ′φ are bounded, the last term is also o(N −1/2).
+By the previous arguments, we also conclude that
+2P
+¨
+(J2
+N)c
+φ(x) − φ(y)
+x − y
+dµP (x)dµP (y) +
+ˆ
+Jc
+N
+(φ′ − V ′φ)dµP = o(N −1/2) ,
+thus
+ˆ
+ΞφNdµP =
+ˆ
+ΞφdµP + o(N −1/2) ,
+and so far we have
+e− t2
+2 qP (φ) = lim
+N EV,P
+N
+�
+1ENeit
+√
+N
+´
+ΞφNdˆµN�
+e−it
+√
+N
+´
+ΞφdµP .
+Finally, on EN, using φN = φ and that ˆµN is supported in JN,
+ˆ
+ΞφNdˆµN = 2P
+¨
+J2
+N
+φ(x) − φ(y)
+x − y
+dµP (x)dˆµN(y) + 2P
+¨
+(J2
+N)c
+φN(x) − φN(y)
+x − y
+dµP (x)dˆµN(y) +
+ˆ
+JN
+(φ′ − V ′φ)dˆµN
+= 2P
+¨ φ(x) − φ(y)
+x − y
+dµP (x)dˆµN(y) +
+ˆ
+(φ′ − V ′φ)dˆµN + o(N −1/2) ,
+Where in the second line we used, using Lemma 4.4 again, that
+¨
+(J2
+N)c
+φN(x) − φN(y)
+x − y
+dµP (x)dˆµN(y) =
+¨
+JN×Jc
+N
+φN(x) − φN(y)
+x − y
+dµP (x)dˆµN(y) = o(N −1/2) ,
+and the same estimate holds for φN replaced by φ. Therefore,
+e− t2
+2 qP (φ) = lim
+N EV,P
+N
+�
+1ENeit
+√
+N
+´
+ΞφdˆµN�
+e−it
+√
+N
+´
+ΞφdµP .
+This establishes that
+lim
+N EV,P
+N
+�
+eit
+√
+N
+´
+ΞφdˆνN�
+= e− t2
+2 qP (φ) ,
+which concludes the proof.
+Remark 5.3. Taking φ such that φ′ satisfies the conditions of Theorem 5.2, we then have
+(48)
+EV,P
+N
+�
+et
+√
+NνN (Lφ)�
+−→
+N→∞ exp
+�t2
+2 qP (φ′)
+�
+,
+21
+
+where the operator L is defined as Lφ := Ξφ′, ie
+Lφ = 2P
+ˆ
+R
+φ′(x) − φ′(y)
+x − y
+dµP (y) + φ′′(x) − V ′(x)φ′(x) .
+(49)
+Note that qV
+P (φ′) =
+�
+σV
+P
+�2(Lφ) where σV
+P is defined in (11). By Theorem 7.1, the class of functions in
+L−1(T ) where
+T :=
+�
+f ∈ C1(R), f = O(1/x), f ′ = O(1/x2),
+ˆ
+R
+fρP = 0
+�
+satisfies (48). This proves Theorem 1.3.
+6
+Inversion of L
+This section is dedicated to the definition of L given by (8) and its domain and then we focus on its
+inversion. We rely heavily on results of Appendix A: the diagonalization of the operator A by the use of
+the theory of Schrödinger operators.
+Let P > 0 be fixed. We introduce the operators A and W, acting on sufficiently smooth functions of
+L2(ρP ), by
+Aφ = −
+�
+φ′ρP
+�′
+ρP
+= −
+�
+φ′′ + ρ′
+P
+ρP
+φ′
+�
+and
+Wφ = −H
+�
+φ′ρP
+�
+.
+(50)
+We first show the following decomposition of L.
+Lemma 6.1. For φ twice differentiable we have the following pointwise identity
+−Lφ = Aφ + 2PWφ .
+(51)
+Proof. We write for x ∈ R
+2P
+ˆ
+R
+φ′(x) − φ′(y)
+x − y
+ρP (y)dy = −2Pφ′(x)H[ρP ](x) + 2PH[φ′ρP ](x) .
+(52)
+Then,
+Lφ = φ′′ − V ′φ′ − 2Pφ′H[ρP ] + 2PH
+�
+φ′ρP
+�
+.
+By (18) we have −V ′ − 2PH[ρP ] = ρ′
+P
+ρP , which concludes the proof.
+In order to state the next theorem, whose proof we detail in the Appendix, we introduce the following
+Sobolev-type spaces. Let
+H2
+V ′(R) :=
+�
+u ∈ H2(R), uV ′ ∈ L2(R)
+�
+.
+We now define
+H2
+V ′(ρP ) := ρ−1/2
+P
+H2
+V ′(R)
+and its homogeneous counterpart
+H2
+V ′,0(ρP ) :=
+�
+u ∈ H2
+V ′(ρP ),
+ˆ
+R
+uρP dx = 0
+�
+.
+Finally, we let L2
+0(ρP ) be the subset of L2(ρP ) of zero mean functions with respect to ρP .
+We detail the proof of the following theorem in Appendix A which is based on Schrödinger operators
+theory.
+22
+
+Theorem 6.2 (Diagonalization of A in L2
+0(ρP )). There exists a sequence 0 < λ1 < λ2 < . . . going to
+infinity, and a complete orthonormal set (φn)n⩾1 of L2
+0(ρP ) of associated eigenfunctions for A, meaning
+that
+• span{φn, n ⩾ 1} is dense in L2
+0(ρP ),
+• For all i, j, ⟨φi, φj⟩L2(ρP ) = ��i,j,
+• For all n ⩾ 1, Aφn = λnφn.
+Furthermore, each φn is in H2
+V ′,0(ρP ), A : H2
+V ′,0(ρP ) → L2
+0(ρP ) is bijective, and we have the writing, for
+u ∈ L2
+0(ρP )
+A−1u =
+�
+n⩾1
+λ−1
+n ⟨u, φn⟩L2(ρP ) φn .
+We see the operators A and W as unbounded operators on the space
+H =
+�
+u ∈ H1(ρP ) |
+ˆ
+R
+uρP dx = 0
+�
+endowed with the inner product ⟨u, v⟩H = ⟨u′, v′⟩L2(ρP ). This defines an inner product on H and makes
+it a complete space: it can be seen that H1(ρP ) is the completion of C∞
+c (R) with respect to the inner
+product ⟨u, v⟩L2(ρP ) +⟨u′, v′⟩L2(ρP ). The space H is then the kernel of the bounded (with respect to ∥·∥H)
+linear form, ⟨�1, ·⟩L2(ρP ) on H1(ρP ), and both inner products are equivalent on H because of the Poincaré
+inequality, Proposition 2.4. The use of H is motivated by the fact that both A and W are self-adjoint
+positive on this space as we show in Lemma 6.4.
+In the next proposition, we deduce from Theorem 6.2 the diagonalization of A in H.
+Proposition 6.3 (Diagonalization of A in H). With the same eigenvalues 0 < λ1 < λ2 < . . . as in
+Theorem 6.2, there exists a complete orthonormal set (ψn)n⩾1 of H formed by eigenfunctions of A.
+Proof. With (φn)n⩾1 of Theorem 6.2,
+δi,j = ⟨φi, φj⟩L2(ρP ) = 1
+λj
+⟨φi, Aφj⟩L2(ρP )
+= 1
+λj
+⟨φ′
+i, φ′
+j⟩L2(ρP )
+= 1
+λj
+⟨φi, φj⟩H.
+With ψn =
+1
+√λn φn, (ψn)n⩾1 is then orthonormal with respect to the inner product of H. To show that
+span{ψn, n ⩾ 1} is dense in H, let u ∈ H be such that for all j ⩾ 1, ⟨u, φj⟩H = 0. In the last series of
+equalities, replace φi by u: we see that u is orthogonal to each φj in L2(ρP ), thus u is a constant as
+shown in the proof of Lemma A.10, and because u ∈ H it has zero mean against ρP . This shows that
+u = 0.
+We set for what follows D(A) =
+�
+u ∈ H2
+V ′,0(ρP ) | Au ∈ H
+�
+and D(W) = {u ∈ H | Wu ∈ H}.
+Lemma 6.4. The following properties hold:
+• The operator W : D(W) → H is positive: for all φ ∈ D(W),
+⟨Wφ, φ⟩H = 1
+2∥φ′ρP ∥2
+1/2 ⩾ 0 ,
+with equality only for φ = 0, where the 1/2-norm of u is given by
+∥u∥2
+1/2 =
+ˆ
+R
+|x|. |F[u](x)|2 dx .
+23
+
+• Both A and W are self-adjoint for the inner product of H.
+Proof. To prove the first point, let φ ∈ D(W). Then,
+2π ⟨Wφ, φ⟩H = −2π ⟨H[φ′ρP ]′, φ′ρP ⟩L2(dx) = −
+�
+ixF
+�
+H[φ′ρP ]
+�
+, F[φ′ρP ]
+�
+L2(dx)
+= π ⟨ | x | F[φ′ρP ], F[φ′ρP ]⟩L2(dx) = π∥φ′ρP ∥2
+1/2 ⩾ 0 ,
+and because φ is in H, this last quantity is zero if and only if φ vanishes.
+For the second point, let u, v ∈ D(W). Using Plancherel’s isometry and i) of Lemma 2.1,
+⟨Wu, v⟩H = ⟨(Wu)′, v′ρP ⟩L2(dx) = 1
+2 ⟨ | x | F[u′ρP ], F[v′ρP ]⟩L2(dx) ,
+and this last expression is symmetric in (u, v).
+The proof of the self-adjointness of A follows from
+integration by partss.
+Definition 6.5 (Quadratic form associated to −L). We define for all u, v ∈ H ∩ C∞
+c (R) the quadratic
+form associated to −L by
+q−L(u, v) = ⟨Au, Av⟩L2(ρP ) + 2P ⟨F[u′ρP ], F[v′ρP ]⟩L2(|x|dx)
+Note that for all u, v ∈ H ∩ C∞
+c (R), q−L(u, v) = ⟨−Lu, v⟩H and that whenever u ∈ D(A) ∩ D(W),
+q−L(u, u) = ⟨Au, u⟩H + 2P ⟨Wu, u⟩H ⩾ λ1(A)∥u∥2
+H
+(53)
+by Proposition 6.3 and Lemma 6.4. After extending the q−L to its form domain Q(L) which is equal
+to
+�
+u ∈ H, Au ∈ L2(ρP ), F[u′ρP ] ∈ L2(|x|dx)
+�
+= H2
+V ′,0(ρP ). The equality comes from the fact that
+A−1�
+L2
+0(ρP )
+�
+= H2
+V ′,0(ρP ), that H ⊂ H2
+V ′,0(ρP ) and that F[u′ρP ] ∈ L2(x2dx) whenever u ∈ H2
+V ′,0(ρP ),
+indeed u′ρP ∈ H1(R) because (u′ρP )′ = −ρP Au ∈ L2(R). We now define D(L) the domain of definition
+of −L by:
+D(L) :=
+�
+u ∈ Q(L), v �→ q−L(u, v) can be extended to a continuous linear form on H
+�
+Proposition 6.6. D(L) = D(A) ∩ D(W).
+Proof. Let u ∈ D(L), by Riesz’s theorem there exists fu ∈ H, such that q−L(u, v) = ⟨fu, v⟩H for all v ∈ H,
+we set −Lu := fu, it is called the Friedrichs extension of −L. Then for all v ∈ H ∩ C∞
+c (R), by integration
+by part we get:
+⟨−Lu, v⟩H = q−L(u, v) = ⟨u, Av⟩H + 2P ⟨u, Wv⟩H ,
+hence we deduce the distributionnal identity −Lu = Au + 2PWu. Since u ∈ H2
+V ′,0(ρP ), Wu ∈ H1(ρP )
+implying that Au ∈ H and then that Wu ∈ H.
+We are now ready to state the main theorem of this section, that is the inversion of L on D(L).
+Theorem 6.7 (Inversion of L). −L : D(L) −→ H is bijective. Furthermore, (−L)−1 is continuous from
+(H, ∥.∥H) to (D(L), q−L).
+Proof. Let f ∈ H, since ⟨f, .⟩H is a linear form on Q(L) = H2
+V ′,0(ρP ) which is, by (53), continuous with
+respect to q−L, one can applies Riesz’s theorem so there exists a unique uf ∈ H2
+V ′,0(R), such that for all
+v ∈ H, ⟨f, v⟩H = q−L(uf, v). Since, uf is clearly in D(L) by definition of the Friedrichs extension of −L,
+we have −Lu = f.
+Remark 6.8. We can diagonalize L by the same argument we used in Appendix A to diagonalize A in
+L2
+0(ρP ).
+24
+
+We now state a result that could allow one to extract more regularity for L−1 by the help of an
+explicit form that uses Fredholm determinant theory for Hilbert-Schmidt operators, the reader can refer
+to [GGK12].
+Definition 6.9 (Fredholm determinant). Let U be a self-adjoint Hilbert-Schmidt operator, we denote the
+Fredholm determinant by det(I + U).
+Theorem 6.10 (Determinant formula for L−1). For all u ∈ H, such that x �→
+1
+ρP (x)
+ˆ +∞
+x
+u(t)ρP (t)dt
+is integrable at +∞, we have:
+L−1u = A−1u − ρ−1/2
+P
+R
+�
+ρ1/2
+P A−1u
+�
+(54)
+where R is the kernel operator defined for all v ∈ L2(R) by:
+R[v](x) =
+ˆ
+R
+R(x, y)v(y)dy
+where
+R(x, y) =
+1
+det(I + K)
+�
+n⩾0
+1
+n!
+ˆ
+Rn det
+n+1
+� K(x, y)
+K(x, λb)
+K(λa, y)
+K(λa, λb)
+�
+a,b=1...n
+dλ1 . . . dλn
+where K is the kernel operator defined for all w ∈ L2(ρP ) by:
+K[v](x) =
+ˆ
+R
+K(x, y)w(y)dy
+(55)
+with
+K(x, y) = −2PρP(x)ρP (y) ln
+���1 − y
+x
+���.
+(56)
+Proof. Let f ∈ H, there exists a unique u ∈ D(A) such that Au = f. Since (u′ρP )′ = ρP Au ∈ L2(R),
+hence u′ρP ∈ H1(R) so u′(x)ρP (x)
+−→
+|x|→+∞ 0 −(u′ρP )′
+ρP
+= f hence
+(A−1f)′(x)ρP (x) = u′(x)ρP (x) =
+ˆ +∞
+x
+f(t)ρP (t)dt.
+(57)
+Using the fact that
+´
+R u(x)ρP (x)dx = 0, integrating again we get:
+u(x) = −
+ˆ +∞
+x
+ds
+ρP (s)
+ˆ +∞
+s
+f(t)ρP (t)dt + C
+where C =
+ˆ
+R
+ρP (x)dx
+ˆ +∞
+x
+ds
+ρP (s)
+ˆ +∞
+s
+f(t)ρP (t)dt. Now let g ∈ H, there exists a unique v ∈ D(L),
+such that −Lv = Av + 2PWv = g and then v + 2PWA−1v = A−1g. using (57), we get:
+WA−1v(x) =
+ 
+R
+ds
+s − x
+ˆ +∞
+s
+dtv(t)ρP (t)
+By Sokhotski-Plejmel formula, we have:
+ 
+R
+ds
+s − x
+ˆ +∞
+s
+dtv(t)ρP (t) =
+lim
+M→+∞ lim
+ε→0
+ˆ M
+−M
+ds
+2
+�
+1
+x − s + iε +
+1
+x − s − iε
+� ˆ +∞
+s
+dtv(t)ρP (t)
+25
+
+We then proceed to an integration by part:
+ 
+R
+ds
+s − x
+ˆ +∞
+s
+dtv(t)ρP (t) =
+lim
+M→+∞ lim
+ε→0
+�
+− ln
+�
+(x − s)2 + ε2�
+2
+ˆ +∞
+s
+dtv(t)ρP (t)
+�M
+−M
+−
+ˆ
+R
+ds ln |x − s|v(s)ρP (s)
+To conclude that WA−1v(x) = −
+´
+R ds ln |x − s|v(s)ρP (s), we just need to show that
+ln(x)
+ˆ +∞
+x
+dtv(t)ρP (t) −→
+|x|→∞ 0
+which can be seen by Cauchy-Schwarz inequality:
+��� ln(x)
+ˆ +∞
+x
+dtv(t)ρP (t)
+��� ⩽ | ln(x)|∥v∥L2(ρP ).ρP (x)1/4� ˆ
+R
+ρP (t)1/2dt
+�1/2
+.
+In this inequality, we used that ρP is decreasing in a neighborhood of +∞, hence
+ln |x|
+ˆ +∞
+s
+dtv(t)ρP (t)
+−→
+x→+∞ 0
+the exact same argument allows us to conclude when x goes to −∞. Using the fact that
+´
+R v(t)ρP (t)dt = 0,
+we obtain the following equality:
+v − 2P
+ˆ
+R
+ds ln |x − s|v(s)ρP (s) = A−1g := h.
+Now setting ˜v(t) = ρ1/2
+P (t)v(t) and ˜h = ρ1/2
+P (t)h(t), we obtain ˜v + K[˜v] = ˜h where K is defined in
+(55). Since its kernel (defined in (56)) belongs to L2(R2), K is Hilbert-Schmidt. Hence by Fredholm
+determinant theory:
+˜v = ˜h − R[˜h]
+or L−1g = A−1g − ρ−1/2
+P
+R
+�
+ρ1/2
+P A−1g
+�
+as expected.
+7
+Regularity of the inverse of L and completion of the proof of
+Theorem 1.3
+Since we have proven the central limit theorem for functions of the type Lφ with φ regular enough and
+satisfying vanishing asymptotic conditions at infinity, we exhibit a class of functions f for which L−1f is
+regular enough to satisfy conditions of Theorem 5.2. We define T the subset of H defined by
+T :=
+�
+f ∈ C1(R), f(x) =
+O
+|x|→∞(x−1), f ′(x) =
+O
+|x|→∞(x−2),
+ˆ
+R
+fρP = 0
+�
+Theorem 7.1. For all f ∈ T , there exists a unique u ∈ C3(R) such that u′ ∈ H2(R) with u(3) bounded
+wich verifies:
+• u′(x) =
+O
+|x|→∞
+�
+1
+xV ′(x)
+�
+• u′′(x) =
+O
+|x|→∞
+�
+1
+xV ′(x)
+�
+26
+
+• u(3)(x) =
+O
+|x|→∞
+� 1
+x
+�
+such that f = Lu.
+Proof. Let f ∈ T ⊂ H, then since −L is bijective from D(L) → H, there exists a unique u ∈ D(L) such
+that −Lu = f ie:
+−u′′ − ρ′
+P
+ρP
+u′ − 2PH[u′ρP ] = f
+(58)
+Hence we have
+−(u′ρP )′ = ρP
+�
+f + 2PH[u′ρP ]
+�
+.
+(59)
+Since u ∈ D(L) ⊂ {u ∈ H2
+V ′,0(R), Au ∈ H}, the functions u′ρP and its derivatives (u′ρP )′ = −ρP Au and
+(u′ρP )′′ = −ρ′
+P
+ρP
+ρ1/2
+P .
+�
+ρ1/2
+P Au
+�
+− ρP
+�
+Au
+�′
+are in L2(dx). In particular u′ρP goes to zero at infinity, and
+H[u′ρP ] ∈ H2(R) ⊂ C1(R). So we can integrate (59) on [x, +∞[ , since by Lemma 2.3, the right-hand
+side behaves like a
+O
+|x|→∞
+�ρP (x)
+x
+�
+, to get the following expression
+u′(x)ρP (x) =
+ˆ +∞
+x
+ρP (t)
+ρ′
+P (t)(f + 2PH[u′ρP ]).ρ′
+P (t)dt
+(60)
+From this expression, we can see that u′ ∈ C2(R) so we just have to check the integrability condition at
+infinity and the fact that u(3) is bounded. After proceeding to an integration by parts, which is permitted
+by the previous argument, we obtain:
+u′(x) = −ρP (x)
+ρ′
+P (x)
+�
+f(x) + 2PH[u′ρP ](x)
+�
+−
+1
+ρP (x)
+ˆ +∞
+x
+�
+ρP (t)
+ρ′
+P (t)(f + 2PH[u′ρP ])
+�′
+ρP (t)dt
+(61)
+and we define R1(x) :=
+1
+ρP (x)
+ˆ +∞
+x
+�
+ρP (t)
+ρ′
+P (t)(f + 2PH[u′ρP ])
+�′
+ρP (t)dt, we will need to show that R1 is
+a remainder of order O
+�
+1
+xV ′(x)2
+�
+at infinity. In this case we will have u′(x) = O
+�
+1
+xV ′(x)
+�
+which will
+be useful for the following. If we reinject (61) in (58), we find:
+u′′ = −(f + 2PH[u′ρP ]) − ρ′
+P
+ρP
+�
+− ρP
+ρ′
+P
+�
+f + 2PH[u′ρP ]
+�
+− R1
+�
+= ρ′
+P
+ρP
+R1
+(62)
+Hence
+u′′(x) = ρ′
+P
+ρ2
+P
+(x)
+ˆ +∞
+x
+ρP (t)dt
+�
+�ρP
+ρ′
+P
+�′
+(t)
+�
+��
+�
+=
+O
+t→+∞
+�
+V ′′(t)
+V ′(t)2
+�
+�
+f + 2PH[u′ρP ]
+�
+(t)
+�
+��
+�
+=
+O
+t→+∞
+�
+1
+t
+�
++
+ρP
+ρ′
+P
+(t)
+� �� �
+=
+O
+t→+∞
+�
+1
+V ′(t)
+�
+�
+f ′ − 2PH[ρP Au]
+�
+(t)
+�
+��
+�
+=
+O
+t→+∞
+�
+1
+t2
+�
+�
+.
+The fact that H[ρP Au](t) =
+O
+t→+∞(t−2) comes again from lemma 2.3. Finally we have that,
+u(3)(x) =
+�ρ′
+P
+ρ2
+P
+�′
+(x)ρP (x)R1(x) −
+�ρ′
+P
+ρ2
+P
+�
+(x)
+�
+ρP
+ρ′
+P
+(f + 2PH[u′ρP ])
+�′
+(x)ρP (x)
+=
+� ρ′′
+P
+ρP
+− 2ρ′2
+P
+ρ2
+P
+�
+(x)
+�
+��
+�
+=
+O
+x→+∞
+�
+V ′(x)2
+�
+R1(x) −
+�ρ′
+P
+ρP
+�
+(x)
+�
+ρP
+ρ′
+P
+(f + 2PH[u′ρP ])
+�′
+(x)
+�
+��
+�
+=
+O
+x→+∞
+�
+V ′′(x)
+xV ′(x) +x−2
+�
+27
+
+The second term is
+O
+x→+∞
+�1
+x
+�
+by the assumption that V ′′
+V ′ (x) =
+O
+|x|→∞(1). Hence, we just have to check
+that R1(x) =
+O
+x→+∞
+�
+1
+xV ′(x)2
+�
+to establish that u′, u′′, u(3) are in L2(R). By a comparison argument,
+we control R1 by controlling
+I1(x) :=
+ˆ +∞
+x
+ρP (t)
+tV ′(t)dt
+By integration by parts:
+I1(x) := −
+ρP (x)
+xV ′(x)
+ρP
+ρ′
+P
+(x)
+�
+��
+�
+=
+O
+x→+∞
+�
+ρP (x)
+xV ′(x)2
+�
+−
+ˆ +∞
+x
+ρP (t)
+� 1
+tV ′
+ρP
+ρ′
+P
+�′
+(t)dt
+=
+O
+x→+∞
+� ρP (x)
+xV ′(x)2
+�
+−
+ˆ +∞
+x
+ρP (t)dt
+�
+−
+1
+t2V ′(t)
+ρP
+ρ′
+P
+(t) − V ′′(t)
+tV ′(t)2
+ρP
+ρ′
+P
++
+1
+tV ′(t)
+�ρP
+ρ′
+P
+�′
+(t)
+�
+(63)
+By the same argument as before, the last integral is of the form
+ˆ +∞
+x
+O
+t→+∞
+� ρP (t)
+tV ′(t)2
+�
+dt so if
+I2(x) :=
+ˆ +∞
+x
+ρP (t)
+tV ′(t)2 dt =
+O
+x→+∞
+� ρP (x)
+xV ′(x)2
+�
+then so is I1. By integration by parts, we obtain:
+I2(x) = ρP (x)
+1
+xV ′(x)2
+ρP
+ρ′
+P
+(x) −
+ˆ +∞
+x
+ρP (t)dt
+��ρP
+ρ′
+P
+�′
+(t)
+1
+tV ′(t)2 − ρP
+ρ′
+P
+(t)
+�
+1
+t2V ′(t)2 + 2V ′′(t)
+tV ′(t)3
+��
+=
+O
+x→+∞
+� ρP (x)
+xV ′(x)2
+�
++
+ˆ +∞
+x
+O
+t→+∞
+� ρP (t)
+tV ′(t)3
+�
+dt
+The last integral is a
+O
+x→+∞
+� ρP (x)
+xV ′(x)2
+�
+because, again, by integration by part:
+ˆ +∞
+x
+ρP (t)
+tV ′(t)3 dt = ρP (x)
+1
+xV ′(x)3
+ρP
+ρ′
+P
+(x) −
+ˆ +∞
+x
+O
+t→+∞
+� ρP (t)
+tV ′(t)4
+�
+and finally
+ˆ +∞
+x
+ρP (t)
+tV ′(t)4 dt ⩽
+ρP (x)
+xV ′(x)2
+ˆ +∞
+x
+dt
+V ′(t)2 =
+O
+x→+∞
+� ρP (x)
+xV ′(x)2
+�
+In the final step, we used the fact that x �→
+ρP (x)
+xV ′(x) is decreasing in a neighborhood of +∞ (which
+can be checked by differentiating) and that x �→
+1
+V ′(x)2 is integrable at ∞ by assumption iii). Hence
+R1(x) =
+O
+x→+∞
+�
+1
+xV ′(x)2
+�
+(the exact same result can be shown at −∞), which leads to the fact
+u′(x) =
+O
+|x|→+∞
+�
+1
+xV ′(x)
+�
+,
+u′′(x) =
+O
+|x|→+∞
+�
+1
+xV ′(x)
+�
+and
+u(3)(x) =
+O
+|x|→+∞
+�1
+x
+�
+(64)
+which establishes that these functions are in L2 in a neighborhood of ∞. Since we already showed that
+u ∈ C3(R) ⊂ H3
+loc(R), it establishes that u ∈ H3(R) ∩ C3(R) with u(3) bounded. To complete the proof
+we just have to show that (u′)2V (3), u′u′′V ′′, (u′)2V ′′ and u′V ′ are bounded which is easily checked by
+(64) and Assumption 1.1 iv).
+28
+
+A
+Appendix: proof of Theorem 6.2
+In order to analyze A, we let, for u ∈ L2(dx),
+Su := ρ1/2
+P Aρ−1/2
+P
+u .
+Note that u ∈
+�
+L2(dx), ∥.∥L2(dx)
+�
+�→ ρ−1/2
+P
+u ∈
+�
+L2(ρP ), ∥.∥L2(ρP )
+�
+is an isometry. It turns out that it will
+be easier to study first the operator S in order to get the spectal properties of A.
+Proposition A.1. The operator S is a Schrödinger operator: it admits the following expression for all
+u ∈ C2
+c (R): Su = −u′′ + wP u with
+wP = 1
+2
+�1
+2V ′2 − V ′′ + 2PV ′H[ρP ] − 2PH[ρ′
+P ] + 2P 2H[ρP ]2
+�
+= 1
+2
+�
+(ln ρP )′′ + 1
+2(ln ρP )′2�
+.
+Furthermore, wP is continuous and we have wP (x) ∼
+∞
+V ′(x)2
+4
+−→
+|x|→∞ +∞.
+Proof. We compute directly
+�
+ρP
+�
+ρ−1/2
+P
+u
+�′�′
+ρP
+=
+�
+ρ−1/2
+P
+u
+�′′ + ρ′
+P
+ρP
+�
+ρ−1/2
+P
+u
+�′
+=
+�
+ρ−1/2
+P
+u′ − 1
+2ρ−3/2
+P
+ρ′
+P u
+�′ + ρ′
+P ρ−3/2
+P
+u′ − 1
+2ρ−5/2
+P
+�
+ρ′
+P
+�2u
+= ρ−1/2
+P
+u′′ + 1
+4ρ−5/2
+P
+�
+ρ′
+P
+�2u − 1
+2ρ−3/2
+P
+ρ′′
+P u
+= ρ−1/2
+P
+�
+u′′ + 1
+4ρ−2
+P
+�
+ρ′
+P
+�2u − 1
+2ρ−1
+P ρ′′
+P u
+�
+= ρ−1/2
+P
+�
+u′′ − 1
+2
+��ρ′′
+P
+ρP
+�
+− 1
+2
+�ρ′
+P
+ρP
+�2�
+u
+�
+= ρ−1/2
+P
+�
+u′′ − 1
+2
+�
+(ln ρP )′′ + 1
+2(ln ρP )′2�
+u
+�
+= ρ−1/2
+P
+�
+u′′ − wP u
+�
+.
+Now, using Lemma 2.2, we have
+wP = 1
+2
+�1
+2V ′2 − V ′′ + 2PV ′H[ρP ] − 2PH[ρ′
+P ] + 2P 2H[ρP ]2
+�
+.
+Notice that H[ρ′
+P ] and H[ρP ] are bounded since they belong to H1(R), as we showed in Lemma 2.2 that
+ρP is H2(R). Along with Assumption 1.1iii) and Lemma 2.3, we deduce wP (x) ∼
+∞
+1
+4V ′2(x).
+Remark A.2. Note that the function wP need not be positive on R. In fact, neglecting the terms involving
+the Hilbert transforms of ρP and ρ′
+P , wP would only be positive outside of a compact set. However, using
+the positivity of A, which will be shown further in the article, we can show that the operator −u′′ + wP u
+is itself positive on L2(dx). It can also be checked that, by integration by partss, S is symmetric on C2
+c(R)
+with the inner product of L2(dx).
+We now introduce an extension of S by defining its associated bilinear form.
+Definition A.3 (Quadratic form associated to S).
+Let α ⩾ 0 such that wP + α ⩾ 1. We define the quadratic form associated to S + αI, defined for all
+u ∈ C2
+c(R) by
+qα(u, u) :=
+ˆ
+R
+u′2dx +
+ˆ
+R
+u2(wP + α)dx
+29
+
+This quadratic form can be extended to a larger domain denoted by Q(S+αI), called the form domain
+of the operator S + αI. By the theory of Schrödinger operators, it is well-known (see [Dav96][Theorem
+8.2.1]) that such a domain is given by
+Q(S + αI) =
+�
+u ∈ H1(R), u(wP + α)1/2 ∈ L2(R)
+�
+=
+�
+u ∈ H1(R), uV ′ ∈ L2(R)
+�
+=: H1
+V ′(R) .
+The space H1
+V ′(R) can be seen to be the completion under the norm qα of C∞
+c . Now that the quadratic
+form associated to S + αI has been extended to its form domain, it is possible to go back to the operator
+and extend it by its Friedrichs extension.
+Theorem A.4 (Friedrichs extension of S + αI).
+There exists an extension (S + αI)F of the operator S + αI, called the Friedrichs extension of S + αI
+defined on H2
+V ′(R) :=
+�
+u ∈ H2(R), uV ′ ∈ L2(R)
+�
+.
+Proof. We denote
+D
+�
+(S + αI)F
+�
+=
+�
+v ∈ H1
+V ′(R), u ∈ H1
+V ′(R) �−→ qα(u, v) can be extended to a continuous linear form on L2(R)
+�
+If v ∈ D
+�
+(S + αI)F
+�
+, by Riesz’s theorem there exists a unique fv ∈ L2(R) such that qα(u, v) =
+⟨u, fv⟩L2(dx) holds for all u ∈ L2(R) and we can set (S + αI)F v := fv. Note that it is indeed a way
+of extending S + αI since for all u, v ∈ C2
+c(R), qα(u, v) = ⟨u, (S + αI)v⟩L2(dx).
+We want to show that D
+�
+(S+αI)F
+�
+= H2
+V ′(R). Let f ∈ D
+�
+(S+αI)F
+�
+and g := (S+αI)F f ∈ L2(R).
+By definition of qα, for all u ∈ C2
+c(R):
+ˆ
+R
+gudx =
+ˆ
+R
+f ′u′dx +
+ˆ
+R
+(wP + α)fudx = −
+ˆ
+R
+fu′′dx +
+ˆ
+R
+(wP + α)fudx
+Therefore in the sense of distributions, we get −f ′′ = g − (wP + α)f which is a function in L2(R), hence
+f ∈ H2
+V ′(R). Conversely, if f ∈ H2
+V ′(R), it is possible to extend u �→ qα(f, u) to a continuous linear form
+on L2(R) by
+u �→ −
+ˆ
+R
+uf ′′dx +
+ˆ
+R
+uf(wP + α)dx
+which concludes the fact that D
+�
+(S + αI)F
+�
+= H2
+V ′(R).
+In the following, we will deal only with (S + αI)F : H2
+V ′(R) −→ L2(R) and denote it (S + αI).
+Remark A.5. Note that in the previous proof, the application of Riesz’s theorem doesn’t allow to say that
+(S + αI) : v ∈
+�
+H2
+V ′(R), ∥.∥qα
+�
+�→ fv ∈
+�
+L2(R), ∥.∥L2(dx)
+�
+, where ∥.∥qα stands for the norm associated
+to the bilinear positive definite form qα, is continuous. It can be seen by the fact that
+v ∈
+�
+D(S + αI), ∥.∥qα
+�
+�→ q(., v) ∈
+�
+L2(R)′, ∥.∥L2(dx)′
+�
+, where L2(R)′ stands for the topological dual of
+L2(R) equipped with its usual norm, is not continuous. Indeed the ∥.∥qα norm doesn’t control the second
+derivative of v and hence doesn’t provide any module of continuity for the L2(R)-extended linear form
+q(., v).
+Theorem A.6 (Inversion of S + αI).
+For every f ∈ L2(R), there exists a unique u ∈ H2
+V ′(R) such that (S + αI)u = f. Furthermore, the map
+(S + αI)−1 is continuous from
+�
+L2(R), ∥.∥L2(dx)
+�
+to
+�
+H2
+V ′(R), ∥.∥qα
+�
+.
+30
+
+Proof. Let f ∈ L2(R), the map u �−→ ⟨u, f⟩L2(dx) is continuous on
+�
+H1
+V ′(R), ∥.∥qα
+�
+which is a Hilbert
+space. Therefore by Riesz’s theorem, there exists a unique vf ∈ H1
+V ′(R) such that for all u ∈ H1
+V ′(R),
+⟨f, u⟩L2(dx) = qα(vf, u) from which we deduce that, in the sense of distributions, f = −v′′
+f + (wP + α)vf
+which implies that vf ∈ H2
+V ′(R). Since vf ∈ D
+�
+S + αI
+�
+, we have then for all u ∈ L2(R), ⟨f, u⟩L2(dx) =
+qα(vf, u) = ⟨(S + α)vf, u⟩L2(dx), hence (S + αI)vf = f. Finally, by Riesz’s theorem, f ∈ L2(R) �→ vf ∈
+H1
+V ′(R) is continuous hence so is (S + αI)−1.
+Remark A.7. It would be tempting to use Banach’s isomorphism theorem to say that since (S + αI)−1
+is bijective and continuous, so must be S + αI. But since
+�
+H2
+V ′(R), ∥.∥qα
+�
+is not a Banach space (it’s not
+closed in H1
+V ′(R)) we can’t apply it.
+We are now able to diagonalize the resolvent of S.
+Theorem A.8 (Diagonalization of (S + αI)−1).
+There exists a complete orthonormal set (ψn)n⩾0 of L2(dx) (meaning that
+span{ψn, n ⩾ 0}
+∥.∥L2(dx) = L2(dx)
+and ⟨ψi, ψj⟩L2(dx) = δi,j), where each ψn ∈ H2
+V ′ and
+�
+µn(α)
+�
+n⩾0 ∈ [0, 1]N with µn(α) −→
+N→∞ 0 such that
+(S + αI)−1ψn = µn(α)ψn for all n ⩾ 0. We also have
+���
+���
+���
+�
+S + αI
+�−1���
+���
+���
+L
+�
+L2(dx)
+� ⩽ 1.
+Proof. By Proposition A.1, wP + α is continuous and goes to infinity at infinity. By Rellich criterion
+[RS78][see Theorem XIII.65], the unit ball of H2
+V ′(R), ie the set
+�
+u ∈ H2
+V ′(R),
+ˆ
+R
+u′2 +
+ˆ
+R
+(wP + α)u2 ⩽ 1
+�
+considered as a subset of L2(R) is relatively compact in
+�
+L2(dx), ∥.∥L2(dx)
+�
+. Hence, we can conclude
+that the injection ι :
+�
+H2
+V ′(R), ∥.∥qα
+�
+−→
+�
+L2(dx), ∥.∥L2(dx)
+�
+is a compact operator. Since (S + αI)−1 :
+�
+L2(dx), ∥.∥L2(dx)
+�
+−→
+�
+H2
+V ′(R), ∥.∥qα
+�
+is continuous then (S+αI)−1 is compact from
+�
+L2(dx), ∥.∥L2(dx)
+�
+to itself. The fact that (S + αI)−1 is self-adjoint and positive allows us to apply the spectral theorem
+to obtain
+�
+µn(α)
+�
+n⩾0 positive eigenvalues verifying µn(α) −→
+N→∞ 0 by compactness and a Hilbertian basis
+(ψn)n⩾0 ∈ L2(R)N, such that for all n ⩾ 0, (S + αI)−1ψn = µn(α)ψn. It is then easy to see that for
+all n, ψn ∈ H2
+V ′(R) since they belong to the range of (S + αI)−1. Finally, since for all φ ∈ L2(R),
+⟨(S + αI)φ, φ⟩L2(dx) ⩾ ∥φ∥2
+L2(dx), the spectrum of (S + αI)−1 is contained in [0, 1].
+It allows us to
+conclude that
+������(S + αI)−1������
+L2(dx) ⩽ 1.
+It is now straightforward to see how to extend A = ρ−1/2
+P
+Sρ1/2
+P
+on H2
+V ′(ρP ) := ρ−1/2
+P
+H2
+V ′(R) equipped
+with the norm ∥.∥qα,ρP to
+�
+L2(ρP ), ∥.∥L2(ρP )
+�
+. The norm ∥.∥qα,ρP is defined for all u ∈ H2(ρP ) by
+∥u∥qα,ρP =
+ˆ
+R
+u′2ρP dx +
+ˆ
+R
+u2(wP + α)ρP dx .
+It is easy to see that (A + αI)−1 is continuous. We stress the fact that H2
+V ′(ρP ) ̸=
+�
+u ∈ H2(ρP ), uV ′ ∈
+L2(ρP )
+�
+. Indeed if u ∈ H2
+V ′(R), even though uρ−1/2
+P
+and its derivative belong to L2(ρP ), (ρ−1/2
+P
+u)′′ /∈
+L2(ρP ). The reader can check that for such a function to be in L2(ρP ), it would be necessary to have
+that u2V ′4 ∈ L2(dx) which is not the case.
+Remark A.9. The kernel of A is generated by the function �1. Indeed if φ ∈ H2
+V ′(ρP ) is in the kernel of
+A then
+31
+
+0 = −
+�
+φ′ρP
+�′
+ρP
+⇒ ∃c ∈ R, φ′ = c
+ρP
+But since φ′ is in L2(ρP ) then c = 0 which implies that φ is constant. We must restrict A to the orthogonal
+of KerA with respect to the inner product of L2(ρP ), ie
+H2
+V ′,0(ρP ) :=
+�
+u ∈ H2
+V ′(ρP ) |
+ˆ
+R
+uρP = 0
+�
+.
+Doing so makes A injective.
+Before inverting A, we need the following lemma:
+Lemma A.10. The following equality holds
+(A + αI)
+�
+H2
+V ′,0(ρP )
+�
+= L2
+0(ρP ) :=
+�
+u ∈ L2(ρP ),
+ˆ
+R
+uρPdx = 0
+�
+Proof. Let φ = �c for c ∈ R, (A + αI)φ = �
+αc then (A + αI)(R.�1) = R�1. Hence since A + αI is self-adjoint
+with respect to the inner product of L2(ρP ) and that R�1 is stable by A + αI, then (A + αI)
+�
+(R.�1)⊥ ∩
+H2
+V ′(ρP )
+�
+⊂ (R.�1)⊥. For the converse, let u ∈ (R.�1)⊥, since A + αI is bijective, there exists v ∈ H2
+V ′(ρP )
+such that u = (A + αI)v. For all w ∈ R.�1,
+0 = ⟨u, w⟩L2(ρP ) = ⟨(A + αI)v, w⟩L2(ρP ) = ⟨v, (A + αI)w⟩L2(ρP )
+Hence v ∈
+�
+(A + αI)(R�1)
+�⊥ = R�1⊥ and so (R.�1)⊥ ⊂ (A + αI)
+�
+(R.�1)⊥�
+.
+It is easy to see that L2
+0(ρP ) is a closed subset of L2(ρP ) as it is the kernel of the linear form
+φ ∈ L2(ρP ) �→
+�
+φ,�1
+�
+L2(ρP ), making it a Hilbert space.
+Proposition A.11 (Diagonalization and invertibility of A). There exists a complete orthonormal set of
+�
+L2
+0(ρP ), ⟨., .⟩L2(ρP )
+�
+, (φn)n∈N ∈ H2
+V ′,0(ρP )N such that Aφn = λnφn (meaning that
+span{φn, n ⩾ 0}
+∥.∥L2(ρP ) = L2
+0(ρP )
+and ⟨φi, φj⟩L2(ρP ) = δi,j). Furthermore, A : H2
+V ′,0(ρP ) −→ L2
+0(ρP ) :=
+�
+u ∈ L2(ρP ),
+´
+R uρPdx = 0
+�
+is
+bijective, A−1 is continuous when considered as an operator of L2
+0(ρP ).
+Proof. Since (S + αI)−1 considered as an operator of L2(dx), is compact so is (A + αI)−1 on L2(ρP )
+and since A is self-adjoint, by the spectral theorem, (A + αI)−1 is diagonalizable. With the notations of
+Theorem A.8, (A+αI)−1 has eigenvalues
+�
+µn(α)
+�
+n⩾0 and corresponding eigenfunctions φn = ρ−1/2
+P
+ψn ∈
+H2
+V ′(ρP ). Hence for all n ∈ N, Aφn = λnφn with λn :=
+�
+1
+µn(α) − α
+�
+. Now,
+λn∥φn∥L2(ρP ) =
+ˆ
+R
+(Aφn)φnρP dx = −
+ˆ
+R
+(ρP φ′
+n)′φn =
+ˆ
+R
+φ′2
+n ρP ⩾ 0 .
+Furthermore, the kernel of A is R.�1, thus the spectrum of A restricted to H2
+V ′,0(ρP ) is positive. But
+since (A + αI)−1 is a compact operator of L2(ρP ) and that (A + αI) maps R.�1⊥ to R.�1⊥ with respect
+to the inner product of L2(ρP ) (see lemma A.10), then
+�
+A + αI
+�−1 is compact as an operator from
+L2
+0(ρP ) to itself. By Fredholm alternative, for every λ ∈ R λ ̸= 0, either (A + αI)−1 − λI is bijective
+32
+
+either λ ∈ Sp
+�
+(A + αI)−1�
+. These conditions are equivalent to: either A + (α − 1
+λ)I is bijective as an
+operator from H2
+V ′,0(ρP ) to L2
+0(ρP ), either −α + 1
+λ ∈ Sp
+�
+A
+�
+. If we set λ = 1
+α then either A is bijective
+either 0 ∈ Sp(A), since the latter is wrong then A : H2
+V ′,0(ρP ) → L2
+0(ρP ) is bijective. The spectrum of
+A is
+�
+1
+µn(α) − α
+�
+n⩾0
+⊂ (λ1, +∞) ⊂ (0, +∞), where λ1 is the smallest eigenvalue. Hence, we deduce
+������A−1������
+L(L2(ρP )) ⩽ λ−1
+1 .
+References
+[ABG12]
+R. Allez, J.P. Bouchaud, and A. Guionnet. Invariant beta ensembles and the gauss-wigner
+crossover. Physical Review Letters, Aug. 2012.
+[AGZ10]
+G. W. Anderson, A. Guionnet, and O. Zeitouni. An introduction to random matrices, volume
+118 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge,
+2010.
+[BBCG08] D. Bakry, F. Barthe, P. Cattiaux, and A. Guillin. A simple proof of the Poincaré inequality
+for a large class of probability measures including the log-concave case. Electron. Commun.
+Probab., 13:60–66, 2008.
+[BFG15]
+F. Bekerman, A. Figalli, and A. Guionnet. Transport maps for β-matrix models and univer-
+sality. Communications in mathematical physics, 338(2):589–619, 2015.
+[BG13a]
+G. Borot and A. Guionnet. Asymptotic expansion of β matrix models in the one-cut regime.
+Comm. Math. Phys., 317(2):447–483, 2013.
+[BG13b]
+G. Borot and A. Guionnet. Asymptotic expansion of beta matrix models in the several-cut
+regime. arxiv 1303.1045, 2013.
+[BGK16]
+G. Borot, A. Guionnet, and K. K. Kozlowski. Asymptotic expansion of a partition function
+related to the sinh-model. Mathematical Physics Studies. Springer, [Cham], 2016.
+[BGP15]
+F. Benaych-Georges and S. Péché. Poisson statistics for matrix ensembles at large tempera-
+ture. Journal of Statistical Physics, 161(3):633–656, 2015.
+[BLS18]
+Florent Bekerman, Thomas Leblé, and Sylvia Serfaty. Clt for fluctuations of β-ensembles with
+general potential. Electronic Journal of Probability, 23:1–31, 2018.
+[BMP22]
+P. Bourgade, K. Mody, and M. Pain. Optimal local law and central limit theorem for β-
+ensembles. Communications in Mathematical Physics, 390(3):1017–1079, 2022.
+[Dav96]
+E. B. Davies. Spectral theory and differential operators, volume 42. Cambridge University
+Press, 1996.
+[DE02]
+I. Dumitriu and A. Edelman. Matrix models for beta ensembles. J. Math. Phys., 43:5830–5847,
+2002.
+[GGK12]
+I. Gohberg, S. Goldberg, and N. Krupnik. Traces and determinants of linear operators, volume
+116. Birkhäuser, 2012.
+[GM22]
+A. Guionnet and R. Memin. Large deviations for gibbs ensembles of the classical toda chain.
+Electron. J. Probab., 27:1–29, 2022.
+[Gui19]
+A. Guionnet.
+Asymptotics of random matrices and related models:
+the uses of Dyson-
+Schwinger equations, volume 130. American Mathematical Soc., 2019.
+33
+
+[GZ19]
+D. García-Zelada. A large deviation principle for empirical measures on Polish spaces: ap-
+plication to singular Gibbs measures on manifolds. Ann. Inst. Henri Poincaré Probab. Stat.,
+55(3):1377–1401, 2019.
+[HL21]
+A. Hardy and G. Lambert. Clt for circular beta-ensembles at high temperature. Journal of
+Functional Analysis, 280(7):108869, 2021.
+[Joh98]
+K. Johansson. On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J.,
+91:151–204, 1998.
+[Kin09]
+F. W. King. Hilbert Transforms: Volume 1 (Encyclopedia of Mathematics and its Applica-
+tions). 1 edition, 2009.
+[Lam21]
+G. Lambert. Poisson statistics for gibbs measures at high temperature. 57(1):326–350, 2021.
+[MMS14]
+M. Maïda and É. Maurel-Segala. Free transport-entropy inequalities for non-convex potentials
+and application to concentration for random matrices. Probability Theory and Related Fields,
+159(1):329–356, 2014.
+[NT18]
+F. Nakano and K. D. Trinh. Gaussian beta ensembles at high temperature: eigenvalue fluc-
+tuations and bulk statistics. J.Stat.Phys, 173:295–321, 2018.
+[NT20]
+F. Nakano and K. D. Trinh. Poisson statistics for beta ensembles on the real line at high
+temperature. J. Stat. Phys., 179(2):632–649, 2020.
+[Pak18]
+C. Pakzad. Poisson statistics at the edge of gaussian beta-ensembles at high temperature.
+arXiv preprint arXiv:1804.08214, 2018.
+[RS78]
+M. Reed and B. Simon. Methods of modern mathematical physics, vol. 4, 1978.
+[Shc13]
+M. Shcherbina. Fluctuations of linear eigenvalue statistics of β matrix models in the multi-cut
+regime. J. Stat. Phys., 151(6):1004–1034, 2013.
+[Shc14]
+M. Shcherbina. Change of variables as a method to study general β-models: bulk universality.
+Journal of Mathematical Physics, 55(4):043504, 2014.
+[Spo20]
+H. Spohn. Generalized Gibbs Ensembles of the Classical Toda Chain. J. Stat. Phys., 180(1-
+6):4–22, 2020.
+[Spo21]
+H. Spohn. Hydrodynamic equations for the toda lattice. ArXiv 2101.06528, 2021.
+34
+

8dE1T4oBgHgl3EQfUAND/content/tmp_files/2301.03084v1.pdf.txt

@@ -0,0 +1,1494 @@
+Dynamic Binary Search Trees: Improved Lower
+Bounds for the Greedy-Future Algorithm
+Yaniv Sadeh � �
+Tel Aviv University, Israel
+Haim Kaplan � �
+Tel Aviv University, Israel
+Abstract
+Binary search trees (BSTs) are one of the most basic and widely used data structures. The best
+static tree for serving a sequence of queries (searches) can be computed by dynamic programming.
+In contrast, when the BSTs are allowed to be dynamic (i.e. change by rotations between searches),
+we still do not know how to compute the optimal algorithm (OPT) for a given sequence. One of the
+candidate algorithms whose serving cost is suspected to be optimal up-to a (multiplicative) constant
+factor is known by the name Greedy Future (GF). In an equivalent geometric way of representing
+queries on BSTs, GF is in fact equivalent to another algorithm called Geometric Greedy (GG). Most
+of the results on GF are obtained using the geometric model and the study of GG. Despite this
+intensive recent fruitful research, the best lower bound we have on the competitive ratio of GF is 4
+3.
+Furthermore, it has been conjectured that the additive gap between the cost of GF and OPT is only
+linear in the number of queries. In this paper we prove a lower bound of 2 on the competitive ratio
+of GF, and we prove that the additive gap between the cost of GF and OPT can be Ω(m · log log n)
+where n is the number of items in the tree and m is the number of queries.
+2012 ACM Subject Classification Theory of computation → Online algorithms; Theory of compu-
+tation → Sorting and searching
+Keywords and phrases Binary Search Trees, Greedy Future, Geometric Greedy, Lower Bounds,
+Dynamic Optimality Conjecture
+Digital Object Identifier 10.4230/LIPIcs.STACS.2023.39
+Funding The work of the authors is partially supported by Israel Science Foundation (ISF) grant
+number 1595-19, German Science Foundation (GIF) grant number 1367 and the Blavatnik research
+fund at Tel Aviv University.
+© Yaniv Sadeh and Haim Kaplan;
+licensed under Creative Commons License CC-BY 4.0
+40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023).
+Editors: Petra Berenbrink, Mamadou Moustapha Kanté, Patricia Bouyer, and Anuj Dawar; Article No. 39;
+pp. 39:1–39:22
+Leibniz International Proceedings in Informatics
+Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
+arXiv:2301.03084v1  [cs.DS]  8 Jan 2023
+
+39:2
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+1
+Introduction
+Binary search trees (BSTs) are one of the most basic and widely used data-structures. They
+are used to store a sorted set of keys from a totally ordered universe. Traversing BSTs is
+usually done by using a single pointer, initially pointing to the root, and moving to the left or
+right child according to the order of the searched key and the key of the item at the current
+node. Therefore, we typically define the cost1 of a search to be the length of the search path.
+The data structure itself may be static, or change dynamically throughout time, in response
+to insertions and deletions of items, and possibly even restructured during queries.
+Static BSTs are well understood. One can guarantee that the longest path from the root
+to a leaf is of length O(log n) if the number of keys is n, by using a balanced tree. If the access
+sequence is known in advance (in fact only the frequency of accesses of each key matters)
+then an O(n2) time algorithm computing the optimal static tree for the particular set of
+frequencies was given by Knuth [13]. It is also notable that the lower bound on the cost when
+the known frequencies are ⃗f = [f1, f2, . . . , fn] and the number of queries is m, is Ω(m · H(⃗f))
+where H(⃗f) = �n
+i=1 fi log 1
+fi is the entropy function. A simple way with O(n log n) running
+time to construct a near-optimal static (centroid) tree whose cost is O(m · H(⃗f)), has been
+described by Mehlhorn [17]. The running time has been improved to O(n) by Fredman [10].
+In contrast to the static case, the dynamic case is less understood. One can, of course,
+serve the sequence with a static tree. But, for many sequences we must change the structure
+of the tree as we make the searches in order to be efficient. For example, the requested items
+may be different in different parts of the sequence so a different set of items has to be placed
+near the root during different parts of the sequence. Restructuring is done by rotations that
+maintain the symmetric order. When rotations are allowed, the cost is defined to be the size
+of the subtree that contains the search path and all edges which we rotate.
+Here, we assume that the set of values stored in the tree does not change (no insertions
+or deletions), yet restructuring the tree is allowed to speed up future searches. One famous
+dynamic algorithm for doing this is the Splay algorithm of Sleator and Tarjan [20]. After
+each query, the splay algorithm moves the queried item to the root of the tree, according to
+three simple rules called zig-zag, zig-zig and zig. The splay algorithm is efficient in the sense
+that it is able to exploit the structure of many families of sequences. In particular splay is
+proven to be as good as the static optimum (up to a constant factor), which also implies that
+the cost of splay on any given sequence is at most O(log n) times the (dynamic) optimum
+cost. Sleator and Tarjan conjectured that splay is in fact dynamically-optimal, meaning that
+its cost is like the cost of an optimal algorithm that knows the whole sequence of queries
+in advance, up to some constant factor. However, this dynamic-optimality conjecture of
+splay is still open. In fact, it is open whether there is any dynamically-optimal online binary
+search tree algorithm. The best competitive ratio achievable to date is O(log log n), and
+it is obtained by Tango [8], Multi-splay [21] and Chain-splay [11] trees, and a geometric
+divide-and-conquer approach of [1].
+While seeking for (better) guaranteed competitiveness, other dynamic algorithms were
+considered. A promising candidate was independently proposed by Lucas [16] and Munro [18],
+which is now commonly referred to as Greedy Future, henceforth: GF in short. As its name
+suggests, GF is a greedy algorithm that rearranges the nodes on the path from the root to
+the current queried item as a treap whose priorities are according to the future accesses2
+1 Our cost model is formally defined in Definition 1, in Section 2.
+2 Each item in a treap has two keys: value and priority. The treap is a binary search tree with respect to
+
+Y. Sadeh and H. Kaplan
+39:3
+(as this paper deals with analyzing GF, we detail it formally in Algorithm 1). Note that
+unlike splay, GF, by definition, is required to know the future in order to restructure the tree.
+Surprisingly however, Demaine et al. [7] showed that one can simulate GF without knowing
+the future by a hierarchy of split-trees while losing only a constant factor in performance.
+Additionally, [7] presented a geometric view of an algorithm serving queries by a dynamic
+binary search tree using a two dimensional grid on which we mark the sequence as well
+as the items accessed by the algorithm. In this presentation there is yet another natural
+promising candidate for dynamic optimality, which is commonly known as Geometric Greedy
+and sometimes simply Greedy, which we shall refer to as GG. [7] showed that GG is in fact
+the same algorithm as GF.
+The geometric view proved useful to obtain new results regarding GG and hence GF.
+Fox [9] proved that an access-lemma that is analogues to the so called access-lemma of splay
+trees holds for GG. From this follows that most of the nice properties that hold for splay also
+hold for GF. In particular, it follows that GF is O(log n) competitive. Chalermsook et al. [3]
+analyzed upper bounds on the cost of GG for access patterns which are permutations, and in
+particular found that for highly structured permutations, which they called k-decomposable,
+the cost is n · 2α(n)O(k) where α(n) is the inverse-Ackermann function. Chalermsook et
+al. [5] study special access patterns that belong to a broader family of pattern-avoiding
+permutations. See [4] for a survey of currently known properties of greedy and splay.
+Our Contributions:
+1. It is known that GF is not exactly optimal, but it is conjectured, like splay, to be
+optimal up to a constant factor. In fact, it has been even more strongly conjectured
+by Demaine et al. [7] to be optimal up to an additive O(m) term, and possibly even
+exactly m. Kozma [14] refuted the second part and gave a specific sequence for which
+this additive gap is m + 1. In this paper we refute the linear gap conjecture and show a
+family of sequences for which the additive gap is at least Ω(m log log n).
+2. The largest lower bound on the competitive ratio of GF is 4
+3 by Demaine et al. [7]. They
+show a family of sequences on which after an initial query, the optimum pays 1.5 on
+average per query while GF pays 2.3 We describe a technique that allows us to improve
+this lower bound to 2. We note that the best known lower bound on the competitive
+ratio of splay is 2 (see [15, Section 2.5]). In both cases, the construction requires a rather
+large number of items (large n).
+3. Based on the multiplicative lower bound described above we show the following two
+interesting properties of GF: (1) There are sequences X such that the cost of GF on the
+reverse sequence is twice larger than the cost of GF on X. (2) There are sequences X
+such that we can remove some queries from them and get a subsequence X′, such that
+the cost of GF on X′ is twice larger than the cost of GF on X.
+4. Based on the additive lower bound described above, the two properties of GF in the
+previous bullet also hold if we replace the (multiplicative) relation of “costs twice more”
+by the (additive) relation “costs Ω(m log log n) more”.
+We study subsequences and reversal (contributions 3-4) since any dynamically-optimal
+the values of the items and a heap with respect to their priorities. That is, the priority of an item is
+no larger than the priorities of its children. In our case, the priorities are deterministically defined by
+future requests in a way that we define precisely in Algorithm 1.
+3 Reddmann [19] found an example in which the cost ratio between GF and the optimum is 26
+17 ≈ 1.53.
+But this is for one particular sequence of a fixed length so it does not rule out any competitive ratio if
+we allow an additive constant.
+STACS 2023
+
+39:4
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+algorithm A must have a “nice” behavior in these cases. Concretely, A must satisfy the
+approximately-monotone property (Definition 8) which states that there is a fixed constant c
+such that the cost of A on any subsequence of any sequence is never more than c times the
+cost on the whole sequence. As for reversal, the optimum can process a sequence and its
+reversal with similar costs up to a difference of n, thus any dynamically-optimal algorithm
+must be able to do so with costs that differ by at most a constant factor. We discuss this
+motivation in more detail in Section 3 (right after stating Theorem 7).
+Our contributions are all based on the same technique, which is quite simple. We enforce
+GF to maintain a static tree and only query the leaves of this tree. Although being dynamic
+in general, there are some access-patterns that cause GF not to change the tree. By studying
+these patterns, we can study GF on a static tree, and the analysis of its cost simplifies to the
+weighted-average of the depth of the queries (weighted by frequency). To lower-bound the
+gap between GF and OPT, we analyze the average cost that can be saved by promoting the
+items in the leaves to locations closer to the root. Note that any other item can be placed
+further away from the root since it is never queried by the sequence.
+2
+Model
+In this section we describe the model which we use, and define our notations. First, we note
+that throughout the paper lg x is used to denote the base two logarithm of x.
+We consider a totally ordered universe of (fixed size) n items. For simplicity, one may
+think of the values {1, . . . , n}. The items are organized in some initial BST which we denote
+by T0. Then, a sequence of queries, denoted by X = [x1, x2, . . . , xm], is given, one query at
+a time. We reserve m to denote the length of the sequence. The tree before serving xt is
+denoted by Tt−1. An algorithm has to find the queried value xt, by traversing Tt−1 from
+its root. After finding xt, the algorithm is allowed to re-structure Tt−1 to get Tt. We define
+the cost of the algorithm at time t to be the total number of nodes that were touched at
+time t, both on the path to xt and for restructuring. The cost of an algorithm for the whole
+sequence is simply the sum of its costs over all times. We define it formally below.
+▶ Definition 1 (Cost). Let X be a sequence of queries, and let T0 be an initial tree. Let A
+be an algorithm that serves X and let Tt be the tree that A has after serving xt. Let Pt be
+the set of nodes on the path from the root to xt in Tt−1 and let Ut be the set of nodes of the
+minimal subtree that contains all the edges that were rotated by A to transform Tt−1 to Tt.
+Then the cost of A for serving X at time t is |Pt ∪ Ut|, and the cost of A for serving X is
+the sum of costs over t = 1, . . . , m. We denote the cost of A to serve X starting with T0 by
+cost(A, X, T0). We denote the average cost per query by ˆc(A, X, T0) = cost(A,X,T0)
+m
+. When T0
+is clear from the context, or immaterial, we write cost(A, X) and ˆc(A, X).
+▶ Definition 2 (Depth). Let T be a tree. The depth of a node v ∈ T, denoted by d(v), is the
+number of edges in the path from the root to v (in particular d(root) = 0). Note that the
+cost of querying v (without restructuring) is d(v) + 1. We also define the depth of the tree,
+denoted by d(T), as the maximum depth of a node in T, that is d(T) = maxv∈T d(v).
+▶ Definition 3 (Competitiveness). We say that an algorithm A is (α, β)-competitive for initial
+tree T0 if for any sequence of queries X, it holds that cost(A, X, T0) ≤ α·cost(OPT, X, T0)+β
+where OPT is a best algorithm to serve X given T0 (with full knowledge of X). When we
+do not specify T0 we mean that the relation holds for all initial trees. We refer to α as the
+multiplicative term and to β as the additive term. For ease of language, we regard the
+multiplicative term as the competitive ratio, and also write “the competitive ratio of” instead
+
+Y. Sadeh and H. Kaplan
+39:5
+of “the multiplicative term of the competitiveness of”. In such cases, we assume that the
+additive term is o(m). It is easiest to think of β = O(n) while assuming that m = ω(n).
+To conclude this section, we give a precise description of the GF algorithm, in Algorithm 1.
+We emphasize that its implementation is complex and probably would not be good in practice.
+However, its main benefit is its theoretical value, as a candidate for dynamic optimality.
+Should it be proven to be dynamically-optimal, then we would get a better understanding
+of the problem and also a stepping-stone to analyze simpler algorithms, such as splay, in
+comparison to GF rather than against some “vague” optimum that depends on the sequence.
+Algorithm 1 GreedyFuture (GF) Algorithm
+Input: A sequence of queries X ∈ [n]m and an initial BST T0. We restructure Tt−1
+to Tt after serving the request xt with Tt−1 for t = 1, . . . , m.
+Function Restructure(query value v, current tree Tt−1, future accesses X′):
+Let v1 < v2 < . . . < vk be the nodes on the path from the root of Tt−1 to the
+queried value v (including v and the root). We also define v0 = −∞ and
+vk+1 = +∞. Denote the subtrees hanging off this path by R0, . . . , Rk.
+For each i = 1, . . . , k, let τ(vi) be the index of the first appearance of a query of a
+value x ∈ (vi−1, vi+1) in X′. Restructure the nodes v1, . . . , vk as a treap:
+maintain a BST ordering, while the heap’s priorities are set to be the τ values,
+where the root’s τ is smallest. Tie-break arbitrarily, e.g. in favor of smaller
+values, or smaller depth prior to restructuring. Then, hang the subtrees
+R0, . . . , Rk unchanged at their appropriate locations. The resulting tree is Tt.
+3
+Stable Sequences and Lower Bounds
+In this section we properly define the family of stable sequences (Definition 11) for which
+the tree maintained by GF is never changed (i.e. the access path of the current query is a
+treap with respect to the suffix of the sequence). To prove our lower bounds we use such
+sequences in which only the items at the leaves of GF are requested, and the internal nodes
+cause some extra cost that OPT avoids. We use a natural way to represent such sequences
+as trees, and use this representation to prove the following lower bounds, which are the main
+results of this section.
+▶ Theorem 4. If GF is (c, d)-competitive where the additive term d is sublinear in the length
+of the sequence, i.e. d = o(m), then c ≥ 2.
+▶ Theorem 5. For every n ≥ 2 there exist sequences X ∈ [n]m such that cost(GF, X) =
+cost(OPT, X) + Ω(m · lg lg n). Among these sequences, there exists a sequence whose length
+is m = nΘ(
+lg lg n
+lg lg lg n ). (There exist other longer sequences too.)
+Theorems 4 and 5 enable us to prove the following two theorems, proven in Appendix A.2.
+▶ Theorem 6. For any ϵ > 0 there exists a sequence X with a subsequence (not necessarily
+consecutive) X′ ⊆ X such that cost(GF, X′) ≥ (2 − ϵ) · cost(GF, X).
+There exists a sequence Y with a subsequence (not necessarily consecutive) Y ′ ⊆ Y such
+that cost(GF, Y ′) − cost(GF, Y ) = Ω(m · lg lg n).
+STACS 2023
+
+39:6
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+▶ Theorem 7. Let S be a sequence, we define rev(S) to be the sequence S in reverse. For
+any ϵ > 0 there exists a sequence X such that cost(GF, rev(X)) ≥ (2 − ϵ) · cost(GF, X).
+There exists a sequence Y such that cost(GF, rev(Y )) − cost(GF, Y ) = Ω(m · lg lg n).
+The motivation for studying subsequences (Theorem 6) is the fact that OPT always saves
+costs when queries are removed from its sequence. Formally, if X′ ⊆ X, then cost(OPT, X′) ≤
+cost(OPT, X). Indeed, OPT can serve X′ by simulating a run on X. More generally, this
+relation of costs when comparing a sequence to a subsequence of it, is an important property
+which even has a name:
+▶ Definition 8 (Approximate-monotonicity [12, 15]). An algorithm A is approximately-
+monotone with a constant c if for any sequence X, subsequence X′ ⊆ X, and initial tree T,
+it holds that cost(A, X′, T) ≤ c · cost(A, X, T).
+▶ Corollary 9. If GF is approximately-monotone with a constant c, then c ≥ 2.
+As noted, OPT is approximately-monotone with c = 1 (strictly monotone). The reason
+that approximate-monotonicity is of interest, in particular for GF, is because it is one of two
+properties that together are necessary and sufficient for any dynamically-optimal algorithm.
+The complementing property, which GF is known to satisfy, is simulation-embedding:
+▶ Definition 10 (Simulation-Embedding [15]). An algorithm A has the simulation-embedding
+property with a constant c if for any algorithm B and any sequence X, there exists a
+supersequence Y ⊇ X such that cost(A, Y ) ≤ c · cost(B, X). (X is a subsequence of Y , not
+necessarily of consecutive queries.)
+An algorithm A which is approximately-monotone with a constant c1 and has the
+simulation-embedding property with a constant c2 is dynamically-optimal with a constant
+c1 · c2. Indeed, for any sequence X, there is some supersequence Y (X) ⊇ X such that
+cost(A, X) ≤ c1 · cost(A, Y (X)) ≤ c1 · c2 · cost(OPT, X). Harmon [12] proved that GG, and
+hence GF, has the simulation-embedding property, hence GF is dynamically-optimal if and
+only if it is approximately-monotone. An alternative indirect proof was given by [6], proving
+that GG is O(1)-competitive versus the move-to-root algorithm, therefore inheriting the
+property from move-to-root.
+The motivation for studying reversal (Theorem 7) is that OPT is oblivious to reversing
+the sequence of queries, up to an additive difference of n. Indeed, to serve a sequence X in
+reverse, we can pay n to restructure the initial tree T0 to the final tree Tm, and then “reverse
+the arrow of time”: when serving query xt, also modify the tree from Tt to Tt−1 where Ti is
+the tree that OPT would get by the end of processing the i-th query of X, in order. This
+means that any dynamically-optimal algorithm must be able to serve a sequence of requests
+and its reverse with the same cost up to a constant factor. Theorem 7 does not disprove
+dynamic-optimality for GF, but gives some insight of how reversal affects GF.
+3.1
+Maintaining a Static Tree for GF
+In this section we describe the basic “tool” which we use to fix a tree structure for GF despite
+its dynamic nature. That is, we describe a class of sequences which we call mixed-stable
+sequences such that GF never restructures its tree when serving a sequence in this class.
+For the sake of simplicity, we assume that the initial tree is structured as we need it to be.
+Appendix A.1 explains how to enforce a specific “initial” tree given an arbitrary initial tree,
+and also argues why this minor issue does not affect the competitive ratio of GF.
+
+Y. Sadeh and H. Kaplan
+39:7
+As noted, our objective is to produce a sequence that “tricks” GF into having unnecessary
+nodes in the core of the tree, such that the requested values are only at the leaves. As
+an example, consider the classic sequence of queries X = [1, 3, 1, 3, . . .] with an initial tree
+containing 2 at the root, 1 as a left child of the root and 3 as a right child of the root.
+Because of the alternating pattern, GF never re-structures the tree, and the cost per query
+is 2 rather than 1.5 on average (e.g. when 1 is in the root, and 3 is its right child).
+▶ Definition 11 (Stable Nodes and Sequences). Let T be a full binary search tree, and let X
+be a sequence of queries over the items in the leaves of T. We define the stability of nodes as
+follows, see also Figure 1.
+We say that an inner node v in T is strongly-stable if it has two children, and the
+subsequence of X consisting only of the items in the subtree of v, alternates between accesses
+to the left and right subtrees of v.
+We say that an inner node v with a left child u in T is weakly-stable with a left-bias if
+both v and u have two children, and the subsequence of X consisting only of the items in the
+subtree of v, repeats the following 3-cycle. First it accesses the left-subtree of u, then the
+right subtree of u, and finally right subtree of v. (It is left-biased because 2
+3 of the accesses
+are to the left of v). Symmetrically, we say that v is weakly-stable with a right-bias if v has
+two children, its right child u has two children, and the restriction of X to accesses in the
+subtree of v repeats a 3-cycle consisting of an access to the right subtree of u, the left subtree
+of u, and the left subtree of v. Notice that u is a strongly-stable node by definition, and we
+refer to it as the favored-child of v.
+We regard the sequence X as being induced by the tree T with stability “attached” to its
+inner nodes. We assume that every node is stable, and refer to X as a mixed-stable sequence
+and to T as a mixed-stable tree. We distinguish two special cases: If all inner nodes are
+strongly-stable then we refer to X and T as strongly-stable, and if exactly half of the inner
+nodes of T are weakly-stable then we refer to X and T as weakly-stable (recall that each
+weakly-stable node has a strongly-stable favored-child).
+Figure 1 Node and sequence stability (Definition 11). First, consider the repeated sequence
+421, i.e. X = 421421421 . . .. Then v is a weakly-stable right-biased node because its visits pattern
+is a repetition of right(u), left(u), left(v). u is a strongly-stable node because its visits pattern
+is right(u), left(u). w is not stable at all, because its visits pattern is always left(w). Second,
+consider the repetition of the access pattern 12141314. One can verify that all three inner nodes
+are strongly-stable. Hence, this is a strongly-stable sequence. Third, note that no weakly-stable
+sequence corresponds to the figure, because it requires an even number of inner nodes, but if we
+make w a leaf (removing 2, 3), then the repeated access pattern of 4w1 is a weakly-stable sequence.
+To motivate Definition 11 a little, note that the sequence X = [1, 3, 1, 3, . . .] is a strongly-
+stable sequence that corresponds to a tree over the items {1, 2, 3} where 2 is in the root.
+X yields a lower-bound of 4
+3 on the competitive ratio of GF. Similarly, the sequence X′ =
+[5, 3, 1, 5, 3, 1, . . .] is a weakly-stable sequence that corresponds to the tree over {1, 2, 3, 4, 5}
+with 2 at the root and 4 its right-child. X′ yields a lower-bound of 8
+5 on the competitive ratio
+of GF, which is already an improvement over the best known lower bound, see also Figure 2.
+The distinction between strongly-stable and weakly-stable nodes is that GF may modify
+STACS 2023
+
+39:8
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+(a) X = [1, 3, 1, 3, . . .]
+(b) X′ = [5, 3, 1, 5, 3, 1, . . .]
+Figure 2 Examples of the simplest strongly-stable (a) and weakly-stable (b) sequences. Their
+corresponding trees are the left tree in each pair while the right tree in each pair is an optimized static
+tree to serve the same sequence. Queried nodes are colored in blue. One can verify that ˆc(X, GF) = 2
+and ˆc(X′, GF) = 8
+3 while based on the optimized tree, ˆc(X, OPT) ≤ 3
+2 and ˆc(X′, OPT) ≤ 5
+3.
+the structure of the tree when a weakly-stable node is considered, but only temporarily and
+without affecting the cost. In our example with X′, after querying 5, GF may put 4 in the
+root instead of 2, but following the query of 3 this change will be reverted.
+Motivated by the power of stable sequences over small trees, we proceed to a more general
+analysis of stable sequences.
+▶ Definition 12 (Atomic Sequence). A tree T, along with stability type (weak/strong) for
+each node, and a subtree of each node to be accessed initially, induce a stable sequence. This
+sequence is unique up to its length, which can be extended indefinitely. We define the “atomic
+unit” of this sequence as the shortest sequence X such that any repetition of X is also a
+stable sequence that corresponds to T.
+Throughout the paper we work with whole multiples of the atomic sequence. Moreover,
+unless stated otherwise, we work with the atomic sequence itself (a single repetition).
+▶ Lemma 13. Let X be a mixed-stable sequence with respect to a tree T. Then every leaf u
+is visited once every 2a(u) · 3b(u) queries where a(u) and b(u) are non-negative integers. In
+particular, the atomic length of X is 2maxleaf u a(u) · 3maxleaf u b(u) (the lcm). Moreover, if X
+is strongly-stable then ∀u : b(u) = 0, and if X is weakly-stable then ∀u : a(u) = 0.
+Proof. Consider a leaf u. Define the frequency of visiting an ancestor w of u to be the
+frequency of accessing a leaf in the subtree of w. If w is a strongly-stable ancestor then the
+frequency of visiting a child of w is 1
+2 of the frequency of visiting w. If w is weakly-stable, v
+is its favored-child, and x is a child of v then the frequency of visiting x is 1
+3 of the frequency
+of visiting w. Similarly if w is weakly-stable, v is its non-favored-child then the frequency
+of visiting v is 1
+3 of the frequency of visiting w. It follows that u is visited exactly once
+every 2a(u) · 3b(u) queries where a(u) is the number of strongly-stable nodes that are not
+favored-children (there are no such nodes if X is weakly-stable), and b(u) is the number of
+weakly-stable nodes (no such nodes if X is strongly-stable), on the path to u. Finally, since
+every leaf u is visited with a specific period, the whole sequence has a period which is the
+lcm of all periods.
+◀
+▶ Lemma 14. Let X be a mixed-stable sequence with respect to a tree T. If GF serves
+X with T as initial tree, and breaks ties in favor of nodes of smaller-depth, then it never
+restructures T.
+Proof. The proof is by induction on the size of the tree. If T has a single node, then it is
+trivial. Otherwise, the root r is an inner-node, and we prove that it always remains the root.
+
+Y. Sadeh and H. Kaplan
+39:9
+It then follows, by restricting the access sequence to values within each subtree, that the rest
+of the tree remains fixed as well. We use the notations of τ(v) and vi as in Algorithm 1.
+First, consider the case that r is a strongly-stable node (Definition 11). Given an access
+to some value x in the left subtree of r, by definition, the next access would be to a value
+in the right subtree of r, hence τ(r) < τ(vi) for any vi ̸= r on the path from r to x, and
+therefore GF will keep r in the root. The same argument holds if x is in the right subtree of
+r, and the next access is in the left subtree.
+Next, consider the case that r is a weakly-stable node. Without loss of generality, assume
+that it is left-biased, and denote its favored-child (left child) by u. Denote the left and right
+subtrees of u by A and B respectively, and the right subtree of r by C. The access pattern
+of subtrees is ABC(ABC . . .).
+If the current access was to some x ∈ A, both r and u have been touched. The next
+access queries in B, so τ(u) = τ(r) < τ(vi) for any vi ̸= u, r on the access path to x.
+Since GF tie-breaks in favor of smaller-depth, it will keep r in the root.4
+If the current access was to some x ∈ B, then both r and u have been touched. The next
+access touches C, so τ(r) < τ(vi) for any vi ̸= r on the access path to x, including u,
+thus r must remain the root.
+If the current access was to some x ∈ C, since the next access touches A, τ(r) < τ(vi) for
+any vi ̸= r on the access path to x, thus r must remain the root. In this case u was not
+touched, but nonetheless it remains the left child of r.
+◀
+▶ Lemma 15. If X is a mixed-stable sequence, the frequency of accessing x ∈ X is in the
+range of [
+1
+3d(x) ,
+1
+3d(x)/2 ]. In particular, if X is strongly-stable then the frequency equals
+1
+2d(x) .
+Proof. The frequency of visiting a node depends on the path to it.
+The frequency is
+multiplied by 1
+2 when passing through a strongly-stable node, and multiplied by either 1
+3 or
+2
+3 when passing through a weakly-stable node. Every factor of 2
+3 is followed by 1
+2, due to
+the strongly-stable favored-child of the weakly-stable node. Thus the frequency is bounded
+between
+1
+3d(x) and
+1
+2d(x)/2 ·
+� 2
+3
+�d(x)/2 =
+1
+3d(x)/2 .
+◀
+▶ Corollary 16. Let X be a strongly-stable sequence, then: ˆc(GF, X) = �
+x∈X
+d(x)+1
+2d(x) .
+3.2
+Promotions and Recursive Trees
+The way in which we show our lower bounds relies on the fact that serving the leaves of a
+static tree is sub-optimal, since a trivial static optimization is to move the leaves closer to
+the root. We refer to this operation as a promotion of the leaf that we move. We emphasize
+that for the purpose of our result, we analyze the improvement one gets from promotions,
+but the actual OPT, which is dynamic, may be able to reduce the cost further.
+▶ Definition 17 (Promotion). Consider trees T and T ′. We say that a node x was promoted
+in T ′ by h (with respect to T), if dT (x) − dT ′(x) = h. Given a mixed-stable sequence X, the
+average promotion of T to T ′ is the weighted average promotion in T ′ of the nodes of T,
+weighted by the query frequencies of the nodes.
+4 This is the reason we defined this kind of access pattern as weakly-stable, because the stability can be
+chosen, but is not forced. We emphasize that putting u as a parent of r will not make the next access
+cheaper as both u and r will be touched anyway, and then r will be reinstated as the root.
+STACS 2023
+
+39:10
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+By definition, static optimization of a tree T to T ′ for a mixed-stable sequence X, implies
+a cost improvement for OPT which is at least the average promotion of T to T ′, per query.
+Intuitively, promoting leaves that are closer to the root contributes more to the average
+promotion than promoting deeper leaves since the access frequencies decrease exponentially
+with depth. That being said, our promotion scheme will be relatively uniform, promoting
+most leaves by roughly the same amount, as in the following example.
+▶ Example 18. To clarify promotions, consider Figure 3. There, we can safely promote every
+node by one, except for one of the deepest nodes. Therefore, we immediately conclude that for
+the corresponding strongly-stable sequence X, we have: ˆc(GF, X) ≥ ˆc(OPT, X) + (1 −
+1
+2n ).
+(a) Before promotions.
+(b) After promotions.
+Figure 3 (a) A tree which induces a strongly-stable sequence X, only blue nodes are queried.
+The frequency of querying an odd number v = 2i − 1 in this tree is
+1
+2i except for v = 2n + 1 which
+has the same frequency as v = 2n − 1. (b) An improved static tree, in which each node except for
+one has been promoted one step closer to the root. The cost of serving X over this tree is cheaper
+by almost 1 per query.
+We define our trees using recursive structures.
+▶ Definition 19. A recursive tree, Tr, of depth r is defined by a specific full binary tree T
+(independent of r) such that at least one of its leaves is an actual leaf, and some of its leaves
+are roots of recursive trees, Tr−1, of depth r − 1. We refer to the inner nodes of T as the
+trunk of Tr, and define T0 to be a single node. See Figure 4 for two examples.5
+Figure 4 Two recursive trees of depth r. Each of the trees T and F is a full binary tree with
+at least one actual leaf (in blue), and some hanging subtrees. At the bottom of the recursion (for
+r = 0), the subtrees are nodes. Note that: (a) Expanding T for r = n results in the tree in Figure 3;
+(b) The pattern F is important for Theorem 4.
+5 The name of the pattern F in Figure 4, stands for Fibonacci: One can verify that for r ≥ 2, the number
+of leaves at depth 1 ≤ d ≤ r − 1 is the (d − 1)th Fibonnaci number Fd−1 (we define F0 = 0). Moreover,
+this can be used to prove the nice equation: �∞
+d=0
+Fd
+2d = 2.
+
+2n- 1
+2n + 12n - 1
+2n :
+2n + 1Y. Sadeh and H. Kaplan
+39:11
+3.3
+Multiplicative Lower Bound for GF
+In this section we prove Theorem 4. We do it by describing a concrete weakly-stable sequence,
+whose average cost per query is 6 while an average promotion of 3 is possible, resulting in an
+optimal cost of at most 3. We start by stating a purely mathematical lemma that will be
+used in the analysis.
+▶ Lemma 20. Let br be a sequence defined by an initial value b0 and the relation br =
+α · br−1 + β + γ · r
+2r for some constants α, β, γ where α ̸= 1
+2, 1. Then br =
+β
+1−α(1 − αr) + αr ·
+b0 +
+2αγ
+(2α−1)2 ·(αr − 1
+2r )−
+γ
+(2α−1) · r
+2r . In particular, when γ = 0 then br =
+β
+1−α(1−αr)+αr ·b0.
+Proof Sketch. Either use induction, or “guess” that a geometric sequence yr with a multiplier
+of α satisfies yr = p · r
+2r + q · 1
+2r + s + br, and determine the fixed coefficients p, q, s.
+◀
+▶ Lemma 21. Let X be a weakly-stable sequence implied by the recursive tree Fr in Figure 4,
+where the root is a weakly-stable node with a right-bias. Then for any ϵ > 0, there is a
+sufficiently large recursive depth r such that (1) ˆc(GF, X) > 6 − ϵ, (2) a static optimization
+of the tree saves an average cost of at least 3 − ϵ, and (3) regardless of r, ˆc(OPT, X) < 3.
+Proof. Let cr denote the average cost of serving X with Fr.
+Then c0 = 1 and cr =
+1
+3(cr−1 + 1) + 1
+3 · 3 + 1
+3(cr−1 + 2) =
+2
+3cr−1 + 2, which yields by Lemma 20 that cr =
+2
+1−2/3(1 − (2/3)r) + (2/3)r · 1 = 6 · (1 − (2/3)r) + (2/3)r. To analyze the average promotion,
+we re-structure Fr to a new static structure F ′
+r as follows, see Figure 5. The leaf is moved
+to the root, whose children are the recursive subtrees, optimized themselves by the same
+logic. The old root is moved to be a right child of the maximal value in the new left subtree,
+and the old right-child (of the old-root) is moved to be a left child of the minimal value in
+the new right subtree. F ′
+r maintains the order of values as was in Fr. The demotions of the
+old root and its right child do not affect the cost, because X does not query these values.
+Denote by pr the average promotion of Fr to F ′
+r. Then p0 = 0 since nothing is promoted for
+a singleton, and pr = 1
+3pr−1 + 1
+3 · 2 + 1
+3(pr−1 + 1) = 2
+3pr−1 + 1. Again by Lemma 20 we get
+that pr =
+1
+1−2/3(1 − (2/3)r) + (2/3)r · 0 = 3 · (1 − (2/3)r). Observe that for r → ∞ we get
+that cr → 6 and pr → 3, thus parts (1) and (2) of the claim follow. For part (3), observe
+that cr − pr = 6 · (1 − (2/3)r) + (2/3)r − 3 · (1 − (2/3)r) = 3 − 2 · (2/3)r < 3.
+◀
+(a) F-tree pattern.
+(b) Promotion scheme.
+Figure 5 The F-tree pattern and its promotion scheme in Lemma 21.
+Only the top-level
+promotions are presented in (b), but more promotions are done recursively within each subtree.
+▶ Theorem 4. If GF is (c, d)-competitive where the additive term d is sublinear in the length
+of the sequence, i.e. d = o(m), then c ≥ 2.
+Proof. Assume by contradiction that GF is (2 − δ, f(m))-competitive for some δ > 0
+and a function f(m) = o(m). Let X′ be a sequence that consists of s repetitions of the
+atomic weakly-stable sequence that corresponds to the recursive tree Fr. It follows that
+ˆc(GF, X′) ≤ (2 − δ) · ˆc(OPT, X′) + f(|X′|)
+|X′| . By Lemma 21, we can choose r large enough such
+STACS 2023
+
+Fr:
+Fr-139:12
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+that ˆc(GF, X′) > 6 − δ, and regardless of r, ˆc(OPT, X′) < 3. Then, since f is sub-linear, we
+can choose the number of repetitions s to be large enough such that f(|X′|)
+|X′|
+< 2δ. But then
+we also get that ˆc(GF, X′) < (2 − δ) · 3 + 2δ = 6 − δ, which is a contradiction.
+◀
+By Analyzing mixed-stable sequences we proved a lower bound of 2 on the competitve
+ratio of GF. Theorem 22 gives an upper bound.
+▶ Theorem 22. Let X be a mixed-stable sequence and let T be the tree that corresponds to
+it. Then cost(GF, X, T) < c · cost(OPT, X, T) for c = 5
+2. If X is strongly-stable, then c = 2.
+We defer the proof of Theorem 22 to Appendix A.2. The upper-bound in Theorem 22 is
+clearly not tight, since in the proof of Theorem 22 we neglected a term using the inequality
+ˆc(GF, X) ≤
+2
+α · ˆc(OPT, X) − 1
+α
+�
+1 − n−1
+2m
+�
+<
+2
+α · ˆc(OPT, X), for a constant α. The lack
+of tightness is more prominent when ˆc(OPT, X) is small, like in the sequence studied in
+Lemma 21 (for Theorem 4). We suspect that the lower bound in Theorem 4 is tight, and
+more strongly, that the F-tree pattern is the best pattern to use. This is based on studying
+several other recursive patterns, including those in Figure 4 and Figure 6: None was stronger,
+and it also seems that patterns with large costs do not “compensate” with large enough
+promotions.
+As a closing remark to the multiplicative results, we note that by the static optimality
+theorem for GG [9], competitive analysis against a static algorithm (i.e. an algorithm that
+does not change its initial tree) cannot show a super-constant lower bound. Concretely,
+the theorem states that cost(GF, X) ≡ cost(GG, X) = O(m + �n
+i=1 ni lg m
+ni ) and one can
+verify that the actual constants are 5m + 6 �n
+i=1 ni lg m
+ni . This bound can be re-written as
+5m + 6m · H2(X) where H2(X) = �n
+i=1
+ni
+m lg m
+ni is the base-2 entropy of the frequencies of
+the values in X. By [17], cost(OPT s, X) ≥ m · H2(X)
+lg 3
+where OPT s is the static optimum,
+and therefore cost(GF, X) ≤ (5 + 6 lg 3) · cost(OPT s, X). Thus, no static argument can show
+a lower bound larger than ≈ 11.59.
+3.4
+Additive Lower Bounds for GF
+In this section we move on to analyze the additive gap between GF and OPT. For this,
+we construct and analyze more elaborate patterns of recursively-defined trees, in order to
+get a large average promotion when optimizing the structure of the trees. The analysis is
+more involved since we cannot simply assume that the depth of the recurrence, r, approaches
+infinity. Here n is a function of r and the difference of cost can be meaningful in terms of n
+only if n is finite.
+▶ Definition 23. For k ≥ 2, and r ≥ 0 we define a (k, r)-tree Tr as follows. The tree is
+recursive of depth r (as in Definition 19), such that its trunk is composed of a root and a
+left-chain of length k − 1 that starts in the right-child of the root. The left child of the deepest
+node of the trunk is an actual leaf, and the rest of the leaves are Tr−1 subtrees. T0 is a single
+node. See Figure 6. When k is clear from the context, we also refer to the tree as Tr.
+Observe that the tree Fr that was used to prove Theorem 4 is in fact a (k, r)-tree with
+k = 2. When we conclude the analysis, we will get the two ends of a “tradeoff” such that
+on the one end we have a relatively high cost ratio, and on the other a relatively high cost
+difference. Moreover, we will show that the higher the difference of costs on a sequence
+induced by (k, r)-tree, the closer the cost ratio is to 1 (comparing GF to OPT).
+▶ Lemma 24. The depth of a (k, r)-tree is k · r, and its left-most node is at depth r.
+
+Y. Sadeh and H. Kaplan
+39:13
+(a) (k, r)-tree pattern.
+(b) Promotion scheme. The main gain is due to the first step.
+Figure 6 (a) The recursive pattern of a (k, r)-tree, Tr. The trunk of the tree has k nodes: the
+root, and a chain of k − 1 nodes leading to an actual leaf. The rest of the leaves are (k, r − 1)-trees.
+(b) The promotion scheme used later in Lemma 26, exemplified for k = 4 (see also Figure 5 for the
+degenerate case of k = 2). The main gain is from the first step of promoting the actual leaf to the
+root, and its sibling subtree (marked with +) one step upwards. Additional gain is achieved by
+promoting the left-most node of each hanging right subtree to the trunk at the expense of demoting
+trunk nodes. More promotions are done recursively within each subtree. The nodes marked 0, 1, 2
+are indeed consecutive, and also: 2 < x and x + 1 = w < y = z − 1.
+Proof. Trivial by induction: For r = 0, the deepest node is the root, at depth 0. For r ≥ 1,
+observe that the deepest node belongs to the deepest subtree Tr−1, which is rooted at depth
+k since the path to it includes k trunk nodes. Similarly, the depth of the left-most node is
+increased by 1 per recursive level of the tree.
+◀
+▶ Lemma 25. Let Tr be a (k, r)-tree. Then |Tr| = (2 +
+2
+k−1)kr − (1 +
+2
+k−1) where |Tr| is the
+number of nodes in Tr. In rougher terms, |Tr| = Θ(kr).
+Proof. Denote nr = |Tr|. By definition, n0 = 1 and nr = (k + 1) + k · nr−1. Hence, by
+Lemma 20 (with γ = 0): nr = k+1
+1−k(1 − kr) + kr = (2 +
+2
+k−1)kr − (1 +
+2
+k−1).
+◀
+▶ Lemma 26. Let X be any mixed-stable sequence corresponding to a (k, r)-tree Tr. Denote
+the average weighted promotion possible in Tr by pr, where weighting is according to the
+frequency of querying each leaf. Then pr > k · (1 − αr) for α = 1 − 1
+3k . In particular, if X is
+a strongly-stable sequence, then pr = (k + 1) · (1 − αr) + δ for α = 1 −
+1
+2k and 0 ≤ δ < αr.
+Proof. We can promote by k every explicit leaf in every Tr′ for all recursive levels 1 ≤ r′ ≤ r,
+from its location to the root of Tr′. Only nodes that are T0 leaves do not contribute an explicit
+promotion of at least k, therefore pr > k · (1 − f) where f is the sum of query-frequencies of
+all T0 leaves (the inequality is strict due to unaccounted subtree promotions). To conclude,
+we argue that f ≤ (1 −
+1
+3k )r. The frequency of accessing the explicit leaf of Tr is at least
+1
+3k by Lemma 15, hence with frequency of at most 1 −
+1
+3k we query a value in some Tr−1
+subtree. Similarly, within the chosen subtree there is again a relative frequency of at most
+1 −
+1
+3k to query within some Tr−2 subtree. Overall, since there are r levels of recursion, we
+conclude that f ≤ (1 −
+1
+3k )r.
+Proving the second part of the claim required a more careful analysis. We define the
+following method of promotion, depicted in Figure 6. In the (k, r)-tree we promote the
+(only) explicit leaf to the root, and promote its sibling subtree by 1. Then we apply similar
+promotions recursively within every (k, r − 1)-subtree. Finally, we promote the left-most
+node within each (k, r − 1)-subtree that hangs as a right-subtree from the trunk to the parent
+of this subtree. Denote the total average (weighted) promotion by pr. Note that it does not
+matter if we promote the left-most nodes of the right subtrees before or after the recursive
+promotions, because the total order on the items guarantees that there is only one value
+STACS 2023
+
+nodes
+Tr-039:14
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+that can be put instead of every demoted trunk node, and the recursive promotions within a
+specific subtree do not change the depth of its leftmost leaf.
+The promotion of the explicit leaf of Tr saves a cost of k weighted by a factor (query
+frequency) of
+1
+2k . The promotion of the sibling subtree saves 1 weighted by a factor of
+1
+2k .
+The recursive promotions are pr−1 weighted by �k
+i=1
+1
+2i (for all the k subtrees), and finally
+the last promotions are technically negligible (as seen in the analysis below), but for the
+sake of completeness we consider them in the analysis as well: promoting the left-most node
+from each subtree saves (r − 1) + 1 = r since the leaf that we promote last is at depth r − 1
+within the recursive subtree, and this promotion is weighted by
+1
+2r · �k−1
+i=2
+1
+2i (factor of
+1
+2r
+follows from Lemma 24). We get that: pr = k+1
+2k + pr−1 ·
+�
+1 −
+1
+2k
+�
++ r
+2r · 1
+2
+�
+1 −
+1
+2k−2
+�
+. Then
+by Lemma 20, with α = 1 −
+1
+2k and γ = 1
+2(1 −
+1
+2k−2 ), we get:
+pr = (k + 1) · (1 − αr) + δ
+,
+δ ≡ αr · p0 +
+2αγ
+(2α − 1)2 ·
+�
+αr − 1
+2r
+�
+−
+γ
+(2α − 1) · r
+2r
+It remains to show that 0 ≤ δ < αr. It is simple to see that δ = 0 for k = 2, because then
+γ = 0 and p0 = 0 is the average weighted promotion in a tree with a single node. For k ≥ 3,
+by the definition of α and γ we have that
+2αγ
+(2α−1)2 = (2k−1)(2k−4)
+(2k−2)2
+= 1 −
+1
+2k−4+ 4
+2k ∈ ( 3
+4, 1) and
+γ
+2α−1 = 1
+2 −
+1
+2k−2 ∈ [ 1
+3, 1
+2). Substituting these bounds and p0 = 0 into the formula for δ gives
+δ < αr. Moreover, δ is positive since δ > 3
+4(αr − 1
+2r )− 1
+2 · r
+2r = 3
+4(α− 1
+2)·�r−1
+i=0 αi ·
+� 1
+2
+�r−1−i−
+r
+2r+1 = 3(α− 1
+2 )
+2r+1
+· �r−1
+i=0 (2α)i −
+r
+2r+1 > 3(α− 1
+2 )
+2r+1
+· r −
+r
+2r+1 = (3( 1
+2 − 1
+2k ) − 1) ·
+r
+2r+1 > 0 for k ≥ 3.
+Note that indeed the gain from promoting the left-most node of each subtree is negligible,
+since the effect is merely having γ ̸= 0, which only contributes 0 ≤ δ < αr < 1.
+◀
+▶ Corollary 27. The average cost-per-query of GF on a strongly-stable sequence induced
+by a (k, r)-tree is larger than the optimal cost by at least (k + 1) ·
+�
+1 −
+�
+1 −
+1
+2k
+�r�
+. On any
+mixed-stable sequence, the difference is at least k ·
+�
+1 −
+�
+1 −
+1
+3k
+�r�
+.
+We are ready to prove Theorem 5.
+▶ Theorem 5. For every n ≥ 2 there exist sequences X ∈ [n]m such that cost(GF, X) =
+cost(OPT, X) + Ω(m · lg lg n). Among these sequences, there exists a sequence whose length
+is m = nΘ(
+lg lg n
+lg lg lg n ). (There exist other longer sequences too.)
+Proof. Let X be the strongly-stable sequence induced by a (k, r)-tree Tr, and for simplicity
+assume that the initial tree is Tr.6 By Lemma 25, n = (2+
+2
+k−1)kr−(1+
+2
+k−1) therefore lg lg n =
+lg r + lg lg k + O(1).7 By Corollary 27, ˆc(GF, X) − ˆc(OPT, X) ≥ ∆ ≡ (k + 1) · (1 − (1 − 1
+2k )r).
+By choosing r = 2k we get that ∆ = (k + 1) · (1 − (1 −
+1
+2k )2k) ≈ (1 − 1
+e) · (k + 1).8 We
+also get that lg lg n = k + lg lg k + O(1), therefore ∆ ≈ (1 − 1
+e) lg lg n and we conclude that
+ˆc(GF, X) − ˆc(OPT, X) ≥ Ω(lg lg n).
+By Lemma 13 the length of the atomic strongly-stable sequence of Tr is m = 2d(Tr), hence
+m = 2rk by Lemma 24. By Lemma 25, n+(1+2/(k−1))
+2+2/(k−1)
+= kr = 2r lg k. Together we get that
+m = 2rk = 2(r lg k)·(k/ lg k) =
+�
+n+(1+2/(k−1))
+2+2/(k−1)
+�(k/ lg k)
+= nΘ(
+lg lg n
+lg lg lg n ).
+◀
+6 We remove this assumption in Remark 35.
+7 k ≥ 2 ⇒ kr ≤ n < 4kr ⇒ lg n = r lg k + c for c ∈ [0, 2), and so lg lg n = lg r + lg lg k + O(1).
+8 The approximation is off by less than 10% for k ≥ 2. (60% and 20% for k = 0, 1 respectively.)
+
+Y. Sadeh and H. Kaplan
+39:15
+▶ Remark 28. In the proof of Theorem 5, the sequence X does not have to be strongly-
+stable, and any mixed-stable sequence X induced by a (k, r)-tree Tr works as well. Indeed,
+Corollary 27 guarantees that ˆc(GF, X)−ˆc(OPT, X) ≥ ∆ for ∆ = k·
+�
+1−
+�
+1− 1
+3k
+�r�
+, and then
+by choosing r = 3k we get that ∆ = Θ(k), and k = Θ(lg lg n), and m = 2Θ(rk) = nΘ(
+lg lg n
+lg lg lg n ).
+▶ Remark 29. The choice of r = 2k in the proof of Theorem 5 maximizes our lower
+bound on the additive gap cost(GF, X) − cost(OPT, X) (up to constants) for our (k, r)-
+trees. Indeed, revisiting the proof, we have that ∆ and n are both functions of k and r,
+and we need to choose r and k to maximize ∆ as a function of n. Note that ∆ = O(k)
+regardless of r, and lg lg(n) = lg r + lg lg k + O(1). To simplify and eliminate a parameter
+we define r = 2k · f(k) for some monotone function f. Now we get simplified relations:
+∆ = (k+1)·(1−(1− 1
+2k )2kf(k)) ≈ (k+1)·(1−e−f(k)) and lg lg n = k+lg f(k)+lg lg k+O(1).
+Consider the following two cases.
+If f(k) = Ω(1): then ∃c ∈ R such that ∀k ≥ 1 : lg f(k) ≥ c, and therefore lg lg n = Ω(k),
+written differently k = O(lg lg n), which yields ∆ = O(k) = O(lg lg n).
+If f(k) = o(1): Being o(1) means that limk→∞ f(k) = 0, so for sufficiently large values
+of k we can use the approximation ex ≈ 1 + x (that holds for small x) to get: ∆ ≈
+(k + 1) · f(k) =
+k+1
+1/f(k). If
+1
+f(k) grows faster than (k + 1), we get ∆ = O(1) which does
+not even grow with n. Therefore
+1
+f(k) is increasing, but at a sub-linear rate. Recall that
+lg lg n = k − lg
+1
+f(k) + lg k + O(1). Since
+1
+f(k) is sub-linear, we get that k = Θ(lg lg n),
+which yields ∆ = O(k) = O(lg lg n).
+▶ Corollary 30. GF is not (1, O(m))-competitive. If the multiplicative term is 1, then the
+additive term is at least Ω(m · lg lg n).
+We have yet to analyze the cost of OPT on a strongly-stable sequence X corresponding to
+a (k, r)-tree that produces the gap of Ω(m·lg lg n) in Theorem 5. Allegedly, if the cost is cheap,
+say linear, we would get a large competitive ratio as well. However, by Theorem 22 we expect
+a competitive ratio of at most 2, and therefore we can conclude without further analysis,
+that cost(OPT, X) = Ω(m · lg lg n). In fact, we prove that cost(OPT, X) = Θ(m ·
+lg n
+lg lg lg n).
+It follows that the competitive ratio deteriorates when the additive gap increases.
+▶ Lemma 31. Define the constants α ≡ 1 −
+1
+2k and β ≡ �k
+j=1
+j
+2j + k+1
+2k . Let X be a
+strongly-stable sequence induced by a (k, r)-tree. Then ˆc(GF, X) = 2k · β · (1 − αr) + αr. In
+asymptotic terms: ˆc(GF, X) = Θ(2k · (1 − αr)).
+Proof. We write a recurrence for the average cost, cr, of GF on the strongly-stable sequence
+induced by Tr. We have c0 = 1, and
+cr+1 = 1 + k
+2k
++
+k
+�
+j=1
+j + cr
+2j
+=
+�
+1 − 1
+2k
+�
+cr +
+k
+�
+j=1
+j
+2j + 1 + k
+2k
+≡ α · cr + β
+( 1+k
+2k
+is due to the actual leaf, and the summation is the contribution of all the Tr subtrees.)
+By Lemma 20 (with γ = 0), cr =
+β
+1−α(1 − αr) + αr · c0 = 2k · β(1 − αr) + αr. Since
+α = 1 −
+1
+2k ∈ [ 3
+4, 1) clearly αr < 1. Furthermore, β = Θ(1). To see this note that β only
+depends on k. Denote β = β(k) and observe that: β(k+1)−β(k) =
+� k+1
+2k+1 + k+2
+2k+1
+�
+− k+1
+2k =
+1
+2k+1 .
+Therefore, β(k) = β(2) + �k
+i=3
+�
+β(i) − β(i − 1)
+�
+=
+�
+1
+2 + 2
+4 + 3
+4
+�
++ �k
+i=3
+1
+2i = 2 −
+1
+2k , and
+β(k) ∈ [ 7
+4, 2) ⇒ β = Θ(1). Because 2k · (1 − αr) ≥ 2k · (1 − α) = 1 > αr, we conclude that
+cr = Θ(2k · (1 − αr)).
+◀
+STACS 2023
+
+39:16
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+▶ Lemma 32. Let X be a sequence from the family of sequences in Theorem 5, then
+cost(OPT, X) = Θ(m ·
+lg n
+lg lg lg n).
+Proof. Let X be a strongly-stable sequence induced by querying a (k, r)-tree. We know that
+1
+2ˆc(GF, X) < ˆc(OPT, X) ≤ ˆc(GF, X) where the lower-bound is by Theorem 22. Therefore,
+ˆc(OPT, X) = Θ(2k · (1 − αr)) by Lemma 31. By Lemma 25, lg n = r · lg k + O(1), or r =
+lg n−O(1)
+lg k
+. When we substitute r = 2k as in the proof of Theorem 5, we get that (1−αr) = Θ(1)
+and 2k = r = lg n−O(1)
+lg k
+= Θ(
+lg n
+lg lg lg n). Therefore, cost(OPT, X) = Θ(m ·
+lg n
+lg lg lg n).
+◀
+As a concluding remark, we recall that the Fr-tree is a (k, r)-tree for k = 2. If we
+substitute k = 2 in the formula of Lemma 31 we get that α = 3
+4, β = 7
+4, and ˆc(GF, X) =
+7 · (1 − (3/4)r) + (3/4)r. By Lemma 26, the average promotion is 3 · (1 − (3/4)r) (for k = 2,
+we have δ = 0). These values are the strongly-stable analogues of Lemma 21, and can be
+used to show a weaker lower bound of 7
+4, on the competitive ratio of GF.
+4
+Conclusions and Open Questions
+In this paper we gave improved lower bounds on the competitiveness of the Greedy Future
+(GF) algorithm for serving a sequence of queries by a dynamic binary search tree (BST). In
+contrast to many of the previous results on GF that are obtained using the geometric-view by
+studying the equivalent Geometric Greedy (GG) algorithm, we used the standard “tree-view”
+and the treap-based definition of GF. We showed that the competitive ratio of GF is at
+least 2, and that there are sequences X ∈ [n]m for which the cost difference (additive gap)
+between GF and OPT is Ω(m · lg lg n). These lower bounds enabled us to show that if GF
+is approximately-monotone (Definition 8) with some constant c then c ≥ 2. Also, the lower
+bounds show that the cost of GF on a sequence compared to its cost on its reverse, may
+differ by a factor as close as we like to 2, or by a difference of Ω(m · lg lg n). In contrast, the
+cost of OPT on a sequence compared to its reverse may differ by at most n.
+Our results give new insights on the “tradeoff” between the additive term and the
+multiplicative term in the competitiveness of GF, showing that the multiplicative term is
+typically larger when the total cost of the algorithm on the sequence is smaller. Indeed, our
+best multiplicative term is achieved for a sequence whose average cost per query is 6. This
+tradeoff is not surprising since a fixed difference implies a larger ratio when the quantities are
+small. It may be interesting to figure out if this tradeoff hints of some underlying property
+of GF, or is just an artifact of our technique that requires high costs on average per query in
+order to increase the additive gap between GF and OPT.
+Clearly, these improved lower bounds still don’t settle the deeper question of whether GF
+(and GG) is dynamically-optimal. Our techniques focused on a smaller family of sequences
+which we named mixed-stable sequences, whereas “most” sequences are not stable. While
+it is possible that an improved lower bound (larger than 2) can be found by a more clever
+pattern of mixed-stable sequences, it seems more likely to be found by analyzing sequences
+for which the tree maintained by GF is not static. In addition, we note that GF was not
+investigated too deeply directly, as most of the work has been done in the geometric view
+with respect to its counterpart GG. Therefore, studying other problems in tree-view may give
+complementing insights. One such problem is the deque conjecture, which has been partially
+settled for GG, in the case when deletions are only allowed on the minimum item [2].
+
+Y. Sadeh and H. Kaplan
+39:17
+References
+1
+Parinya Chalermsook, Julia Chuzhoy, and Thatchaphol Saranurak. Pinning down the Strong
+Wilber 1 Bound for Binary Search Trees. In Approximation, Randomization, and Combinatorial
+Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 33:1–33:21, 2020.
+2
+Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol
+Saranurak. Greedy is an almost optimal deque. In 14th International Conference on Algorithms
+and Data Structures (WADS), pages 152–165, 2015.
+3
+Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol
+Saranurak. Pattern-avoiding access in binary search trees. In IEEE 56th Annual Symposium
+on Foundations of Computer Science (FOCS), pages 410–423, 2015.
+4
+Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol
+Saranurak. The landscape of bounds for binary search trees. arXiv, abs/1603.04892, 2016.
+5
+Parinya Chalermsook, Manoj Gupta, Wanchote Jiamjitrak, Nidia Obscura Acosta, Akash
+Pareek, and Sorrachai Yingchareonthawornchai. Improved pattern-avoidance bounds for greedy
+BSTs via matrix decomposition. In ACM-SIAM Symposium on Discrete Algorithms (SODA),
+2023.
+6
+Parinya Chalermsook and Wanchote Po Jiamjitrak.
+New binary search tree bounds via
+geometric inversions. In 28th Annual European Symposium on Algorithms (ESA), pages
+28:1–28:16, 2020.
+7
+Erik D. Demaine, Dion Harmon, John Iacono, Daniel Kane, and Mihai Pătraşcu. The geometry
+of binary search trees. In 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA),
+page 496–505, 2009.
+8
+Erik D. Demaine, Dion Harmon, John Iacono, and Mihai Pătraşcu.
+Dynamic optimal-
+ity—almost. SIAM Journal on Computing, 37(1):240–251, 2007.
+9
+Kyle Fox. Upper bounds for maximally greedy binary search trees. In 12th International
+Conference on Algorithms and Data Structures (WADS), page 411–422, 2011.
+10
+Michael L. Fredman. Two applications of a probabilistic search technique: Sorting X+Y and
+building balanced search trees. In 7th Annual ACM Symposium on Theory of Computing
+(STOC), page 240–244, 1975.
+11
+George F. Georgakopoulos.
+Chain-splay trees, or, how to achieve and prove loglogN-
+competitiveness by splaying. Information Processing Letters, 106(1):37–43, 2008.
+12
+Dion Harmon. New Bounds on Optimal Binary Search Trees. PhD thesis, Massachusetts
+Institute of Technology, 2006.
+13
+Donald E. Knuth. Optimum binary search trees. Acta Informatica, 1:14–25, 1989.
+14
+László Kozma. Binary search trees, rectangles and patterns. PhD thesis, Saarland University,
+2016.
+15
+Caleb Levy and Robert Tarjan. New Paths from Splay to Dynamic Optimality. PhD thesis,
+Princeton, 2019.
+16
+Joan M. Lucas. Canonical forms for competitive binary search tree algorithms. In Tech. Rep.
+DCS-TR-250. Rutgers University, 1988.
+17
+Kurt Mehlhorn. Nearly optimal binary search trees. Acta Informatica, 5:287–295, 1975.
+18
+J. Ian Munro. On the competitiveness of linear search. In 8th Annual European Symposium
+on Algorithms (ESA), page 338–345. Springer-Verlag, 2000.
+19
+Hauke Reddmann.
+On the geometric equivalent of instance optimal binary search tree
+algorithms. Master’s thesis, Universität Hamburg, 2021.
+20
+Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal
+of ACM, 32(3):652–686, 1985.
+21
+Chengwen Chris Wang, Jonathan Derryberry, and Daniel Dominic Sleator. O(log log n)-
+competitive dynamic binary search trees. In 17th Annual ACM-SIAM Symposium on Discrete
+Algorithm (SODA), page 374–383, 2006.
+22
+Robert Wilber. Lower bounds for accessing binary search trees with rotations. SIAM Journal
+on Computing, 18(1):56–67, 1989.
+STACS 2023
+
+39:18
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+A
+Appendix: Deferred Proofs and Discussions
+A.1
+Enforcing a Stable Tree for GF
+We describe how to restructure any initial tree, to a desired tree, when GF is considered.
+The initial tree cannot simply be re-organized since GF updates the tree in a specific way
+following each query. Moreover, when we are given a sequence X and add a prefix P to it,
+denote the concatenation by P ◦ X, even if P enforces the desired tree when served alone,
+serving P ◦X may give a different tree following P when X starts. The reason for this is that
+GF restructures the tree while serving P according to future queries, therefore the existence
+of X may affect its decisions while serving P. Nevertheless, the idea is to restructure the
+tree top-down from the root, such that we “propagate” stability, as in Definition 11, over the
+nodes that have already been fixed in their correct places. See Figure 7 for an example.
+Figure 7 Example of enforcing a tree as detailed in Theorem 33. Consider the mixed-stable
+sequence X = [11, 7, 1, 13, 9, 3, 11, 7, 5, 15, 9, 1, 11, 7, 3, 17, 9, 5]. Its desired tree T is the right-most
+tree in the figure. Nodes 6, 4, 16 are weakly-stable biased towards their starred-edge (leading to
+10, 2, 14 respectively), all other internal nodes are strongly-stable. Left-to-right: Initially, there are
+no stabilized nodes (left). The first step queries only Y1 = [A] as if it was a leaf. Duplicated six times
+and substituted for the pattern of a weakly-stable node, we get Z1 = [10, 10, 6, 6, 10, 6], stabilizing
+{6, 10}. In the second step, the next nodes that are stabilized are {2, 4, 8, 12}. The weakly-stable
+sequence of the subtrees is Y2 = [C, B, A]. Duplicated six times, and substituted 2, 2, 4, 4, 2, 4 for
+A, 8, . . . , 8 for B and 12, . . . , 12 for C, we get: Z2 = [12, 8, 2, 12, 8, 2, 12, 8, 4, 12, 8, 4, 12, 8, 2, 12, 8, 4].
+Next Y3 = [F, D, A, G, E, B, F, D, C, G, E, A, F, D, B, G, E, C], and Z3 is the result of duplication
+and substitution. Since A−F are leaves, only the substitution of G requires the non-trivial pattern
+(14, 14, 16, 16, 14, 16). This stabilizes the nodes {14, 16} and is the last step of this example. In total,
+the enforcing sequence is Z1 ◦ Z2 ◦ Z3 (◦ for concatenation). In general, there may be more steps,
+each stabilizes at least one more node until all inner nodes are stable.
+▶ Theorem 33. Let X be a mixed-stable sequence that corresponds to a tree T.9 There is a
+sequence S(X) such that when GF serves the concatenation S(X) followed by X, then when
+it finishes serving S(X) its current tree is T regardless of the initial tree T0. We refer to S(X)
+as the enforcing sequence of X. Additionally, if X is the atomic sequence corresponding to T
+(Definition 12), then |S(X)| < 3n · |X|. If X is strongly-stable or weakly-stable (and atomic),
+then |S(X)| < 2|X| and |S(X)| < 3|X| respectively.
+Proof. We construct S(X) in steps. Initially, the tree of GF (which is the initial tree T0) and
+the desired mixed-stable tree T may be completely different. Each step of the construction
+adds a sequence of queries to S(X) that extends a rooted and connected subtree that belongs
+9 The type of stability of each node of T can be deduced from X.
+
+Y. Sadeh and H. Kaplan
+39:19
+both to T and to the current tree of GF and will not change subsequently. We regard nodes
+that already joined this subtree as stabilized. Once all inner nodes have been stabilized then
+S(X) is complete and the tree of GF is exactly T. Every stabilized node remains stable
+with respect to the continuation of the sequence as in Definition 11. See Figure 7 for an
+illustration. We first describe how to stabilize the desired root (of T), r, which is the base
+of the construction. We emphasize that stabilizing a weakly-stable node also stabilizes its
+favored-child (Definition 11). We split into three cases.
+1. If there is only a single node, then it is r which is also already the root. We do nothing.
+2. If r is strongly-stable in T: We query [r, r] to make it the root. The second query
+guarantees that r is placed at the root after the first query.
+3. If r is weakly-stable in T: Let z be r’s favored-child. We add the queries [z, z, r, r, z, r]
+to S(X). This stabilizes r and z, and by the end of these six queries, r is the root and
+z is its child. To see why this happens, consider how GF works: The second query to
+z guarantees that it is placed at the root after the first query. The third query touches
+both z (the root) and r (queried), and due to the fourth and fifth queries places r at
+the root and z as its child. The purpose of the sixth query is to ensure that z does not
+become the root due to future queries when the fifth query is processed.
+Assume that we already have a tree with a connected subtree of stabilized internal nodes.
+The subtrees that hang off the stable nodes are not empty since stable nodes are internal
+nodes. If each of these subtrees is a leaf then we are done and S(X) is complete. Otherwise,
+we stabilize the root of every subtree that is not a leaf as we describe next. Recall that if the
+root of a subtree is weakly-stable, we also stabilize its favored-child.
+For convenience, denote the index of the current step by ℓ. We regard each unstabilized
+subtree as a leaf, and generate an atomic mixed-stable sequence over the current stabilized
+connected subtree, according to the stability types of the inner nodes. Denote this sequence
+by Yℓ. Note that Yℓ is a sequence of the leaves that correspond to the unstabilized subtrees.
+We derive from Yℓ the stabilizing sequence, Zℓ, for the new nodes, by repeating Yℓ six times,
+and replacing each leaf in Yℓ
+6 by an appropriate node from the subtree that it represents.
+This replacement is done as follows. If the root r of the subtree is a leaf or a strongly-stable
+node then we simply access r. If r is weakly-stable and its favored-child is z then we
+replace each query of the leaf (subtree) by the next query of the sequence z, z, r, r, z, r, while
+cyclically repeating it. Since Yℓ was repeated six times, the sequence z, z, r, r, z, r repeats
+an integral number of times in Zℓ. Because Zℓ is based on the mixed-stable sequence Yℓ,
+the already stabilized nodes remain stable. Only the subtrees hanging off them are affected.
+The restriction of Zℓ to each hanging subtree is either z, z, r, r, z, r or r, r, r, r, r, r so by an
+argument analogous to the one in the base case the root of each hanging subtree is stabilized,
+as well as each favored-child of a weakly-stable root.
+We repeat stabilizing steps until all the inner nodes of T have been stabilized. If the last
+step is d, then S(X) = Z1 ◦ Z2 ◦ . . . ◦ Zd where ◦ denotes concatenation. We emphasize that
+when GF serves S(X) ◦ X, by the end of serving S(X) its tree is indeed T, because we can
+think of X as just another step of the stabilization process, in which all the subtrees are leaves.
+It remains to analyze the length of S(X). The length of any Yi is at most that of X.
+Every step stabilizes at least one inner node, hence there are at most n−1
+2
+steps (the rest
+n+1
+2
+nodes are leaves), so we get that |S(X)| = 6 �d
+i=1 |Yi| ≤ 6 · n−1
+2
+· |X| < 3n · |X|.
+For the refined analysis of the length of S(X) in case X is strongly-stable, or weakly-stable,
+we investigate the relation between |Yi| and |Yi+1|. For convenience we define Yd+1 = X.
+By Lemma 13, |Yi| = 2ai · 3bi, where ai = maxleaf u a(u) and bi = maxleaf u b(u). Observe
+that when we extend the stability from a node v which is a leaf of the stabilized subtree at
+STACS 2023
+
+39:20
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+step i, one or two levels deeper, then we have for every new stabilized leaf descendant u of
+v that either a(u) = a(v) + 1 and b(u) = b(v), or a(u) = a(v) and b(u) = b(v) + 1. However,
+the lcm of the preiods of all leaves of Yi+1 may be affected by two different branches, thus
+allowing any of the combinations of ai ≤ ai+1 ≤ ai + 1 and bi ≤ bi+1 ≤ bi + 1. So we get
+that |Yi| divides |Yi+1| and |Yi| ≤ |Yi+1| ≤ 6|Yi|. In the general case of mixed-stability it
+could be that the length grows initially six-fold (starting from the second step, two differ-
+ent branches might each increase a and b, respectively) while the second half of the steps
+satisfies |Yd/2+1| = |Yd/2+2| = . . . = |X| (another branch that “mixes” the increase in a and
+b “catches up” with the lcm), so |S(X)| = Θ(n · |X|) is possible. As an example, revisit
+Figure 7: there we have |Y1| = 1, |Y2| = 3, |Y3| = 18 and |Y4| = |X| = 18. However, when
+X is strongly-stable, we have |Yi+1| = 2|Yi| for every i because bi = 0 (Lemma 13), and
+ai+1 = ai + 1 for every i. Therefore, |S(X)| = 6 �d
+i=1 |Yi| = 6 �d
+i=1
+|X|
+2i < 6|X|. We can
+optimize further by noting that we can define Zi to be only twice longer than Yi (no need for
+six repetitions). So in fact we can define S(X) for strongly-stable X such that |S(X)| < 2|X|.
+If X is weakly-stable, similarly |Yi+1| = 3|Yi| because for every i we have ai = 0 (Lemma 13)
+and bi+1 = bi + 1, therefore |S(X)| = 6 �d
+i=1 |Yi| = 6 �d
+i=1
+|X|
+3i < 3|X|.
+◀
+▶ Theorem 34. For any tree T there is a sequence S(T) such that for any suffix of queries
+Y , when GF serves S(T) ◦ Y , its tree when is it done with the last query of S(T) is T.
+Proof. We rely on the ideas from the proof of Theorem 33. To generate an oblivious enforcing
+prefix, we concatenate several enforcing sequences, each enforcing higher nodes in the tree.
+Define T [1] ≡ T and T [i+1] is the tree T [i] stripped of all of its leaves, until the final tree T [h]
+contains only the root. For each tree T [i], denote by Xi its corresponding strongly-stable
+sequence Xi, and by S1 the enforcing sequence of X1. Note that if T [i] is not full, we relax
+the definition of the corresponding strongly-stable sequence and instead of querying a missing
+leaf we query its unary parent. This modification works as-well because we can imagine that
+the query proceeds to the missing leaf, which would anyway remain below the parent. We
+claim that S(T) ≡ S1 ◦ X2 ◦ . . . ◦ Xh satisfies the theorem. Indeed, S1 enforces the desired
+tree, and in particular the position of the leaves. Following S1, all the nodes except for the
+leaves are touched again, so these leaves can never become parents, regardless of the suffix Y .
+The argument holds similarly for the following steps, and since the structure of T is already
+in place, it suffices to use Xi instead of their enforcing sequences.
+◀
+Adding a prefix to our sequence may affect the competitive ratio. However, once we fixed
+the stable tree, we can repeat the corresponding stable sequence to “amplify” the original
+competitive ratio making the effect of the prefix negligible. One difficulty raised by repetitions
+is when we care about the length of the sequence in our claim. This is the case in Theorem 5
+where we claim the existence of a sequence of length nΘ(
+lg lg n
+lg lg lg n ). In the proof of this theorem
+we assumed for simplicity that we can choose the initial tree. The following remark shows
+that indeed we can start with an arbitrary initial tree without weakening the theorem.
+▶ Remark 35. Let X be an atomic mixed-stable sequence used to prove Theorem 5. Consider
+the sequence Z = S(X) ◦ Xn, where S(X) is the prefix (guaranteed by Theorem 33) that is
+enforcing the desired “initial” tree T, Xn are n repetitions of X, and ◦ represents concaten-
+ation. By Theorem 33, n|X| ≤ |Z| < 4n|X| therefore we have: |Z| = Θ(n|X|) = nΘ(
+lg lg n
+lg lg lg n )
+(the second equality is by Theorem 5). Since after processing S(X) the tree of GF is fixed:
+cost(GF, Z, T0)−cost(OPT, Z, T0) ≥ n·(cost(GF, X, T)−cost(OPT, X, T)). By the proof of
+Theorem 5 and Remark 28: cost(GF, X, T) − cost(OPT, X, T) = Ω(|X| · lg lg n), and putting
+everything together we get that: cost(GF, Z, T0) − cost(OPT, Z, T0) = Ω(|Z| · lg lg n). Note
+
+Y. Sadeh and H. Kaplan
+39:21
+that if X is strongly-stable, or weakly-stable, then it suffices to define Z = S(X) ◦ X without
+repetitions and we get that |Z| = Θ(|X|), and the rest of the arguments remain the same.
+A.2
+Omitted Proofs
+In this subsection we restate and prove Lemmas and Theorems that were omitted from the
+main text. For convenience, we restate the claims in their original numbering.
+The proof of Theorem 22 makes use of Wilber’s first bound [22]. We use the original
+presentation of this bound which is a bit tighter than later simplified versions such as [14].
+▶ Definition 36 (Wilber’s First Bound [22]). Let X be a sequence of queries, and let T be a
+static reference tree such that every query of X is in a leaf of T. An alternation at an inner
+node u of T is defined to be two queries closest in time such that one accesses either the left
+or right subtree of u and the other accesses the other subtree of u. Define ALT(u) to be the
+number of alternations at node u. Then: cost(OPT, X) ≥ m + 1
+2
+�
+inner u∈T ALT(u).
+▶ Theorem 22. Let X be a mixed-stable sequence and let T be the tree that corresponds to
+it. Then cost(GF, X, T) < c · cost(OPT, X, T) for c = 5
+2. If X is strongly-stable, then c = 2.
+Proof. We use the tree that corresponds to the mixed-stable sequence as the reference tree
+for Wilber’s first bound. Arithmetic manipulations will yield an expression that we can tie
+to the cost of GF, according to the claim.
+Let X be a mixed-stable sequence, with a corresponding tree T. Let S be the set of
+values that are in the leaves of T, and let U be the set of inner nodes, |U| = n−1
+2 . We also
+denote by A(i) the set of proper ancestors of i. By the definition of the cost of a static tree,
+we know that ˆc(GF, X) = �
+i∈S (d(i) + 1) · f(i) where d(i) is the depth of i and f(i) is the
+frequency of accessing i. We extend f(u) to refer to the frequency of visiting any node u.
+Note that f(u) = �
+i∈S∧u∈A(i) f(i) and that �
+i∈S f(i) = 1.
+Now consider Wilber’s bound for X, with T as the reference tree. We can use T as the
+reference tree since X only accesses leaves of T, by definition. We also denote αu ≡ ALT (u)+1
+f(u)·m
+(ALT(u) is defined in Defintion 36, and note that 0 ≤ ALT(u) ≤ f(u) · m − 1). We have
+αu ∈ (0, 1], where αu = 1 corresponds to fully alternating accesses to the subtree rooted
+at u. The lower bound is cost(OPT, X) ≥ m + 1
+2
+�
+u∈U (ALT(u) + 1) − |U|
+2 = m
+2 + m
+2 (1 +
+�
+u∈U αu · f(u)) − n−1
+4
+=
+� m
+2 − n−1
+4
+�
++ m
+2
+�
+i∈S (1 + �
+u∈A(i) αu)f(i). Let α ≤ minu∈U αu,
+we get that ˆc(OPT, X) ≥
+� 1
+2 − n−1
+4m
+�
++ α
+2
+�
+i∈S (d(i) + 1) · f(i) = α
+2 ˆc(GF, X) +
+� 1
+2 − n−1
+4m
+�
+where the equality holds since GF maintains a static tree. Thus ˆc(GF, X) ≤ 2
+α · ˆc(OPT, X)−
+1
+α
+�
+1 − n−1
+2m
+�
+< 2
+α · ˆc(OPT, X).
+In order to choose a suitable α, recall that a strongly-stable node u has a coefficient of
+αu = 1, which means that for strongly-stable sequences, in which all inner nodes are stable,
+we can pick α = 1 and conclude that ˆc(GF, X) < 2 · ˆc(OPT, X). If u is a weakly-stable node,
+then its coefficient is αu = 2
+3. So for a mixed-stable sequence we can naively pick α = 2
+3,
+resulting in ˆc(GF, X) < 3 · ˆc(OPT, X).
+In order to improve from 3 to 5
+2, we observe that by definition, every weakly-stable node has
+a strongly-stable child. Let u be a weakly-stable node and let w be its (strongly-stable) favored-
+child (recall Definition 11). Since ALT(u) = ALT(w) (by definition of the access pattern in u),
+we can present Wilber’s bound differently, summing (ALT(u)+1)·(1+β)+(ALT(w)+1)·(1−β)
+instead of (ALT(u)+1)+(ALT(w)+1). We get modified coefficients α′
+u = (ALT (u)+1)·(1+β)
+m·f(u)
+=
+αu · (1 + β) = 2(1+β)
+3
+and similarly α′
+w = αw(1 − β) = (1 − β). Choosing β = 1
+5 balances the
+coefficients: α′
+u = α′
+w = 4
+5. Now we can choose α = 4
+5, and get ˆc(GF, X) < 5
+2 · ˆc(OPT, X)
+for mixed-stable sequences.
+◀
+STACS 2023
+
+39:22
+Dynamic BSTs: Improved Lower Bounds for Greedy-Future
+▶ Theorem 6. For any ϵ > 0 there exists a sequence X with a subsequence (not necessarily
+consecutive) X′ ⊆ X such that cost(GF, X′) ≥ (2 − ϵ) · cost(GF, X).
+There exists a sequence Y with a subsequence (not necessarily consecutive) Y ′ ⊆ Y such
+that cost(GF, Y ′) − cost(GF, Y ) = Ω(m · lg lg n).
+Proof. Denote the initial tree by T0. Let Z be the weakly-stable sequence used for proving
+Theorem 4. Let TP be the tree that corresponds to Z and TQ the optimized tree, in which
+the leaves are promoted as in Lemma 21. Let P and Q be the sequences that enforce TP
+and TQ by Theorem 34, respectively. Note that ϵ determines Z, P and Q since it tells us
+how close to a ratio of 2 we need to get.
+Revisit Figure 5 to see the (recursive) structures of TP (on the left) and TQ (on the right,
+post-promotions). Observe that TP remains static when GF serves Z with it, by definition.
+Moreover, TQ remains static when GF serves Z with it. Indeed, let r be the root of TQ. Z
+queries the item in r every third access and the other accesses are alternating between its left
+and right subtrees, hence r remains the root of TQ. The rest of TQ remains static recursively.
+Define X = P ◦ Q ◦ Zk for a large k, and X′ = P ◦ Zk ⊂ X (◦ for concatena-
+tion). Since GF does not change TP and TQ while serving Z we get that cost(GF,X′)
+cost(GF,X) =
+cost(GF,P,T0)+k·cost(GF,Z,TP )
+cost(GF,P ◦Q,T0)+k·cost(GF,Z,TQ). This ratio approaches cost(GF,Z,TP )
+cost(GF,Z,TQ) for large enough k, and
+since Tp and TQ are exactly the trees used in the proof of Lemma 21, we conclude that we can
+make the resulting ratio as close to 2 as we like (choosing Z, P, Q according to the desired ϵ).
+The proof for Y and Y ′ is similar. We define Z to be the strongly-stable sequence used
+in Theorem 5, and define the appropriate tree-enforcing sequences P and Q by Theorem 34.
+Revisit Figure 6 to see the (recursive) structures of TP (on the left) and TQ (on the right,
+post-promotions). We set Y = P ◦ Q ◦ Zk and Y ′ = P ◦ Zk. The main difference is in the
+argument of why GF does not change TQ when serving Z (TP is static by definition). For this,
+observe that the root of TQ, denote it r, is the left-most leaf in the right subtree of TP . This
+means that the access pattern at r is alternating between its left subtree and its right subtree
+including itself, thus again we conclude that r remains at the root of TQ, and the rest of TQ
+remains static recursively. Therefore, cost(GF, Y ′) − cost(GF, Y ) ≥ k ·
+�
+cost(GF, Z, TP ) −
+cost(GF, Z, TQ)) − cost(GF, P ◦ Q, T0) = Ω(m · lg lg n) for large enough k.
+◀
+▶ Theorem 7. Let S be a sequence, we define rev(S) to be the sequence S in reverse. For
+any ϵ > 0 there exists a sequence X such that cost(GF, rev(X)) ≥ (2 − ϵ) · cost(GF, X).
+There exists a sequence Y such that cost(GF, rev(Y )) − cost(GF, Y ) = Ω(m · lg lg n).
+Proof. The proof is similar to that of Theorem 6, and we define Z, T0, P, TP , Q and TQ the
+same way. Here we define X = Q ◦ (rev(Z))k+1 ◦ rev(P) and Y = Q ◦ (rev(Z))k+1 ◦ rev(P),
+for a large k. Recall that Z is different between X (by Theorem 4) and Y (by Theorem 5).
+We claim that TQ remains static when GF serves rev(Z) over it, rather than Z, by the
+same argument as in the proof of Theorem 6, because the interleaving pattern in the root
+is preserved under reversal. Moreover, cost(GF, rev(Z), TQ) = cost(GF, Z, TQ) because the
+cost on a static tree depends only on the access frequencies. Putting everything together, we
+get:
+cost(GF,rev(X))
+cost(GF,X)
+=
+cost(GF,P,T0)+k·cost(GF,Z,TP )+cost(GF,Z◦rev(Q),TP )
+cost(GF,Q,T0)+k·cost(GF,rev(Z),TQ)+cost(GF,rev(Z)◦rev(P ),TQ). Note that
+the suffix contains one repetition of Z so that the rest of it (rev(P) or rev(Q)) does not
+affect the restructuring decisions of GF during the earlier repetitions of Z. The limit of this
+ratio for large k is cost(GF,Z,TP )
+cost(GF,Z,TQ). We finish the argument as in the proof of Theorem 6.
+In the case of Y , we get that cost(GF, Y ′) − cost(GF, Y ) ��� k ·
+�
+cost(GF, Z, TP ) −
+cost(GF, Z, TQ)) −
+�
+cost(GF, Q, T0) + cost(GF, rev(Z) ◦ rev(P), TQ)
+�
+= Ω(m · lg lg n) for
+large enough k.
+◀
+

9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c76c92754fd97ce725448660803db600374581abb016b3d309d03918ea16cdbf
+size 1517122

9dE3T4oBgHgl3EQfrApq/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:48c100d5c6e87de7ddf3ea30373eac3e2a8ce6f1809b89cdc9ab3837e8401080
+size 310941

C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d35d04c175d951b09f3155f464d27d8593050fd996fa909d27938a06e4b837c2
+size 355392

C9AzT4oBgHgl3EQfwf5z/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7e5930e77fdda57923306cf394b85d1c3da067256635baaf08488ec895bf49c8
+size 1835053

C9AzT4oBgHgl3EQfwf5z/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:af2310ebff51ecf8eb971698ccd9f76b3fcc0e5600e97fb7400ebc7d74c7c2ae
+size 72803

DNFQT4oBgHgl3EQfPTY7/content/2301.13278v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:37b5e45135b05e850a6d4159f20974022a6cbcc3fabd5f070f478d98c83a79a1
+size 1840835

DNFQT4oBgHgl3EQfPTY7/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:feccce4e544ba265c02f061ac539f712192edc061adaba856cbae92be195aa18
+size 4849709

DtAzT4oBgHgl3EQfwv4v/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:296d87fbde1e0094363136bb4463f6993428932369272fea3d84f3ee962c8f38
+size 235954

DtE4T4oBgHgl3EQf6Q5X/content/tmp_files/2301.05330v1.pdf.txt

@@ -0,0 +1,932 @@
+Strain-induced Landau levels of Majorana fermions in an anisotropically interacting
+Kitaev model on a honeycomb lattice
+Takuto Yamada1 and Sei-ichiro Suga1
+1Graduate School of Engineering, University of Hyogo, Himeji 671-2280, Japan
+(Dated: January 16, 2023)
+The low-energy states of an anisotropically interacting Kitaev model on a honeycomb lattice
+under triaxial strain are investigated. A numerical calculation shows that quantized states appear
+in the low-energy region and their energy is proportional to the square root of the quantum number.
+Furthermore, the quantized state at zero energy appears only on one sublattice. The obtained results
+are characteristic of the Landau levels of Dirac fermions with time-reversal symmetry, indicating
+the emergence of the strain-induced Landau levels of Majorana fermions, which is also determined
+in the anisotropic Kitaev model by an analytical calculation.
+I.
+INTRODUCTION
+The Kitaev model is an S = 1/2 quantum spin model
+that has bond-dependent Ising-type interactions on a
+honeycomb lattice1, called Kitaev interactions. A Ma-
+jorana representation of the spin operators has shown
+that this model is described by noninteracting itiner-
+ant Majorana fermions coupled with Z2 gauge fluxes1.
+In the ground state, the model is equivalent to the
+noninteracting itinerant Majorana fermions on a hon-
+eycomb lattice.
+Therefore, the low-lying dispersion is
+described by the type of Dirac fermions.
+Fascinating
+properties related with Majorana fermions have been
+revealed by intensive theoretical studies.
+Since Majo-
+rana fermions are charge-neutral particles acting as their
+own antiparticles, they are difficult to interact directly
+to electromagnetic fields. Materials exhibiting Kitaev in-
+teractions, called Kitaev candidate materials have been
+found, including A2IrO3 (A = Na, Ir)2–10, α-RuCl310–18,
+and H3LiIr2O619.
+The behavior caused by Majorana
+fermions in these materials has been studied using various
+methods20–24. In their results, half-integer thermal quan-
+tum Hall effect can be a conclusive evidence for the emer-
+gent Majorana fermions. This phenomenon has been first
+pointed out theoretically1 and then observed experimen-
+tally in α-RuCl325,26.
+Strain fields can induce an artificial vector potential,
+which has opposite signs at two Dirac points due to time-
+reversal symmetry27. Experiments on artificial graphene
+have revealed a strong pseudomagnetic field in the range
+of 10 T–100 T and the presence of Landau levels28–32.
+The strain-induced pseudomagnetic field can interact di-
+rectly with itinerant Majorana fermions. Indeed, numer-
+ical calculations have shown the emergence of Landau
+levels of itinerant Majorana fermions in the isotropically
+interacting Kitaev model under triaxial strain33. Thus,
+the phenomena related to the strain-induced Landau lev-
+els in the Kitaev candidate materials can be a hallmark
+of itinerant Majorana fermions.
+According to the ab-initio calculations for the Kitaev
+candidate materials, the Kitaev interactions include a
+spatial anisotropy6,7,15.
+So far, the Landau levels of
+itinerant Majorana fermions and the related phenom-
+ena have been investigated for the strained Kitaev model
+with isotropic interactions33,34, while whether these Lan-
+dau levels could emerge in the anisotropically interact-
+ing strained Kitaev model is still unclear. Here, in the
+present study, we explore the low-energy properties of
+the anisotropically interacting Kitaev model on a hon-
+eycomb lattice under triaxial strain.
+We focus on the
+parameter region where the itinerant Majorana fermions
+exhibit a gapless dispersion relation in the absence of a
+strain field. Through a numerical calculation, we demon-
+strate that the strain-induced Landau levels of Majorana
+fermions emerge also in the anisotropically interacting
+Kitaev system, which is confirmed also by an analytical
+calculation.
+The rest of the paper is organized as follows.
+Sec-
+tion II outlines the deformation of the Kitaev model for
+the numerical calculation using a singular-value decom-
+position method. We then determine the Z2 gauge-flux
+sector of the ground state. Section III presents the nu-
+merical results for the local density of states (LDOS) of
+the itinerant Majorana fermions; we show the presence
+of the Landau levels typical of massless Dirac fermions
+with time-reversal symmetry in the low-energy region of
+the considered model, indicating the emergence of the
+strain-induced Landau levels of Majorana fermions in the
+anisotropically interacting Kitaev model. Section IV dis-
+cusses the low-energy states of the system based on the
+analytical calculation, illustrating results consistent with
+the numerical outcomes. Finally, the study is summa-
+rized in Sec. V.
+II.
+MODEL AND METHOD
+A.
+Formulation for numerical calculations
+The Hamiltonian is described by
+H = −
+�
+⟨jk⟩x
+Jx
+jkσx
+j σx
+k −
+�
+⟨jk⟩y
+Jy
+jkσy
+j σy
+k −
+�
+⟨jk⟩z
+Jz
+jkσz
+j σz
+k, (1)
+where σα
+j (α = x, y, z) is an α component of the Pauli
+matrix at the j site and Jα
+jk is the coupling constant be-
+tween the nearest-neighbor atoms on the α bond in the
+arXiv:2301.05330v1  [cond-mat.str-el]  12 Jan 2023
+
+2
+C
+C
+C
+rz
+ry
+rx
+Jz
+Jy
+Jx
+A
+B
+R = 3
+R = 2
+R = 1
+Jz
+Jy
+Jx
+(a)
+(b)
+FIG. 1: (Color online) (a) Unstrained honeycomb flakes ex-
+pressed by R: R = 1 is a central hexagon (a cross denotes its
+center.), R = 2 consists of a central hexagon and six surround-
+ing hexagons, R = 3 consists of the R = 2 and twelve sur-
+rounding hexagons, and so on. Thus, R honeycomb flake in-
+cludes 2N = 6R2 spins. The A and B sublattices are shown in
+black and white, respectively. (b) Central hexagon of the un-
+strained honeycomb lattice. The coupling constants, Jx, Jy,
+and Jz, on the X, Y , and Z bonds are represented in blue, red,
+and green, respectively. The vectors connect correspondingly
+the nearest-neighbor sites along these bonds. Triaxial strain
+C is represented schematically using three brown arrows.
+honeycomb lattice. We use a zigzag-terminated honey-
+comb lattice with an open boundary condition. The size
+of the honeycomb flakes is expressed by R [Fig. 1(a)]33,
+and it includes 2N = 6R2 spins, where N is the number of
+the unit cells. The triaxial strain originates at the center
+of the central hexagon marked by an cross in Fig. 1(a).
+In the unstrained honeycomb lattice, the coupling con-
+stants are independent of the site: Jα
+jk = Jα(> 0). When
+a weak triaxial strain is applied as schematically shown
+in Fig. 1(b), the coupling constant Jα
+jk becomes33,35–37
+Jα
+jk ≈ Jα [1 − β (1 − |rj − rk|/a0)], where β is the mag-
+netoelastic coupling and a0 is the unstrained bond length.
+The position vector of an atom is given by rj = r0
+j + uj,
+where r0
+j = (x0
+j, y0
+j ) is the position vector in the un-
+strained lattice and uj is the displacement vector; they
+are expressed respectively as r0
+j = |r0
+j |(cos θ0
+j, sin θ0
+j) and
+uj = (C/a0) |r0
+j |2(cos 3θ0
+j, sin 3θ0
+j) using the polar coordi-
+nate, where C is the triaxial strain strength. Jα
+jk must be
+positive on the whole nearest-neighbor bonds. According
+to our numerical calculation, this condition is satisfied for
+CR ⪅ 0.3. We thus set CR = 0.2 in the following nu-
+merical calculation. In the honeycomb flakes possessing
+the same constant CR, a scaling holds concerning the
+honeycomb flake shapes for different R values38.
+To
+diagonalize
+the
+Hamiltonian,
+four
+Majorana
+fermions, cj and bα
+j , are set at each site1, satisfying
+{cj, ck} = 2δjk, {cj, bα
+k} = 0, and {bα
+j , bβ
+k} = 2δαβδjk.
+To project the enlarged Hilbert space into the physi-
+cal Hilbert space, the constraint cjbx
+j by
+j bz
+j = 1 is im-
+posed.
+In this procedure, the spin operator is repre-
+sented as σα
+j
+= icjbα
+j and the Hamiltonian reads as
+Hu = i �
+α∈{x,y,z}
+�
+⟨jk⟩α Jα
+jkuα
+jkcjck, where uα
+jk = ibα
+j bα
+k
+is a bond operator with an eigenvalue of ±1 and satisfies
+[Hu, uα
+jk] = 0. Thus, uα
+jk is identified with a static Z2
+gauge field between the nearest-neighbor j and k sites on
+the α bond. We then introduce a relevant gauge-flux op-
+erator defined as a product of the six Z2 gauge fields sur-
+rounding a hexagon1. The gauge-flux operator commutes
+with Hu and its eigenvalue becomes ±1. Therefor, the
+system can be mapped to itinerant Majorana fermions
+coupled with the Z2 gauge fluxes on the hexagonal pla-
+quettes. For every configurations of the Z2 gauge fluxes,
+the Hamiltonian Hu can be expressed as33
+Hu = i
+2
+�
+¯cT
+A ¯cT
+B
+� �
+0
+M
+−M T
+0
+� �
+¯cA
+¯cB
+�
+,
+(2)
+where Mjk = Jα
+jkuα
+jk and ¯cA(B) is an N-component vec-
+tor representing the itinerant Majorana fermions on the
+A(B) sublattice. We call the Z2 gauge-flux having −1
+‘flux’. When at least two of the three coupling constants
+are equal in the unstrained system, the Lieb’s theorem39
+states that the exact ground state is in the sector where
+all the Z2 gauge fluxes take unity (the flux-free sector)1.
+The sector where the n gauge fluxes become −1 is called
+the n-flux sector.
+By using a singular-value decomposition method, we
+calculate the eigenvalues ϵm,n (m = 1, 2, · · · , N) and the
+eigenvectors for a given n-flux configuration n; then we
+obtain the LDOS, ρj,A(B)(E), of the itinerant Majorana
+fermions on the A(B) sublattice in the j-th unit cell. The
+magnetoelastic coupling is set as β = 1 for simplicity.
+The coupling constants in the unstrained lattice satisfy
+Jx + Jy + Jz = 1, forming the triangle in the parameter
+space expressed by Jx, Jy, and Jz [left panel of Fig.
+3(a)]1, while the central downward triangle enlarged in
+the right panel represents the gapless phase.
+B.
+One-flux gap and ground-state sector
+In the strained honeycomb lattice, the translational
+invariance is broken, and hence the Lieb’s theorem can-
+not be adopted.
+Thus, we must confirm whether the
+ground state is in the flux-free sector for the given R
+with CR = 0.2. The ground-state energy for a n-flux
+configuration, n, is given by EGS,n = − �
+m ϵm,n. In the
+open boundary system, the one-flux state is possible and
+can be a candidate competing with the flux-free state38.
+We calculate the one-flux gap ∆1 = EGS,1 − EGS,0 for
+all the one-flux configurations at various R up to 90 for
+the given Jx, Jy, and Jz. Figure 2 depicts the typical
+behavior of the minimum one-flux gap ∆min
+1
+for a given
+R. For R ≥ 35, ∆min
+1
+is well described by the follow-
+ing polynomial: ∆min
+1
+= aR−4 + bR−2 + c, where a, b,
+and c are the constants. The extrapolated c values for
+R → ∞ are 1.61 × 10−3, 4.44 × 10−4, and 3.73 × 10−4 in
+Figs. 2(a)-2(c), respectively. We perform the same cal-
+culations for the given coupling constants used in Figs.
+3(b)-3(g), finding that all the extrapolated c values for
+R → ∞ positive. Thus, we can deduce that the ground
+state of the anisotropically interacting Kitaev model for
+CR = 0.2 is in the flux-free sector. In the following nu-
+
+3
+0.000
+0.005
+0.010
+∆min
+1
+(a) Jx = 1/3, Jy = 1/3, Jz = 1/3
+∆min
+1
+|R→∞ = 1.61 × 10−3
+R ≥ 35
+R < 35
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+∆min
+1
+(b) Jx = 7/24, Jy = 7/24, Jz = 5/12
+∆min
+1
+|R→∞ = 4.44 × 10−4
+R ≥ 35
+R < 35
+0
+1
+202
+1
+252
+1
+302
+1
+352
+1
+502
+1
+802
+1/R2
+0.00000
+0.00025
+0.00050
+0.00075
+0.00100
+∆min
+1
+(c) Jx = 11/48, Jy = 17/48, Jz = 5/12
+∆min
+1
+|R→∞ = 3.73 × 10−4
+R ≥ 35
+R < 35
+FIG. 2: (Color online) (a) Minimum one-flux gap (∆min
+1
+) for
+a given R. The dashed lines reprsent the polynomial, ∆min
+1
+=
+aR−4 + bR−2 + c, that well describes ∆min
+1
+for R ≥ 35 for
+the constants a, b, and c having the following values: (a)
+−2.95 × 103, 1.04 × 101, 1.61 × 10−3; (b) 6.56 × 101, 2.81 ×
+10−1, 4.44 × 10−4; (c) −6.77 × 101, 2.42 × 10−1, 3.73 × 10−4.
+merical calculations, we set R = 60 (2N = 21600) and
+C = 1/300.
+III.
+NUMERICAL RESULTS
+Figures 3(b)-3(g) illustrate the LDOS, ρj,A(E) and
+ρj,B(E), at the site in the central hexagon of the sys-
+tem. The left and right panels show the results for the
+A and B sublattices, respectively. We also evaluate the
+integral value: Ij,A(B)(E) =
+� E
+0 ρj,A(B)(E′)dE′. Figure
+3(b) displays the results for the isotropic interactions that
+are plotted using the small open circle in the right panel
+of Fig. 3(a). The coupling constants of the top, middle,
+and bottom panels in Fig. 3(c)-3(g) correspond to the
+black dots from close to the center toward the edge along
+the lines A-E [right panel of Fig. 3(a)], respectively.
+We find that Ij,A(B)(E) forms plateaus.
+We mea-
+sure them in units of the lowest-energy plateau in the
+A sublattice, finding that the pronounced plateaus are
+described as 2n + 1 (n = 0, 1, 2, · · · ) on the A sublattice
+and 2n (n = 0, 1, 2, · · · ) on the B sublattice from the
+low energy in turn. At or in the vicinity of the bound-
+ary between pronounced neighboring plateaus, ρj,A/B(E)
+reaches a peak, as indicated by the vertical dashed lines
+in Figs.
+3(b)-3(g).
+The peak at E = 0 generally ap-
+pears only on the A sublattice, and it is called the sub-
+lattice polarization40. We then plot the peak energies,
+En (n = 0, 1, 2, · · · ), on the A sublattice (Fig. 4), whose
+coupling constants correspond to the black dots along the
+lines A-E [right panel of Fig. 3(a)]. As shown in Fig. 4,
+En satisfies the relation En ∝ √n. These results are char-
+acteristic of the Landau levels of massless Dirac fermions
+with time-reversal symmetry29,36,37,41; thus, the itinerant
+Majorana fermions under triaxial strain are quantized to
+the Landau levels.
+Figure 3(c)-3(g) indicate that as the system leaves the
+isotropically interacting point, the Landau levels of Ma-
+jorana fermions are smeared at the higher energies and
+their number is reduces at the lower energies. Within the
+shaded areas on the lines A-E in the right panel in Fig.
+3(a), at least three Landau levels of Majorana fermions
+from n = 0 appear on the A sublattice, confirming the
+relation En ∝ √n. From the permutation of Jx, Jy, and
+Jz, there are six equivalent regions in the phase diagram
+shown in Fig. 3(a). We apply our results to the five ad-
+ditional regions and summarize the results in the right
+panel of Fig. 3(a). In the shaded area, the Landau lev-
+els of Majorana fermions emerge. In the outer unshaded
+area, instead, one or two peaks appear at and next to
+E = 0 in ρj,A(E), and the sublattice polarization is sat-
+isfied.
+We perform the same calculations for R = 40,
+45, and 50 while keeping CR = 0.2. As R increases, the
+region where the Landau levels of Majorana fermions ap-
+pear expands toward the boundary between gapless and
+gapped phases. We, therefore, expect the Landau levels
+of Majorana fermions to emerge in the whole unstrained
+gapless phase, when the system becomes large enough.
+IV.
+EFFECTIVE LOW-ENERGY THEORY
+A.
+Formulation, eigenenergy, and sublattice
+polarization
+We now discuss the low-energy states of the itiner-
+ant Majorana fermions on the triaxially strained honey-
+comb lattice through an analytical calculation. Following
+Refs.40,42, we adopt the effects of weak triaxial strain as
+Jα(r) = Jα �
+1 + τ
+�
+r · rα/3a02��
+, where rz = a0(0, 1),
+rx = a0(−
+√
+3/2, −1/2), and ry = a0(
+√
+3/2, −1/2) are
+vectors that connect the unstrained nearest-neighbor
+sites, as shown in Fig. 1(b), and τ controls the strain
+strength. Also in the effective low-energy theory, the cou-
+pling constants in the unstrained lattice satisfy Jx+Jy +
+Jz = 1. The strain effect is considered around the Dirac
+points, K and K′, of the isotropically interacting system
+in the unstrained honeycomb lattice. Therefore, this for-
+mulation is effective for a system with weakly anisotropic
+
+4
+A B C D E
+(a) Phase diagram
+(c) A
+(d) B
+(e) C
+(f) D
+(g) E
+(b) Isotropic point
+FIG. 3: (Color online) (a) Phase diagram of the unstrained Kitaev model on the plane Jx +Jy +Jz = 1, where Jα (α = x, y, z)
+are the coupling constants of the unstrained system with Jα ≥ 0. In the left panel, the inner triangle represents the gapless
+phase and the three outer triangles are gapped phases.
+A gapless phase is enlarged in the right panel, where over three
+Landau levels of Majorana fermions appear in the inner shaded region and the sublattice polarization is satisfied for R = 60
+and C = 1/300. In the outer unshaded area, one or two peaks appear at and next to E = 0 in ρj,A(E), and the sublattice
+polarization is satisfied. (b)-(g) Local density of states, ρj,A/B(E), at the central hexagon of the system and the integral values,
+Ij,A/B(E) =
+� E
+0 ρj,A/B(E′)dE′, for R = 60 and C = 1/300; (b) ρj,A/B(E) and Ij,A/B(E) for the isotropic interactions plotted
+using the small open circle in the right panel of (a); (c)-(g) the coupling constants of the top, middle, and bottom panels
+correspond to the black dots from close to the center toward the edge along the lines A-E [right panel of (a)], respectively.
+Kitaev interactions.
+The pseudovector potential, Aξ = (ξAξ
+x, ξAξ
+y), induced
+by triaxial strain is given as
+Aξ
+x = vξ
+x
+−1 ��
+Jz − 1
+2Jx − 1
+2Jy
+�
++ (Jx − Jy)
+xτ
+4
+√
+3a0
++
+�
+Jz + 1
+4Jx + 1
+4Jy
+� yτ
+3a0
+�
+,
+(3)
+
+5
+(a) A
+(b) B
+(c) C
+(d) D
+(e) E
+FIG. 4: (Color online) Peak energies, En (n = 0, 1, 2, · · · ), of ρj,A(E) as functions of √n. The coupling constants in (a)-(e)
+correspond to those in Figs. 3(c)-3(g), respectively.
+Aξ
+y = vξ
+y
+−1
+√
+3
+2
+�
+(Jx − Jy) − (Jx + Jy)
+xτ
+2
+√
+3a0
+− (Jx − Jy) yτ
+6a0
+�
+,
+(4)
+where ξ = ±1 for K/K′ and
+vξ
+x =
+√
+3a0
+4ℏ
+�√
+3 (Jx + Jy) + iξ (Jx − Jy)
+�
+≡ |vx|eiξφx,
+vξ
+y =
+√
+3a0
+4ℏ
+� 1
+√
+3 (3Jz + 1) − iξ (Jx − Jy)
+�
+≡ |vy|eiξφy.
+The pseudomagnetic field, Bξ = (0, 0, ξBξ
+z), is given as
+Bξ
+z = cos φx∂xAξ
+y−cos φy∂yAξ
+x. The Hamiltonian around
+K and K′ reads
+Hξ = ξ
+�
+|vxvy|
+�
+0
+Πξ
+x
+∗ − iΠξ
+y
+∗
+Πξ
+x + iΠξ
+y
+0
+�
+,
+(5)
+where Πξ
+x
+=
+�
+|vx/vy|
+�
+eiξφxpx + ξAξ
+x
+�
+and Πξ
+y
+=
+�
+|vy/vx|
+�
+eiξφypy + ξAξ
+y
+�
+. By defining the annihilation
+operators as
+aK =
+lB
+√
+2ℏ
+�
+ΠK
+x
+∗ − iΠK
+y
+∗�
+, aK′ =
+lB
+√
+2ℏ
+�
+ΠK′
+x
++ iΠK′
+y
+�
+(6)
+with lB
+2
+= ℏ/|Bξ
+z|, the eigenenergy is obtained as
+En =
+�
+2
+√
+2ℏ/lB
+� �
+|vxvy|√n, (n = 0, 1, 2, · · · ).
+The
+n = 0 eigenenstates are |Ψ+
+0 ⟩ = (0, |ψ0⟩)T for K and
+|Ψ−
+0 ⟩ = (|ψ0⟩, 0)T for K′. Since the components of the
+two-dimensional spinor in the sublattice basis are ex-
+changed in K and K′, the eigenstates at n = 0 are
+nonzero only on the A sublattice (sublattice polariza-
+tion), while they are nonzero on both the A and B sub-
+lattices at n ̸= 0. These eigenenergy and eigenstate fea-
+tures agree with the numerical results, providing the evi-
+dence of the emergence of the Landau levels of Majorana
+fermions.
+B.
+Numerical results in terms of the effective
+low-energy theory
+FIG. 5: (Color online) Peak energy, E1, of ρj,A(E) as a func-
+tion of
+√
+C at the isotropically interacting system obtained in
+the numerical calculation. R =40, 50, and 60 are set. Corre-
+sponding C is obtained for the fixed CR =1/25, 2/25, 3/25,
+4/25, 5/25, 6/25, and 7/25, respectively.
+Let us discuss the numerical results in terms of the
+effective low-energy theory.
+As the system leaves the
+isotropically interacting point in the numerical calcula-
+tion, the Landau levels of Majorana fermions are smeared
+at the higher energies and their number is reduced at the
+lower energies, as shown in Figs. 3(c)-3(g). When the
+anisotropy of the interactions becomes strong, the Dirac
+points deviate considerably from those of the isotropi-
+cally interacting unstrained system. This situation is op-
+
+6
+TABLE I: Parameters a and b to fit the coefficient (E1) of En ∝ √n using a linear regression, E1 = a
+√
+8C +b. The coefficients
+(E1) are obtained by the numerical calculation for R =40, 50, and 60 at the fixed CR =1/25, 2/25, 3/25, 4/25, 5/25, 6/25,
+and 7/25, respectively. The coupling constants are for the isotropically interacting system and for the system closest to the
+isotropically interacting point on each line A-E in Fig. 3(a). The third line (below the results for a) denotes the ratios of a to
+0.99951 at the isotropic point are denoted.
+(Jx, Jy, Jz)
+( 1
+3, 1
+3, 1
+3)
+A ( 31
+96, 31
+96, 17
+48) B ( 121
+384, 127
+384, 17
+48) C ( 59
+192, 65
+192, 17
+48)
+D ( 115
+384, 133
+384, 17
+48)
+E ( 7
+24, 17
+48, 17
+48)
+a
+0.99951
+0.99966
+1.00109
+1.00420
+1.00720
+1.00989
+Ration of a to 0.99951
+1
+1.00015
+1.00016
+1.00047
+1.00769
+1.01039
+b
+9.03667 × 10−5 1.48732 × 10−4 2.51854 × 10−5 −2.13822 × 10−4 −3.83096 × 10−4 −4.69701 × 10−4
+posed to the condition for the effective low-energy the-
+ory to hold. Thus, as the system leaves the isotropically
+interacting point, the higher-order terms of Aξ become
+relevant, reducing the number of the Landau levels of
+Majorana fermions.
+We next investigate the relation between the control
+parameters of the strain strength, C and τ, in the numer-
+ical and analytical calculations. To this end, we evaluate
+the coefficient of En ∝ √n obtained in the numerical cal-
+culation for R =40, 50, and 60 at the seven values of the
+fixed CR within CR < 0.3. Figure 5 illustrates the coef-
+ficient (E1) as a function of
+√
+C in the isotropically inter-
+acting system. The data follows the line E1 = a
+√
+8C + b
+with a ≈ 0.99951 and b ≈ 9.03667×10−5, indicating that
+the relation, En =
+√
+8Cn, is deduced within our numer-
+ical accuracy. The coefficient E1 in the system closest
+to the isotropically interacting point on each line A-E is
+evaluated in the same way. The evaluated a and b are
+summarized in Table I, indicating that a ≈ 1.0 and b ≈ 0,
+and thus En ≈
+√
+8Cn.
+The coefficient of En ∝ √n in the effective low-energy
+theory,
+�
+2
+√
+2ℏ/lB
+� �
+|vxvy|, is given by using τ and one
+of the three coupling constants, leading to the expression
+for En as
+En =
+�
+(Jz − 1)2 +
+�
+Jz + 1
+3
+�2� 1
+2 �
+3τn
+2 .
+(7)
+The eigenenergy in the isotropically interacting system
+is obtained as En = 2
+�
+τn/3.
+Comparing En in the
+numerical and analytical calculations for the isotropi-
+cally interacting system, the relation, τ = 6C, is de-
+rived. This relation is consistent with that derived by
+comparing the pseudomagnetic field in the numerical43
+and analytical42 calculations for the isotropically inter-
+acting system, where |Bξ
+z| = 4Cℏ/a02 and 2ℏτ/(3a02),
+respectively. Since Jz’s in the anisotropically interacting
+system in Table I are the same, their coefficients of √τ
+take the same value,
+�
+1025/768, according to Eq. (7).
+The ratio of this coefficient to that in the isotropically
+interacting system is 1.00049. This ratio is close to that
+denoted in Table I. Therefore, the relation τ = 6C is
+approximately satisfied in the anisotropically interacting
+systems denoted in Table I.
+V.
+SUMMARY
+We have investigated the low-energy states of an
+anisotropically interacting Kitaev model under triaxial
+strain. Based on the numerical results, we argue that the
+Landau levels of Majorana fermions emerge in the wide
+gapless region around the isotropically interacting point
+in the phase diagram of the unstraind system. The emer-
+gence of the strain-induced Landau levels of Majorana
+fermions in the anisotropically interacting Kitaev system
+is confirmed also by an analytical calculation. When uni-
+axial strain is applied, the same features (i.e., En ∝ √n
+and sublattice polarization) are expected to appear. To
+observe the features of the strain-induced Landau lev-
+els of Majorana fermions, scanning tunneling microscopy
+(STM) is promising. According to the STM theory for
+the Kitaev model, the differential conductance through
+a single site provides direct information on the LDOS
+of Majorana fermions at low temperatures44.
+In fact,
+the two features of the strain-induced Landau levels (i.e.,
+En ∝ √n and sublattice polarization) have been observed
+in artificial graphene under simulated triaxial strain eval-
+uated pseudomagnetic fields up to 60 T29. The fabrica-
+tion of the α-RuCl3 thin films has been studied actively
+in recent years. This situation favors experiments for in-
+vestigating whether the strain-induced Landau levels of
+Majorana fermions emerge in Kitaev candidate materials
+such as α-RuCl3. We hope that our results contribute to
+such studies.
+Acknowledgments
+We would like to thank T. Suzuki and R. Taniguchi for
+valuable discussions. This work was supported by JSPS
+KAKENHI (Grant No. 19K03721) from MEXT, Japan.
+1 A. Kitaev, Ann. Phys. 321, 2 (2006).
+2 J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev.
+
+7
+Lett. 105, 027204 (2010).
+3 Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale,
+W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108,
+127203 (2012).
+4 R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veen-
+stra, J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker,
+J. N. Hancock, D. van der Marel, I. S. Elfimov, and A.
+Damascelli, Phys. Rev. Lett. 109, 266406 (2012).
+5 K. Foyevtsova, H. O. Jeschke, I. I. Mazin, D. I. Khomskii,
+and R. Valent´ı, Phys. Rev. B 88, 035107 (2013).
+6 Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M.
+Imada, Phys. Rev. Lett. 113, 107201 (2014).
+7 Y. Sizyuk, C. Price, P. W¨olfle, and N. B. Perkins, Phys.
+Rev. B 90, 155126 (2014).
+8 V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoy-
+anova, H. Kandpal, S. Choi, R. Coldea, I. Rousochatzakis,
+L. Hozoi, and J. van den Brink, New J. Phys. 16, 013056
+(2014).
+9 S.-H. Chun, J.-W. Kim, J. Kim, H. Zheng, C. C. Stoumpos,
+C. D. Malliakas, J. F. Mitchell, K. Mehlawat, Y. Singh,
+Y. Choi, T. Gog, A. Al-Zein, M. M. Sala, M. Krisch, J.
+Chaloupka, G. Jackeli, G. Khaliullin, and B. J. Kim, Na-
+ture Physics 11, 462 (2015).
+10 S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent´ı, Phys.
+Rev. B 93, 214431 (2016).
+11 K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V.
+Shankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J.
+Kim, Phys. Rev. B 90, 041112(R) (2014).
+12 Y. Kubota, H. Tanaka, T. Ono, Y. Narumi, and K. Kindo,
+Phys. Rev. B 91, 094422 (2015).
+13 M. Majumder, M. Schmidt, H. Rosner, A. A. Tsirlin, H.
+Yasuoka,and M. Baenitz, Phys. Rev. B 91, 180401(R)
+(2015).
+14 L. J. Sandilands, Y. Tian, A. A. Reijnders, H.-S. Kim, K.
+W. Plumb, Y.-J. Kim, H.-Y. Kee, and K. S. Burch, Phys.
+Rev. B 93, 075144 (2016).
+15 H.-S. Kim, and H.-Y. Kee, Phys. Rev. B 93, 155143 (2016).
+16 R. Yadav, N. A. Bogdanov, V. M. Katukuri, S. Nishimoto,
+J. van den Brink, and L. Hozoi, Sci. Rep. 6, 37925 (2016)
+17 S. Sinn, C.-H. Kim, B.-H. Kim, K.-D. Lee, C.-J. Won, J.-S.
+Oh, M. Han, Y.-J. Chang, N. Hur, Hi. Sato, B.-G. Park,
+C. Kim, H.-D. Kim, and T.-W. Noh, Sci. Rep. 6, 39544
+(2016).
+18 W. Wang, Z.-Y. Dong, S.-L. Yu, and J.-X. Li, Phys. Rev.
+B 96, 115103 (2017).
+19 K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R.
+Takano, Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli,
+and H. Takagi, Nature 554, 341 (2018).
+20 M. Hermanns, I. Kimchi, and J. Knolle, Annu. Rev. Con-
+dens. Matter. Phys. 9, 17 (2018).
+21 J. Knolle and and R. Moessner, Annu. Rev. Condens. Mat-
+ter. Phys. 10, 451 (2019).
+22 H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S.
+E. Nagler, Nat. Rev. Phys. 1, 264 (2019).
+23 Y. Motome and J. Nasu, J. Phys. Soc. Jpn. 89, 1 (2020).
+24 S. Trebst and C. Hickey, Phys. Rep. 950, 012002 (2022).
+25 Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, Sixiao
+Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Motome,
+T. Shibauchi, and Y. Matsuda, Nature 559, 227 (2018)
+26 Y. Kasahara, K. Sugii, T. Ohnishi, M. Shimozawa, M. Ya-
+mashita, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, T.
+Shibauchi, and Y. Matsuda, Phys. Rev. Lett. 120, 217205
+(2018).
+27 M. A. H. Vozmediano, M. I. Katsnelson, and F. Guinea,
+Phys. Rep. 496, 109 (2010).
+28 N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigui, A.
+Zettl, F. Guinea, A. H. C. Neto, M. F. Crommie, Science
+329, 544 (2010).
+29 K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C.
+Manoharan, Nature 483, 306 (2012).
+30 M. C. Rechtsman, J. M. Zeuner, A. Tuennermann, S.
+Nolte, M. Segev, A. Szameit, Nat. Photonics 7, 153 (2013).
+31 Y. Liu, J. N. B. Rodrigues, Y. Z. Luo, L. Li, A. Carvalho,
+M. Yang, E. Laksono, J. Lu, Y. Bao, H. Xu, S. J. R. Tan,
+Z. Qiu, C. H. Sow, Y. P. Feng, A. H. C. Neto, S. Adam, J.
+Lu, K. P. Loh,Nat. Nanotechnol. 13, 828 (2018).
+32 P. Nigge, A. C. Qu, ´E. Lantagne-Hurtubise, E. M˚arsell, S.
+Link, G. Tom, M. Zonno, M. Michiardi, M. Schneider, S.
+Zhdanovich, G. Levy, U. Starke, C. Guti´errez, D. Bonn, S.
+A. Burke, M. Franz, A. Damascelli, Sci. Adv. 5, eaaw5593
+(2019).
+33 S. Rachel, L. Fritz, and M. Vojta, Phys. Rev. Lett. 116,
+167201 (2016).
+34 B. Perreault, S. Rachel, F. J. Burnell, and J. Knolle, Phys.
+Rev. B 95, 184429 (2017).
+35 F. Guinea, M. I. Katsnelson, and A. K. Geim, Nat. Phys.
+6, 30 (2010).
+36 M. Neek-Amal, L. Covaci, Kh. Shakouri, and F. M.
+Peeters, Phys. Rev. B 88, 115428 (2013).
+37 M. Settnes, S. R. Power, and A.-P. Jauho, Phys. Rev. B
+93, 035456 (2016).
+38 M. Fremling and L. Fritz, Phys. Rev. B 105, 085147
+(2022).
+39 E. Lieb, Phys. Rev. Lett. 73, 2158 (1994).
+40 C. Poli, J. Arkinstall, and H. Schomerus, Phys. Rev. B 90,
+155418 (2014).
+41 J. W. F. Venderbos and L. Fu, Phys. Rev. B 93, 195126
+(2016).
+42 G. Salerno, T. Ozawa, H. M. Price, and I. Carusotto, Phys.
+Rev. B 95, 245418 (2017).
+43 M. Settnes, S. R. Power, and A.-P. Jauho, Phys. Rev. B
+93, 035456 (2016).
+44 M. Udagawa, S. Takayoshi, and T. Oka, Phys. Rev. Lett.
+126, 127201 (2021).
+

E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9528208f019c8979a05d3e9d8d627fcbc5030efe3230287dc5a44d4d400c78fb
+size 2587314

EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b5bdab1f9d4e20c987d7efb50b8d186aa9a65138d4baf12e4951e56534995a2b
+size 332458

EdE0T4oBgHgl3EQfywKq/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4cd11b62fd4bda40b4068b2d2d200c56a0923aa5773f85895841ee7c97c210e4
+size 1900589

EdE0T4oBgHgl3EQfywKq/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:3ca651c6f39d86a78233f94455d192c8437c692e157912787ec631c68b34cff3
+size 72983

EdE1T4oBgHgl3EQf-gYV/content/2301.03568v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d6a55a142812e9eab9e4b29b5890c61d6e07eefa57a5a4facd6b234853c8062d
+size 954409

EdE1T4oBgHgl3EQf-gYV/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:732d452c9fde683dedc11ca489ef965162718d7e468b432985d9950bca65833a
+size 185115

I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:104b5e16fb32c7c3e0b0c17792927bab9486d6745ee902ff4ac9556a6d935a6b
+size 39827982

INAzT4oBgHgl3EQfxv6_/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5d25f10ab2957a5acba0b4b65c3f873f3acefd5831e6b86ceaac04e584d7069e
+size 17236013

INAzT4oBgHgl3EQfxv6_/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7b3a768fa53be07e8a4e9dd6a028b6741f18d9352c3f133e008be0f7f9c7c8ac
+size 652592

IdE4T4oBgHgl3EQfIQyU/content/tmp_files/2301.04911v1.pdf.txt

@@ -0,0 +1,2480 @@
+arXiv:2301.04911v1  [math.AP]  12 Jan 2023
+Multi-bubble nodal solutions to slightly subcritical
+elliptic problems with Hardy terms in symmetric
+domains
+Thomas Bartsch∗, Qianqiao Guo†
+Abstract We consider the slightly subcritical elliptic problem with Hardy term
+
+
+
+
+
+−∆u − µ u
+|x|2 = |u|2∗−2−εu
+in Ω ⊂ RN,
+u = 0
+on ∂Ω,
+where 0 ∈ Ω and Ω is invariant under the subgroup SO(2) × {±EN−2} ⊂ O(N); here En denots the n × n
+identity matrix. If µ = µ0εα with µ0 > 0 fixed and α > N−4
+N−2 the existence of nodal solutions that blow up,
+as ε → 0+, positively at the origin and negatively at a different point in a general bounded domain has been
+proved in [5]. Solutions with more than two blow-up points have not been found so far. In the present paper
+we obtain the existence of nodal solutions with a positive blow-up point at the origin and k = 2 or k = 3
+negative blow-up points placed symmetrically in Ω ∩ (R2 × {0}) around the origin provided a certain function
+fk : R+ ×R+ ×I → R has stable critical points; here I = {t > 0 : (t, 0, . . . , 0) ∈ Ω}. If Ω = B(0, 1) ⊂ RN is the
+unit ball centered at the origin we obtain two solutions for k = 2 and N ≥ 7, or k = 3 and N large. The result
+is optimal in the sense that for Ω = B(0, 1) there cannot exist solutions with a positive blow-up point at the
+origin and four negative blow-up points placed on the vertices of a square centered at the origin. Surprisingly
+there do exist solutions on Ω = B(0, 1) with a positive blow-up point at the origin and four blow-up points
+on the vertices of a square with alternating positive and negative signs. The results of our paper show that
+the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well
+understood.
+2010 Mathematics Subject Classification: 35B44, 35B33, 35J60.
+Key words: Hardy term; Critical exponent; Slightly subcritical problems; Nodal solutions; Multi-bubble
+solutions.
+∗Mathematisches Institut, Justus-Liebig-Universit¨at Giessen, Arndtstr. 2, 35392 Giessen, Germany
+†School of Mathematics and Statistics, Northwestern Polytechnical University, 710129 Xi’an, China
+1
+
+2
+T. Bartsch, Q. Guo
+1
+Introduction
+The paper is concerned with the semilinear singular problem
+(1.1)
+
+
+
+
+
+−∆u − µ u
+|x|2 = |u|2∗−2−εu
+in Ω,
+u = 0
+on ∂Ω,
+where Ω ⊂ RN, N ≥ 7, is a smooth bounded domain with 0 ∈ Ω; 2∗ :=
+2N
+N−2 is the critical Sobolev exponent.
+In [5], we obtained the existence of two-bubble nodal solutions to problem (1.1) that blow up positively at the
+origin and negatively at a different point in a general bounded domain, as ε → 0+ and µ = µ0εα with µ0 > 0
+and α > N−4
+N−2. The location of the negative blow-up point is determined by the geometry of the domain.
+The existence of nodal bubble tower solutions has been proved in [6]. These are superpositions of positive
+and negative bubbles with different scalings, all blowing up at the origin. It seems to be a difficult and open
+problem whether solutions with a blow-up point at the origin and more than one blow-up point outside the
+origin exist in a general domain. In the present paper we investigate this problem in symmetric domains, in
+particular for the model case of the ball Ω = B(0, 1).
+In the case of a ball it is natural to place one blow-up point, say positive, at the origin and k blow-up
+points, say negative, at the vertices of a regular k-gon with center at the origin. We shall prove that solutions
+of this shape exist for 2 ≤ k ≤ 3 but, somewhat surprisingly, not for k = 4. On the other hand, we prove the
+existence of solutions with four blow-up points, two positive and two negative ones, at the vertices of a square,
+centered symmetrically around the positive blow-up point at the origin. Our results show that the existence
+of solutions of (1.1) with three or more blow-up points is interesting and far from being understood.
+When µ = 0 the blow-up phenomenon for positive and for nodal solutions to problem (1.1) has been
+studied extensively, see for instance [2–4, 7, 9, 14, 19, 22, 24, 26, 28–31] and the references therein.
+However
+for µ ̸= 0, few results are known about the existence of positive or nodal solutions with multiple bubbles to
+problem (1.1). Positive solutions have been obtained in [12]. Related results, though for different equations,
+can be found in [16,17,27]. We also want to mention the papers [10,11,15,18,21,23,25,32,33,35] dealing with
+the critical exponent, i.e. ε = 0.
+An important role will be played by the limit problem
+(1.2)
+
+
+
+
+
+−∆u − µ u
+|x|2 = |u|2∗−2u
+in RN,
+u → 0
+as |x| → ∞
+which has been investigated in [13,35]. Positive solutions of (1.2) in the range 0 ≤ µ < µ := (N−2)2
+4
+are given
+by
+Vµ,σ(x) = Cµ
+�
+σ
+σ2|x|β1 + |x|β2
+� N−2
+2
+
+Nodal solutions to problems with Hardy terms
+3
+with σ > 0, β1 := (√µ − √µ − µ)/√µ, β2 := (√µ + √µ − µ)/√µ, and Cµ :=
+�
+4N(µ−µ)
+N−2
+� N−2
+4 . These solutions
+minimize
+Sµ :=
+min
+u∈D1,2(RN )\{0}
+�
+RN(|∇u|2 − µ u2
+|x|2 )dx
+(
+�
+RN |u|2∗dx)2/2∗
+,
+and there holds
+�
+RN
+�
+|∇Vµ,σ|2 − µ|Vµ,σ|2
+|x|2
+�
+dx =
+�
+RN |Vµ,σ|2∗dx = S
+N
+2
+µ .
+In the range 0 < µ < µ these are all positive solutions of (1.2). In the case µ = 0 these are all solutions with
+maximum at x = 0. Of course, if µ = 0 also translates of Vµ,σ are solutions of
+(1.3)
+
+
+
+−∆u = |u|2∗−2u
+in RN,
+u → 0
+as |x| → ∞.
+We will write
+Uδ,ξ(x) = C0
+�
+δ
+δ2 + |x − ξ|2
+� N−2
+2
+for the solutions of (1.3) where δ > 0, ξ ∈ RN and C0 := (N(N − 2))
+N−2
+4 .
+These are the well known
+Aubin-Talenti instantons (see [1,34]).
+Now we state our main results. We consider domains satisfying the condition
+(A1) Ω ⊂ RN is a bounded domain with 0 ∈ Ω, and it is invariant under the subgroup SO(2) × {±EN−2} ⊂
+O(N).
+We use the notation x = (x′, x′′) ∈ Ω ⊂ R2 × RN−2 and write A(x′, x′′) = (Ax′, x′′) for A ∈ SO(2). For k ∈ N
+let Rk =
+
+cos 2π
+k
+− sin 2π
+k
+sin 2π
+k
+cos 2π
+k
+
+ ∈ SO(2), and set I = {t > 0 : (t, 0, . . . , 0) ∈ Ω} ⊂ R.
+Our first results are concerned with the existence of nodal solutions with k +1 bubbles, one being positive
+and k being negative. Let G(x, y) =
+1
+|x−y|N−2 − H(x, y) be the Green function (up to a coefficient involving
+the volume of the unit ball) for the Dirichlet Laplace operator in Ω with regular part H. For k = 2, 3 we define
+the function fk : R+ × R+ × I → R by
+fk(λ0, λ1, t) := b1
+�
+H(0, 0)λN−2
+0
++ kH(ξ(t), ξ(t))λN−2
+1
++ 2kG(ξ(t), 0)λ
+N−2
+2
+0
+λ
+N−2
+2
+1
+− 2
+�k
+2
+�
+G(ξ(t), Rkξ(t))λN−2
+1
+�
+− b2
+N − 2
+2
+ln
+�
+λ0λk
+1
+�
+.
+where ξ(t) = (t, 0, . . . , 0) and
+b1 = 1
+2C0
+�
+RN U 2∗−1
+1,0
+and
+b2 = 1
+2∗
+�
+RN U 2∗
+1,0.
+Finally we call a critical point of fk stable if it is isolated and has nontrivial local degree. This is the case, for
+instance, if it is non-degenerate or an isolated local maximum or minimum.
+
+4
+T. Bartsch, Q. Guo
+Theorem 1.1. Suppose Ω satisfies (A1), and suppose (λ0, λ1, t) ∈ R+ × R+ × I is a stable critical point of
+fk, k = 2 or k = 3. Let µ0 > 0 and α > N−4
+N−2 be fixed. Then there exists ε0 > 0 such that for every ε ∈ (0, ε0)
+problem (1.1) with µ = µε = µ0εα has a pair of solutions ±uε satisfying
+(1.4)
+uε(x) = Vµε,σε(x) −
+k
+�
+i=1
+Uδε,Ri
+kξ(tε)(x) + o(1)
+= Cµε
+�
+σε
+(σε)2|x|β1 + |x|β2
+� N−2
+2
+− C0
+k
+�
+i=1
+�
+δε
+(δε)2 + |x − Ri
+kξ(tε)|2
+� N−2
+2
++ o(1),
+where
+σε =
+�
+λ0 + o(1)
+�
+ε
+1
+N−2 , δε =
+�
+λ1 + o(1)
+�
+ε
+1
+N−2 , ξ(tε) = (tε, 0, . . . , 0) = (t + o(1), 0, . . . , 0)
+as ε → 0.
+These solutions satisfy the following symmetries:
+(1.5)
+uε(x′, x′′) = uε(x′, −x′′) = uε(Rkx′, x′′)
+for (x′, x′′) ∈ Ω ⊂ R2 × RN−2.
+As Proposition 1.4 below shows, for k = 4 a family uε as in Theorem 1.1 need not exist even in the case
+of the ball. It is a challenging problem to find critical points of fk for general domains satisfying (A1). We
+consider the special case where Ω = B(0, 1) ⊂ RN is the unit ball.
+Theorem 1.2. If Ω = B(0, 1) ⊂ RN, k = 2 and N ≥ 7, or k = 3 and N is large enough, then fk has two stable
+critical points, one is a local minimum, the other a mountain pass point with Morse index 1. As a consequence,
+problem (1.1) has two families of solutions ±u1
+ε, ±u2
+ε as in Theorem 1.1. They have the additional symmetry
+uε(x′, x′′) = uε(x′, Ax′′)
+for all A ∈ SO(N − 2).
+Remark 1.3. a) In the proof of the case k = 3 we provide an explicit inequality for N, so that the solutions
+as in Theorem 1.2 exist if this inequality holds. Numerical computations show that this inequality is not
+satisfied for N = 7. We do not know the optimal value for N such that Theorem 1.2 is true for k = 3; see also
+Remark 4.2.
+b) We conjecture that Theorem 1.2 holds for other domains satisying (A1), for instance for Ω = B2(0, 1)×
+Ω′ ⊂ R2 × RN−2 with Ω′ = −Ω′ ⊂ RN−2 a bounded symmetric neighborhood of 0. Our proof of Theorem 1.2
+uses the explicit knowledge of the Green function for the ball, hence it does not extend immmediately to other
+domains.
+The next result shows that Theorem 1.2 is optimal.
+Proposition 1.4. For Ω = B(0, 1) ⊂ RN, N ≥ 7 and k = 4 there does not exist a family of solutions ±uε as
+in Theorem 1.1.
+Remark 1.5. a) We conjecture that Proposition 1.4 can be generalized to k ≥ 4.
+
+Nodal solutions to problems with Hardy terms
+5
+b) We cannot exclude the existence of solutions with a positive bubble at the origin and k = 4 negative
+bubbles placed somewhere in the ball and with possibly different blow-up parameters δ. However, we can show
+that there do not exist solutions with four negative bubbles at the vertices of a square centered at the origin
+even if one allows different blow-up speeds, i.e. if one replaces the δε in (1.4) by δi,ε, i = 1, . . . , 4. In fact,
+it is not difficult to prove that the blow-up parameters δi,ε have to be independent of i if the vertices are at
+Ri
+4ξ(tε), i = 1, . . . , 4.
+Considering Proposition 1.4 the following existence results of nodal solutions with five bubbles, three
+being positive and two being negative, is somewhat surprising.
+Theorem 1.6. Let Ω = B(0, 1) ⊂ RN, N ≥ 7, µ0 > 0 and α > N−4
+N−2 be fixed. Then there exists ε0 > 0 such
+that for any ε ∈ (0, ε0), there exist a pair of 5-bubble solutions ±uε to problem (1.1) with µ = µε = µ0εα of
+the shape
+uε(x) = Vµε,σε(x) +
+4
+�
+i=1
+(−1)iUδi,ε,Ri
+4ξ(tε)(x) + o(1)
+= Cµε
+�
+σε
+(σε)2|x|β1 + |x|β2
+� N−2
+2
++ C0
+4
+�
+i=1
+(−1)i
+�
+δi,ε
+(δi,ε)2 + |x − Ri
+4ξ(tε)|2
+� N−2
+2
++ o(1),
+where σε =
+�
+λ0 + o(1)
+�
+ε
+1
+N−2 , δ1,ε = δ3,ε =
+�
+λ1 + o(1)
+�
+ε
+1
+N−2 , δ2,ε = δ4,ε =
+�
+λ2 + o(1)
+�
+ε
+1
+N−2 , ξ(tε) =
+(tε, 0, . . . , 0) = (t + o(1), 0 . . . , 0) as ε → 0, for some λ0, λ1, λ2 > 0, t ∈ (0, 1).
+These solutions satisfy
+the symmetries:
+(1.6)
+uε(x′, x′′) = uε(x′, Ax′′) = −uε(R4x′, x′′)
+for (x′, x′′) ∈ B(0, 1) ⊂ R2 × RN−2, A ∈ SO(N − 2).
+Remark 1.7. a) The parameters (λ0, λ1, λ2, t) ∈ R+ × R+ × R+ × (0, 1) in Theorem 1.6 are obtained as a
+critical point of a suitable limit function f5. We conjecture that there exists a second solution in Theorem 1.6
+but the computations for finding a second critical point of f5 are intimidating.
+b) It seems that for k = 2 in Theorem 1.2, it is still possible to obtain the information on the nodal sets
+of the solutions as in [4]. For k = 3 in Theorem 1.2 and for Theorem 1.6, it is an interesting problem to study
+the profile of the nodal sets of the solutions.
+c) As stated in [5], the assumption α > N−4
+N−2 is essential in our theorems.
+The paper is organized as follows. In Section 2, we collect some notations and preliminary results. Section 3
+is devoted to the method of finite dimensional reduction. Section 4 contains the proof of Theorems 1.1 and
+1.2. Proposition 1.4 is proved in Section 5, and finally Theorem 1.6 is proved in Section 6.
+
+6
+T. Bartsch, Q. Guo
+2
+Notations and preliminary results
+Throughout this paper, positive constants will be denoted by C, c. By Hardy’s inequality the norm
+∥u∥µ :=
+��
+Ω
+(|∇u|2 − µ u2
+|x|2 )dx
+� 1
+2
+is equivalent to the norm ∥u∥0 =
+��
+Ω |∇u|2dx
+�1/2 on H1
+0(Ω) provided 0 ≤ µ < µ. This inequality is of course
+satisfied for µ = µ0εα with α > 0 and ε > 0 small. We write Hµ(Ω) for the Hilbert space consisting of the
+H1
+0(Ω) functions with the inner product
+(u, v)µ :=
+�
+Ω
+�
+∇u∇v − µ uv
+|x|2
+�
+dx.
+As in [5, 16] let ι∗
+µ : L2N/(N+2)(Ω) → Hµ(Ω) be the adjoint operator of the inclusion ιµ : Hµ(Ω) →
+L2N/(N−2)(Ω), that is,
+ι∗
+µ(u) = v
+⇐⇒
+(v, φ)µ =
+�
+Ω
+u(x)φ(x)dx,
+for all φ ∈ Hµ(Ω).
+There exists c > 0 such that
+∥ι∗
+µ(u)∥µ ≤ c∥u∥2N/(N+2).
+Then problem (1.1) is equivalent to the fixed point problem
+u = ι∗
+µ(fε(u)),
+u ∈ Hµ(Ω),
+where fε(s) = |s|2∗−2−εs.
+The following proposition is from [5, Proposition 3.1].
+Proposition 2.1. Let 0 < µ < µ, and let Λi, i = 1, 2, . . . , be the eigenvalues of
+
+
+
+
+
+−∆u − µ u
+|x|2 = Λ|Vσ|2∗−2u
+in RN,
+|u| → 0
+as |x| → +∞
+in increasing order. Then Λ1 = 1 with eigenfunction Vσ, Λ2 = 2∗ − 1 with eigenfunction ∂Vσ
+∂σ .
+Our main results will be proved using variational and singular limit methods applied to the energy func-
+tional
+Jε(u) := 1
+2
+�
+Ω
+�
+|∇u|2 − µ u2
+|x|2
+�
+dx −
+1
+2∗ − ε
+�
+Ω
+|u|2∗−εdx
+defined on Hµ(Ω).
+Let us also recall that the Green’s function of the Dirichlet Laplacian G(x, y) =
+1
+|x−y|N−2 − H(x, y) and
+its regular part H are symmetric: G(x, y) = G(y, x) and H(x, y) = H(y, x). If Ω is invariant under some
+A ∈ O(N) then G(Ax, Ay) = G(x, y), and the same holds for H.
+
+Nodal solutions to problems with Hardy terms
+7
+3
+The finite dimensional reduction
+First we recall some notation from [5].
+Fix µ0 > 0, α >
+N−4
+N−2 and an integer k ≥ 0.
+For λ =
+(λ0, λ1, . . . , λk) ∈ Rk+1
++
+and ξ = (ξ1, ξ2, . . . , ξk) ∈ Ωk we define
+Wε,λ,ξ :=
+k
+�
+i=1
+Ker
+�
+−∆ − (2∗ − 1)U 2∗−2
+δi,ξi
+�
++ Ker
+�
+−∆ − µε
+|x|2 − (2∗ − 1)V 2∗−2
+µε,σε
+�
+⊂ H1(RN)
+where δi = λiε
+1
+N−2 , µε = µ0εα, σε = λ0ε
+1
+N−2 . By Proposition 2.1 and [8] we know that
+Wε,λ,ξ = span
+�
+Ψj
+i, Ψ0
+i , Ψ0, i = 1, 2, . . . , k, j = 1, 2, . . ., N
+�
+,
+where for i = 1, 2, . . . , k and j = 1, 2, . . . , N:
+Ψj
+i := ∂Uδi,ξi
+∂ξi,j
+,
+Ψ0
+i := ∂Uδi,ξi
+∂δi
+,
+Ψ0 := ∂Vµε,σε
+∂σ
+with ξi,j the j-th component of ξi. For η ∈ (0, 1) we define
+Oη :=
+�
+(λ, ξ) ∈ Rk+1
++
+× Ωk : λi ∈ (η, η−1) for i = 0, . . . , k, dist(ξi, ∂Ω) > η,
+|ξi| > η, |ξi1 − ξi2| > η, for i, i1, i2 = 1, . . . , k, i1 ̸= i2
+�
+.
+The projection P : H1(RN) → H1
+0(Ω) is defined by ∆Pu = ∆u in Ω and Pu = 0 on ∂Ω. We also need
+the spaces
+Kε,λ,ξ := PWε,λ,ξ
+and
+K⊥
+ε,λ,ξ := {φ ∈ Hµ(Ω) : (φ, PΨ)µε = 0, for all Ψ ∈ Wε,λ,ξ},
+as well as the (·, ·)µε-orthogonal projections
+Πε,λ,ξ : Hµε(Ω) → Kε,λ,ξ
+and
+Π⊥
+ε,λ,ξ := Id − Πε,λ,ξ : Hµε(Ω) → K⊥
+ε,λ,ξ.
+Then solving problem (1.1) is equivalent to finding η > 0, ε > 0, (λ, ξ) ∈ Oη and φε,λ,ξ ∈ K⊥
+ε,λ,ξ such that:
+(3.1)
+Π⊥
+ε,λ,ξ
+�
+Vε,λ,ξ + φε,λ,ξ − ι∗
+µ(fε(Vε,λ,ξ + φε,λ,ξ))
+�
+= 0,
+and
+Πε,λ,ξ
+�
+Vε,λ,ξ + φε,λ,ξ − ι∗
+µ(fε(Vε,λ,ξ + φε,λ,ξ))
+�
+= 0,
+where in the case of Theorem 1.2
+(3.2)
+Vε,λ,ξ = −
+k
+�
+i=1
+PUδi,ξi + PVµε,σε
+
+8
+T. Bartsch, Q. Guo
+with k = 2, 3, and in the case of Theorem 1.6
+(3.3)
+Vε,λ,ξ =
+k
+�
+i=1
+(−1)iPUδi,ξi + PVµε,σε
+with k = 4.
+The following two propositions have been proved in [5].
+Proposition 3.1. For every η > 0 there exist ε0 > 0 and c0 > 0 with the following property. For every
+(λ, ξ) ∈ Oη and for every ε ∈ (0, ε0) there exists a unique solution φε,λ,ξ ∈ K⊥
+ε,λ,ξ of equation (3.1) satisfying
+∥φε,λ,ξ∥µε ≤ c0
+�
+ε
+N+2
+2(N−2) + ε
+1+2α
+4
+�
+.
+The map Φε : Oη → K⊥
+ε,λ,ξ defined by Φε(λ, ξ) := φε,λ,ξ is C1.
+Now we can define the reduced functional
+Iε : Oη → R,
+Iε(λ, ξ) := Jε(Vε,λ,ξ + φε,λ,ξ).
+Proposition 3.2. If (λ, ξ) ∈ Oη is a critical point of Iε then Vε,λ,ξ + φε,λ,ξ is a solution to problem (1.1) for
+ε > 0 small.
+So far everything works on a general bounded domain.
+Now we will use the invariance of Iε under
+certain symmetries for further reductions. For A ∈ O(N), ξ = (ξ1, . . . , ξk) ∈ (RN)k and u ∈ Lp(RN) we set
+Aξ := (Aξ1, . . . , Aξk) and A ∗ u := u ◦ A−1. This induces isometric actions of O(N) on (RN)k as well as on
+Lp(RN) and, if AΩ = Ω, on Lp(Ω) and on Hµ(Ω) such that ιµ and ι∗
+µ are equivariant. Moreover we have
+Uδ,Aξ = A ∗ Uδ,ξ
+and
+Wε,λ,Aξ = {A ∗ u : u ∈ Wε,λ,ξ},
+and analogously for Kε,λ,ξ, Πε,λ,ξ, Vε,λ,ξ.
+As a consequence, the uniqueness statement in Proposition 3.1
+implies
+(3.4)
+φε,λ,Aξ = A ∗ φε,λ,ξ,
+hence Iε is invariant with respect to the action A ∗ (λ, ξ) = (λ, Aξ) of O(N) on Oη:
+Iε(λ, Aξ) = Iε(λ, ξ).
+Now we apply the principle of symmetric criticality using the matrix AN :=
+
+E2
+0
+0
+−EN−2
+
+ ∈ O(N). By
+assumption AN(Ω) = Ω, hence AN acts on Oη as above leaving Iε invariant. The principle of symmetric
+criticality implies that critical points of Iε constrained to the fixed point set
+OAN
+η
+= {(λ, ξ) ∈ Oη : ANξ = ξ} = {(λ, ξ) ∈ Oη : ξi = (ξ′
+i, 0) ∈ R2 × RN−2, i = 1, . . . , k}
+
+Nodal solutions to problems with Hardy terms
+9
+are critical points of Iε. We also need the invariance of Iε with respect to permutations of the blow-up points.
+Here we need to distinguish between the cases where Vε,λ,ξ is of the form (3.2) or of the form (3.3). Let Sk
+denote the group of permutations of {1, . . ., k}. For π ∈ Sk and (λ, ξ) ∈ Rk+1 × (RN)k we define
+π ∗ (λ0, λ1, . . . , λk) := (λ0, λπ(1), . . . , λπ(k))
+and
+π ∗ (ξ1, . . . , ξk) := (ξπ(1), . . . , ξπ(k)).
+In the case when Vε,λ,ξ is of the form (3.2) it is obvious that
+Iε(π ∗ λ, π ∗ ξ) = Iε(λ, ξ)
+for all π ∈ Sk, (λ, ξ) ∈ Oη.
+It follows that Iε is invariant under the map
+τ : OAN
+η
+→ OAN
+η
+,
+τ(λ0, λ1, . . . , λk, ξ1, . . . , ξk) := (λ0, λk, λ1, . . . , λk−1, Rkξk, Rkξ1, . . . , Rkξk−1),
+which induces an action of Z/kZ on OAN
+η
+; here Rk(ξ′, ξ′′) := (Rkξ′, ξ′′) where Rk ∈ SO(2) is the rotation from
+Theorem 1.2. Therefore critical points of Iε constrained to the fixed point set of the above map, i.e. to
+OAN,τ
+η
+= {(λ, ξ) ∈ OAN
+η
+: λi = · · · = λ1, ξi = Ri−1
+k
+ξ1, i = 2, . . . , k},
+are critical points of Iε.
+In conclusion, for the proofs of Theorems 1.1 and 1.2 it remains to find critical points of Iε constrained
+to OAN ,τ for ε > 0 small.
+The additional symmetry of the solutions stated in Theorem 1.2, and also in
+Theorem 1.6, is obtained as follows. Since the ball is invariant under the action of A ∈ SO(N − 2) defined by
+A(x′, x′′) := (x′, Ax′′) and since A ∗ (λ, ξ) = (λ, Aξ) = (λ, ξ) for (λ, ξ) ∈ OAN
+η
+it follows from (3.4) that
+A ∗ φε,λ,ξ = φε,λ,Aξ = φε,λ,ξ
+for all A ∈ SO(N − 2),
+hence uε = Vε,λ,ξ + φε,λ,ξ satisfies A ∗ uε = uε, i.e. uε(x′, Ax′′) = uε(x′, x′′), for all A ∈ SO(N − 2).
+In Theorem 1.6 Vε,λ,ξ is of the form (3.3) with k = 4. Here Iε is invariant under the map
+�τ(λ1, λ2, λ3, λ4, λ0, ξ1, ξ2, ξ3, ξ4) = (λ3, λ4, λ1, λ2, R4ξ4, R4ξ1, R4ξ2, R4ξ3),
+so, applying the principle of symmetric criticality once more, a critical point of Iε constrained to the set
+OAN ,�τ
+η
+= {(λ, ξ) ∈ OAN
+η
+: λ1 = λ3, λ2 = λ4, ξi = Ri−1
+k
+ξ1, i = 2, 3, 4}
+is an unconstrained critical point of Iε. This can of course be generalized to any even integer k ≥ 4.
+4
+Proof of Theorems 1.1 and 1.2
+In this section we consider Vε,λ,ξ = −
+k�
+i=1
+PUδi,ξi + PVµε,σε for k = 2 and k = 3. The reduced energy is
+expanded as follows; see [5, Proposition 5.1].
+
+10
+T. Bartsch, Q. Guo
+Lemma 4.1. For ε → 0+ there holds
+Iε(λ, ξ) = a1 + a2ε − a3εα − a4ε ln ε + ψ(λ, ξ)ε + o(ε)
+C1-uniformly with respect to (λ, ξ) in compact sets of Oη. The constants are given by
+a1 = 1
+N (k + 1)S
+N
+2
+0 , a2 = (k + 1)
+2∗
+�
+RN U 2∗
+1,0 ln U1,0 − k + 1
+(2∗)2 S
+N
+2
+0 , a3 = 1
+2S
+N−2
+2
+0
+Sµ0, a4 = k + 1
+2 · 2∗
+�
+RN U 2∗
+1,0,
+where S > 0 is defined by Sµ = S0 − Sµ + O(µ2); see [5, Lemma A.10]. The function ψ : Oη → R is given by
+ψ(λ, ξ) = b1
+�
+H(0, 0)λN−2
+0
++
+k
+�
+i=1
+H(ξi, ξi)λN−2
+i
++ 2
+k
+�
+i=1
+G(ξi, 0)λ
+N−2
+2
+i
+λ
+N−2
+2
+0
+− 2
+k
+�
+i,j=1,i<j
+G(ξi, ξj)λ
+N−2
+2
+i
+λ
+N−2
+2
+j
+�
+− b2
+N − 2
+2
+ln(λ1λ2 . . . λkλ0),
+with
+b1 = 1
+2C0
+�
+RN U 2∗−1
+1,0
+,
+b2 = 1
+2∗
+�
+RN U 2∗
+1,0.
+It is well known that a stable critical point (λ, ξ) of ψ implies the existence of a critical point (λε, ξε) of
+Iε for ε > 0 small, and that (λε, ξε) → (λ, ξ) as ε → 0. This applies in particular if (λ, ξ) is an isolated critical
+point of ψ with nontrivial local degree.
+Proof of Theorem 1.1. Since the symmetries of Iε carry over to ψ, for the existence of solutions uε as stated
+in Theorem 1.1 it is sufficient to find stable critical points of ψ constrained to
+OAN ,τ
+η
+= {(λ, ξ) ∈ OAN
+η
+: λi = λ1, ξi = Ri−1
+k
+ξ1, i = 2, . . . , k},
+where k = 2, 3. Observe that Iε and ψ are also invariant with respect to the action of A ∈ SO(2) given by
+(x′, 0) �→ (Ax′, 0) acting on the ξi. Therefore in the case k = 2, setting ξ1 = (t, 0, . . . , 0) for 0 < t < 1 and
+ξ2 = R2ξ1 = −ξ1, it is sufficient to find stable critical points of the function f2 : R+ × R+ × (0, 1) → R defined
+by
+f2(λ0, λ1, t) = ψ(λ0, λ1, λ1, ξ1, −ξ1)
+= b1
+�
+H(0, 0)λN−2
+0
++ 2H(ξ1, ξ1)λN−2
+1
++ 4G(ξ1, 0)λ
+N−2
+2
+1
+λ
+N−2
+2
+0
+− 2G(ξ1, −ξ1)λN−2
+1
+�
+− b2
+N − 2
+2
+ln
+�
+λ2
+1λ0
+�
+.
+This proves Theorem 1.1 in the case k = 2. For k = 3 we set ξ1 = (t, 0, . . . , 0) for 0 < t < 1, ξ2 = R3ξ1 =
+�
+− t
+2,
+√
+3t
+2 , 0, . . . , 0
+�
+, ξ3 = R3ξ2 =
+�
+− t
+2, −
+√
+3t
+2 , 0, . . . , 0
+�
+. As above it is sufficient to find stable critical points
+of the function f3 : R+ × R+ × (0, 1) → R defined by
+f3(λ0, λ1, t) = ψ(λ0, λ1, λ1, λ1, ξ1, ξ2, ξ3)
+= b1
+�
+H(0, 0)λN−2
+0
++ 3H(ξ1, ξ1)λN−2
+1
++ 6G(ξ1, 0)λ
+N−2
+2
+0
+λ
+N−2
+2
+1
+− 6G(ξ1, ξ2)λN−2
+1
+�
+− b2 ln
+�
+λ3
+1λ0
+� N−2
+2
+.
+
+Nodal solutions to problems with Hardy terms
+11
+Here we used that G(ξ1, 0) = G(ξ2, 0) = G(ξ3, 0) and G(ξ1, ξ2) = G(ξ1, ξ3) = G(ξ2, ξ3), as well as H(ξ1, ξ1) =
+H(ξ2, ξ2) = H(ξ3, ξ3). This proves Theorem 1.1 also in the case k = 3.
+□
+Proof of Theorem 1.2. Here we need to find stable critical points of f2 and f3 if G is the Green function of
+the Dirichlet Laplace operator in the unit ball in RN. Our proof uses the explicit knowledge of G. In the case
+k = 2 the proof of [4, Lemma 3.1] applies almost verbatim and shows that f2 has two isolated critical points:
+a local saddle point with Morse index 1, hence with local degree −1, and an isolated local minimum, hence
+with local degree 1. These are stable critical points, giving rise to critical points of Iε for ε > 0 small.
+In the case k = 3 we set
+γ1(t) := H(ξ1, ξ1) − 2G(ξ1, ξ2) =
+1
+(1 − t2)N−2 −
+2
+(
+√
+3t)N−2 +
+2
+(t4 + t2 + 1)
+N−2
+2
+and
+(4.1)
+τ1(t) := G(ξ1, 0) =
+1
+tN−2 − 1
+so that
+f3(λ0, λ1, t) = b1
+�
+H(0, 0)λN−2
+0
++ 3γ1(t)λN−2
+1
++ 6τ1(t)λ
+N−2
+2
+0
+λ
+N−2
+2
+1
+�
+− b2
+N − 2
+2
+ln
+�
+λ3
+1λ0
+�
+.
+One easily checks that γ′
+1(t) > 0, γ1(t) → −∞ as t → 0+, and γ1( 1
+2) > 0. Thus there exists t∗ ∈ (0, 1
+2) such
+that
+γ1(t∗) = 0
+and
+γ1(t) > 0 for all t ∈ (t∗, 1).
+A direct computation shows that for t ∈ (t∗, 1) there exist unique λ0(t), λ1(t) such that
+∂f3(λ0(t), λ1(t), t)
+∂λ0
+= 0
+and
+∂f3(λ0(t), λ1(t), t)
+∂λ1
+= 0.
+In fact one obtains
+(4.2)
+λ0(t)
+N−2
+2
+= α(ξ1, ξ2)λ1(t)
+N−2
+2
+and
+λ1(t)
+N−2
+2
+=
+�
+1
+β(ξ1, ξ2) · b2
+2b1
+,
+where
+α(x, y) = −2G(x, 0) +
+�
+4G2(x, 0) + 4H(0, 0)(H(x, x) − 2G(x, y))
+2H(0, 0)
+and
+β(x, y) = H(x, x) − 2G(x, y) + G(x, 0)α(x, y).
+
+12
+T. Bartsch, Q. Guo
+Moreover, continuing the computation one obtains
+∂2f3(λ0(t), λ1(t), t)
+∂λ2
+1
+=
+3(N − 2)b1
+�
+(N − 3)γ1(t)λN−4
+1
++ N − 4
+2
+τ1(t)λ
+N−6
+2
+0
+λ
+N−2
+2
+1
+�
++ 3(N − 2)b2
+2λ2
+1
+=
+3(N − 2)b1
+�
+(N − 2)γ1(t)λN−4
+1
++ N − 2
+2
+τ1(t)λ
+N−6
+2
+0
+λ
+N−2
+2
+1
+�
+,
+∂2f3(λ0(t), λ1(t), t)
+∂λ2
+0
+=
+(N − 2)b1
+�
+(N − 3)H(0, 0)λN−4
+0
++ 3(N − 4)
+2
+τ1(t)λ
+N−2
+2
+0
+λ
+N−6
+2
+1
+�
++(N − 2)b2
+2λ2
+0
+=
+(N − 2)b1
+�
+(N − 2)H(0, 0)λN−4
+0
++ 3(N − 2)
+2
+τ1(t)λ
+N−2
+2
+0
+λ
+N−6
+2
+1
+�
+,
+∂2f3(λ0(t), λ1(t), t)
+∂λ0∂λ1
+=
+3(N − 2)2
+2
+b1τ1(t)λ
+N−4
+2
+0
+λ
+N−4
+2
+1
+.
+It follows that the Hessian matrix D2
+λ0,λ1f3(λ0(t), λ1(t), t) is positive definite, hence nondegenerate. Therefore
+it is sufficient to find stable critical points of the function
+ν1(t) := f3 (λ0(t), λ1(t), t) = 2b2 − b2
+N − 2
+2
+ln
+�
+λ3
+1(t)λ0(t)
+�
+.
+As in [4, (3.4)] one sees that
+(4.3)
+lim
+t→(t∗)+ ν1(t) = −∞
+and
+lim
+t→1− ν1(t) = +∞.
+Now we prove ν′
+1( 1
+2) < 0 for N large. Set
+α(t) := α(ξ1, ξ2) = −τ1(t) +
+�
+τ 2
+1 (t) + γ1(t),
+where we used H(0, 0) = 1. We obtain
+ν′
+1(t) = ∂f3(λ0(t), λ1(t), t)
+∂t
+= 3b1
+�
+γ′
+1(t) + 2α(t)τ ′
+1(t)
+�
+λN−2
+1
+.
+Then setting ι1(t) := γ′
+1(t) + 2α(t)τ ′
+1(t), it is enough to show ι1
+� 1
+2
+�
+< 0 for N large. In fact, since γ1( 1
+2 )
+τ 2
+1 ( 1
+2 ) < 1
+for N large we see as in [4, (3.9)] that
+ι1( 1
+2) ≤ γ′
+1( 1
+2) + 4γ1( 1
+2)
+5τ1( 1
+2) τ ′
+1( 1
+2).
+A direct computation gives for N large the inequalites
+γ′
+1( 1
+2) = (N − 2)
+�
+( 4
+3)N−1 + 4( 2
+√
+3)N−2 −
+3
+2
+( 1
+16 + 1
+4 + 1)
+N
+2
+�
+< 11(N − 2)
+10
+( 4
+3)N−1
+and
+τ ′
+1( 1
+2) = −(N − 2)2N−1
+and
+γ1( 1
+2)
+τ1( 1
+2) =
+( 4
+3)N−2 − 2( 2
+√
+3)N−2 +
+2
+( 1
+16 + 1
+4 +1)
+N−2
+2
+2N−2 − 1
+> 11
+12 · ( 4
+3)N−2
+2N−2
+
+Nodal solutions to problems with Hardy terms
+13
+which yield ι1( 1
+2) < 0, hence ν′
+1( 1
+2) < 0 for N large enough. This together with (4.3) implies that ν1 has a
+local maximum t1 ∈ (t∗, 1
+2) and a local minimum t2 ∈ ( 1
+2, ∞). These are nondegenerate because ν1 is analytic.
+In conclusion, f3 has two critical points: (λ0(t1), λ1(t1), t1) with Morse index 1 and (λ0(t2), λ1(t2), t2) with
+Morse index 0. This concludes the proof of Theorem 1.2.
+□
+Remark 4.2. a) For k = 3, N = 7, numerical computations show that one cannot find t0 ∈ (t∗, 1) such that
+ν′
+1(t0) = 0. Therefore it is necessary to assume N large here.
+b) For k = 4, the idea above cannot give the existence of nodal solutions with five bubbles, one positive
+at the origin and four negative as in Theorem 1.1. This is the content of Proposition 1.4.
+5
+Proof of Proposition 1.4
+It follows from Lemma 4.1 that Iε does not have critical points for ε > 0 small if ψ does not have
+critical points. This also holds if we constrain Iε and ψ to OAN,τ
+η
+. Setting ξ1 = (t, 0, . . . , 0) for 0 < t < 1,
+ξ2 = R4ξ1 = (0, t, 0, . . . , 0), ξ3 = R4ξ2 = (−t, 0, . . . , 0), and ξ4 = R4ξ3 = (0, −t, . . . , 0) we need to consider the
+function
+f4(λ0, λ1, t) := ψ(λ0, λ1, λ1, λ1, λ1, ξ1, ξ2, ξ3, ξ4)
+= b1
+�
+H(0, 0)λN−2
+0
++ 4H(ξ1, ξ1)λN−2
+1
++ 8G(ξ1, 0)λ
+N−2
+2
+0
+λ
+N−2
+2
+1
+− 8G(ξ1, ξ2)λN−2
+1
+− 4G(ξ1, ξ3)λN−2
+1
+�
+− b2
+N − 2
+2
+ln(λ4
+1λ0),
+where we use
+H(ξ1, ξ1) = H(ξ2, ξ2) = H(ξ3, ξ3) = H(ξ4, ξ4)
+and
+G(ξ1, 0) = G(ξ2, 0) = G(ξ3, 0) = G(ξ4, 0)
+as well as
+G(ξ1, ξ2) = G(ξ2, ξ3) = G(ξ3, ξ4) = G(ξ4, ξ1),
+G(ξ1, ξ3) = G(ξ2, ξ4).
+Proposition 1.4 follows if we can prove that f4 does not have critical points. Let τ1(t) be as in (4.1) and define
+γ2(t)
+:=
+H(ξ1, ξ1) − 2G(ξ1, ξ2) − G(ξ1, ξ3)
+=
+1
+(1 − t2)N−2 −
+2
+(
+√
+2t)N−2 +
+2
+(t4 + 1)
+N−2
+2
+−
+1
+(2t)N−2 +
+1
+(t2 + 1)N−2
+so that
+f4(λ0, λ1, t) = b1
+�
+H(0, 0)λN−2
+0
++ 4γ2(t)λN−2
+1
++ 8τ1(t)λ
+N−2
+2
+0
+λ
+N−2
+2
+1
+�
+− b2
+N − 2
+2
+ln
+�
+λ4
+1λ0
+�
+.
+A direct computation shows that
+γ′
+2(t) = (N − 2)
+�
+2t
+(1 − t2)N−1 +
+2
+(
+√
+2)N−2tN−1 −
+4t3
+(t4 + 1)
+N
+2 +
+1
+2N−2tN−1 −
+2t
+(t2 + 1)N−1
+�
+> 0.
+
+14
+T. Bartsch, Q. Guo
+Clearly γ2(t) → −∞ as t → 0+, and γ2
+�
+1
+√
+2
+�
+> 0. Then there exists t∗ ∈ (0,
+1
+√
+2) such that
+(5.1)
+γ2(t∗) = 0
+and
+γ2(t) > 0 for all t ∈ (t∗, 1).
+Notice that
+λ
+N−2
+2
+0
+= α1(ξ1, ξ2, ξ3)λ
+N−2
+2
+1
+,
+λ
+N−2
+2
+1
+=
+�
+1
+β1(ξ1, ξ2, ξ3) · b2
+2b1
+,
+H(ξ1, ξ1) �� 2G(ξ1, ξ2) − G(ξ1, ξ3) > 0,
+where
+α1(x, y, z) = −3G(x, 0) +
+�
+9G2(x, 0) + 4H(0, 0)(H(x, x) − 2G(x, y) − G(x, z))
+2H(0, 0)
+,
+and
+β1(x, y, z) = H(x, x) − 2G(x, y) − G(x, z) + G(x, 0)α1(x, y, z).
+Setting α1(t) := α1(ξ1, ξ2, ξ3) =
+−3τ1(t)+√
+9τ 2
+1 (t)+4γ2(t)
+2
+and ι2(t) := γ′
+2(t) + 2α1(t)τ ′
+1(t), a similar argument as
+above shows that problem (1.1) admits a solution with 5 bubbles, one positive at the origin and 4 negative as
+in Theorem 1.1 only if ι2(t) has a zero in (t∗, 1). The following claim implies that this is not the case.
+Claim: If N ≥ 7 then ι2(t) > 0 for any t ∈ (t∗, 1).
+We first show that t∗ >
+√
+6−
+√
+2
+2
+, where t∗ is from (5.1). In order to see this, it is enough to prove γ2
+� √
+6−
+√
+2
+2
+�
+<
+0. Since 22/5 · 2(
+√
+6−
+√
+2
+2
+)2 < 1 < (
+√
+6−
+√
+2
+2
+)4 + 1, we have
+1
+(
+√
+2 ·
+√
+6−
+√
+2
+2
+)N−2 >
+2
+((
+√
+6−
+√
+2
+2
+)4 + 1)
+N−2
+2
+,
+for all N ≥ 7.
+On the other hand, it is easy to see that
+1
+(1 − (
+√
+6−
+√
+2
+2
+)2)N−2 =
+1
+(
+√
+2(
+√
+6−
+√
+2
+2
+))N−2
+and
+1
+(2(
+√
+6−
+√
+2
+2
+))N−2 >
+1
+((
+√
+6−
+√
+2
+2
+)2 + 1)N−2 .
+It follows that γ2(
+√
+6−
+√
+2
+2
+) < 0.
+Now we prove ι2(t) > 0 for t ∈ (t∗, 1) ⊂ (
+√
+6−
+√
+2
+2
+, 1). It is easy to see that
+γ′
+2(t) ≥ (N − 2) ·
+2t
+(1 − t2)N−1
+and
+γ2(t) ≤
+1
+(1 − t2)N−2
+for all t ∈
+�√
+6 −
+√
+2
+2
+, 1
+�
+.
+Then we have for all t ∈ (t∗, 1) and N ≥ 7:
+ι2(t)
+N − 2
+≥
+2t
+(1 − t2)N−1 − 3
+�
+1
+tN−2 − 1
+�
+·
+1
+tN−1
+
+
+
+�
+�
+�
+�1 +
+4 ·
+1
+(1−t2)N−2
+9(
+1
+tN−2 − 1)2 − 1
+
+
+
+≥
+2t
+(1 − t2)N−1 −
+3
+t2N−3 ·
+��
+1 + 4
+9(
+t2
+1 − t2 )N−2 ·
+1
+(1 − tN−2)2 − 1
+�
+.
+Setting T :=
+t2
+1−t2 it is enough to prove that
+2
+3 · T N−1 + 1 >
+�
+1 + 4
+9 · T N−2 ·
+1
+(1 − tN−2)2
+
+Nodal solutions to problems with Hardy terms
+15
+which is equivalent to
+(5.2)
+(T N + 3T ) · (1 − tN−2)2 > 1.
+It is obvious that if t ∈ [ 1
+√
+2, 4
+5), then
+(5.3)
+3T · (1 − tN−2)2 ≥ 3(1 − t5)2 > 1,
+and if t ∈ [ 4
+5, 1), then
+(5.4)
+T N · (1 − tN−2)2 ≥ T N · (1 − t)2 =
+t4
+(1 + t)2 · T N−2 > ( 4
+5)4
+4
+(
+( 4
+5)2
+1 − ( 4
+5)2 )5 > 1.
+Now we are left to prove (5.2) for t ∈
+�
+t∗,
+1
+√
+2
+�
+. First of all, if t ∈
+� √
+6−
+√
+2
+2
+,
+1
+√
+2
+�
+, then T ∈
+� √
+3−1
+2
+, 1
+�
+. Setting
+f(T ) := 3T
+�
+1 − tN−2�2 = 3T
+�
+1 −
+�
+T
+1 + T
+� N−2
+2 �2
+,
+a direct computation shows that
+f ′(T )
+=
+�
+1 −
+�
+T
+1 + T
+� N−2
+2 � �
+3 − 3
+�
+T
+1 + T
+� N−2
+2
+− 3(N − 2)
+�
+T
+1 + T
+� N−2
+2
+1
+1 + T
+�
+≥
+�
+1 −
+�1
+2
+� N−2
+2 � �
+3 − 3
+�1
+2
+� N−2
+2
+− 3(N − 2)
+�1
+2
+� N−2
+2
+1
+1 +
+√
+3−1
+2
+�
+≥
+�
+1 −
+�1
+2
+� N−2
+2 � �
+3 − 3
+�1
+2
+� 5
+2
+− 15
+�1
+2
+� 5
+2
+1
+1 +
+√
+3−1
+2
+�
+> 0,
+where in the second inequality we use the fact that 3 − 3( 1
+2)
+N−2
+2
+− 3(N − 2)( 1
+2)
+N−2
+2
+1
+1+
+√
+3−1
+2
+is increasing in N.
+Now we conclude that
+f(T ) > 3 ·
+√
+3 − 1
+2
+
+1 −
+�
+√
+3−1
+2
+1 +
+√
+3−1
+2
+� 5
+2 
+
+2
+> 1
+for all N ≥ 7.
+(5.5)
+The claim, hence Proposition 1.4, follows combining (5.2), (5.3), (5.4), and (5.5).
+6
+Proof of Theorem 1.6
+In this section we consider solutions of the form Vε,λ,ξ =
+k�
+i=1
+(−1)iPUδi,ξi + PVσ.
+Then the reduced
+function in Lemma 4.1 becomes
+�ψ(λ, ξ) = b1
+�
+H(0, 0)λN−2
+0
++
+k
+�
+i=1
+H(ξi, ξi)λN−2
+i
++ 2
+k
+�
+i=1
+(−1)i−1G(ξi, 0)λ
+N−2
+2
+0
+λ
+N−2
+2
+i
++2
+k
+�
+i,j=1,i<j
+(−1)i+j−1G(ξi, ξj)λ
+N−2
+2
+i
+λ
+N−2
+2
+j
+
+ − b2
+N − 2
+2
+ln(λ0λ1λ2 . . . λk),
+
+16
+T. Bartsch, Q. Guo
+where b1, b2 are as in Lemma 4.1.
+Proof of Theorem 1.6. Let k = 4. Using the symmetry again we set ξ1 = (t, 0, . . . , 0) for 0 < t < 1,
+ξ2 = R4ξ1 = (0, t, 0, . . ., 0), ξ3 = R4ξ2 = (−t, 0, . . ., 0), and ξ4 = R4ξ3 = (0, −t, . . . , 0). As in the proof of
+Theorem 1.2, it is sufficient to find stable critical points of �ψ constrained to OAN ,�τ
+η
+. Since
+H(ξ1, ξ1) = H(ξ2, ξ2) = H(ξ3, ξ3) = H(ξ4, ξ4),
+G(ξ1, 0) = G(ξ2, 0) = G(ξ3, 0) = G(ξ4, 0),
+and
+G(ξ1, ξ2) = G(ξ2, ξ3) = G(ξ3, ξ4) = G(ξ4, ξ1),
+G(ξ1, ξ3) = G(ξ2, ξ4),
+we need to find a critical point of the function
+f5(λ0, λ1, λ2, t) := �ψ(λ0, λ1, λ2, λ1, λ2, ξ1, ξ2, ξ3, ξ4)
+= b1
+�
+H(0, 0)λN−2
+0
++ 2H(ξ1, ξ1)(λN−2
+1
++ λN−2
+2
+) + 4G(ξ1, 0)λ
+N−2
+2
+0
+�
+λ
+N−2
+2
+1
+− λ
+N−2
+2
+2
+�
++ 8G(ξ1, ξ2)λ
+N−2
+2
+1
+λ
+N−2
+2
+2
+− 2G(ξ1, ξ3)λN−2
+1
+− 2G(ξ1, ξ3)λN−2
+2
+�
+− b2
+N − 2
+2
+ln
+�
+λ0λ2
+1λ2
+2
+�
+.
+Claim 1: There exist t∗
+1 ∈ (0, 1
+2) and t∗
+2 ∈ ( 1
+2, 1) such that for t ∈ (0, t∗
+1) ∪ (t∗
+2, 1) the equation
+(6.1)
+∇λ0,λ1,λ2f5(λ0, λ1, λ2, t) = 0
+has a unique solution (λ0(t), λ1(t), λ2(t), t).
+Observe that (6.1) is equivalent to the equations
+(6.2)
+H(0, 0)λN−2
+0
++ 2G(ξ1, 0)λ
+N−2
+2
+0
+�
+λ
+N−2
+2
+1
+− λ
+N−2
+2
+2
+�
+= b2
+2b1
+and
+(6.3)
+(H(ξ1, ξ1) − G(ξ1, ξ3))λN−2
+1
++ G(ξ1, 0)λ
+N−2
+2
+0
+λ
+N−2
+2
+1
++ 2G(ξ1, ξ2)λ
+N−2
+2
+1
+λ
+N−2
+2
+2
+= b2
+2b1
+,
+and
+(6.4)
+(H(ξ1, ξ1) − G(ξ1, ξ3))λN−2
+2
+− G(ξ1, 0)λ
+N−2
+2
+0
+λ
+N−2
+2
+2
++ 2G(ξ1, ξ2)λ
+N−2
+2
+1
+λ
+N−2
+2
+2
+= b2
+2b1
+.
+From (6.3) and (6.4) we deduce
+(6.5)
+λ
+N−2
+2
+2
+− λ
+N−2
+2
+1
+=
+G(ξ1, 0)
+H(ξ1, ξ1) − G(ξ1, ξ3)λ
+N−2
+2
+0
+,
+which combined with (6.2) implies:
+(6.6)
+λN−2
+0
+=
+H(ξ1, ξ1) − G(ξ1, ξ3)
+H(ξ1, ξ1) − G(ξ1, ξ3) − 2G2(ξ1, 0) · b2
+2b1
+.
+
+Nodal solutions to problems with Hardy terms
+17
+As a consequence of (6.5) we get
+λN−2
+1
++ λN−2
+2
+− 2λ
+N−2
+2
+1
+λ
+N−2
+2
+2
+= λN−2
+0
+·
+�
+G(ξ1, 0)
+H(ξ1, ξ1) − G(ξ1, ξ3)
+�2
+hence using (6.3) and (6.4) we deduce:
+(6.7)
+λ
+N−2
+2
+1
+λ
+N−2
+2
+2
+=
+1
+H(ξ1, ξ1) − G(ξ1, ξ3) + 2G(ξ1, ξ2) · b2
+2b1
+and
+(6.8)
+λN−2
+1
++ λN−2
+2
+=
+1
+H(ξ1, ξ1) − G(ξ1, ξ3) + 2G(ξ1, ξ2) · b2
+b1
++ λN−2
+0
+·
+�
+G(ξ1, 0)
+H(ξ1, ξ1) − G(ξ1, ξ3)
+�2
+.
+Let τ1(t) = G(ξ1, 0) be as in (4.1) and set
+γ3(t) := H(ξ1, ξ1) − G(ξ1, ξ3) =
+1
+(1 − t2)N−2 −
+1
+(2t)N−2 +
+1
+(t2 + 1)N−2
+and
+γ4(t) := G(ξ1, ξ2) =
+1
+(
+√
+2t)N−2 −
+1
+(t4 + 1)
+N−2
+2
+so that
+f5(λ0, λ1, λ2, t) = b1
+�
+H(0, 0)λN−2
+0
++ 2γ2(t)
+�
+λN−2
+1
++ λN−2
+2
+�
++ 8γ4(t)λ
+N−2
+2
+1
+λ
+N−2
+2
+2
++ 4τ1(t)λ
+N−2
+2
+0
+�
+λ
+N−2
+2
+1
+− λ
+N−2
+2
+2
+� �
+− b2
+N − 2
+2
+ln
+�
+λ4
+1λ0
+�
+.
+A direct computation shows that γ′
+3(t) > 0, γ3(t) → −∞ as t → 0+, γ3(t) → +∞ as t → 1−, and
+γ3( 1
+2) > 0. Thus there exists t∗
+1 ∈ (0, 1
+2) such that
+γ3(t∗
+1) = 0
+and
+γ3(t) < 0 for all t ∈ (0, t∗
+1).
+On the other hand,
+�
+γ3(t) − 2τ 2
+1 (t)
+�′ > 0, γ3(t) − 2τ 2
+1 (t) → −∞ as t → 0+, γ3(t) − 2τ 2
+1 (t) → +∞ as t → 1−,
+and γ3( 1
+2) − 2τ 2
+1 ( 1
+2) < 0. Thus there exists t∗
+2 ∈ ( 1
+2, 1) such that
+γ3(t∗
+2) − 2τ 2
+1 (t∗
+2) = 0
+and
+γ3(t) − 2τ 2
+1 (t) > 0 for all t ∈ (t∗
+2, 1).
+It follows that for every t ∈ (0, t∗
+1) ∪ (t∗
+2, 1) there exist unique λ0(t), λ1(t), λ2(t) such that
+∇λ0,λ1,λ2f5(λ0(t), λ1(t), λ2(t), t) = 0,
+where λ0(t), λ1(t), λ2(t) satisfy (6.5), (6.6), (6.7) and (6.8). This proves Claim 1.
+Claim 2: The Hessian matrix D2
+λ0,λ1,λ2f5(λ0(t), λ1(t), λ2(t), t) is nondegenerate for any t ∈ (0, t∗
+1) ∪ (t∗
+2, 1).
+
+18
+T. Bartsch, Q. Guo
+A direct computation using (6.2), (6.3), and (6.4) shows that, writing λi instead of λi(t),
+∂2f5(λ0, λ1, λ2, t)
+∂λ2
+0
+= (N − 2)b1
+�
+(N − 3)H(0, 0)λN−4
+0
++ (N − 4)τ1(t)λ
+N−6
+2
+0
+�
+λ
+N−2
+2
+1
+− λ
+N−2
+2
+2
+� �
++ (N − 2)b2
+2λ2
+0
+= (N − 2)2b1
+�
+H(0, 0)λN−4
+0
++ τ1(t)λ
+N−6
+2
+0
+�
+λ
+N−2
+2
+1
+− λ
+N−2
+2
+2
+� �
+,
+∂2f5(λ0, λ1, λ2, t)
+∂λ2
+1
+= (N − 2)b1
+�
+2(N − 3)γ3(t)λN−4
+1
++ (N − 4)τ1(t)λ
+N−2
+2
+0
+λ
+N−6
+2
+1
++ 2(N − 4)γ4(t)λ
+N−6
+2
+1
+λ
+N−2
+2
+2
+�
++ (N − 2)b2
+λ2
+1
+= (N − 2)2b1
+�
+2γ3(t)λN−4
+1
++ τ1(t)λ
+N−2
+2
+0
+λ
+N−6
+2
+1
++ 2γ4(t)λ
+N−6
+2
+1
+λ
+N−2
+2
+2
+�
+,
+∂2f5(λ0, λ1, λ2, t)
+∂λ2
+2
+= (N − 2)b1
+�
+2(N − 3)γ3(t)λN−4
+2
+− (N − 4)τ1(t)λ
+N−2
+2
+0
+λ
+N−6
+2
+2
++ 2(N − 4)γ4(t)λ
+N−6
+2
+1
+λ
+N−2
+2
+2
+�
++ (N − 2)b2
+λ2
+2
+= (N − 2)2b1
+�
+2γ3(t)λN−4
+2
+− τ1(t)λ
+N−2
+2
+0
+λ
+N−6
+2
+2
++ 2γ4(t)λ
+N−2
+2
+1
+λ
+N−6
+2
+2
+�
+,
+∂2f4(λ0, λ1, λ2, t)
+∂λ0∂λ1
+= (N − 2)2b1τ1(t)λ
+N−4
+2
+0
+λ
+N−4
+2
+1
+,
+∂2f4(λ1, λ2, λ0, t)
+∂λ0∂λ2
+= −(N − 2)2b1τ1(t)λ
+N−4
+2
+0
+λ
+N−4
+2
+2
+,
+∂2f4(λ1, λ2, λ0, t)
+∂λ1∂λ2
+= 2(N − 2)2b1γ4(t)λ
+N−4
+2
+1
+λ
+N−4
+2
+2
+.
+For simplicity, we introduce the notation
+X := λ
+N−2
+2
+0
+,
+Y := λ
+N−2
+2
+1
+,
+Z := λ
+N−2
+2
+2
+.
+In order to prove that D2
+λ0,λ1,λ2f5(λ0(t), λ1(t), λ2(t), t) is nondegenerate for any t ∈ (0, t∗
+1) ∪ (t∗
+2, 1), it suffices
+to show that the matrix
+
+
+
+
+X + τ1(t)(Y − Z)
+τ1(t)X
+2
+N−2 Y
+N−4
+N−2
+−τ1(t)X
+2
+N−2 Z
+N−4
+N−2
+τ1(t)X
+N−4
+N−2 Y
+2
+N−2
+2γ3(t)Y + τ1(t)X + 2γ4(t)Z
+2γ4(t)Y
+2
+N−2 Z
+N−4
+N−2
+−τ1(t)X
+N−4
+N−2 Z
+2
+N−2
+2γ4(t)Y
+N−4
+N−2 Z
+2
+N−2
+2γ3(t)Z − τ1(t)X + 2γ4(t)Y
+
+
+
+
+is nondegenerate. Using (6.2), (6.3) and (6.4) this is equivalent to showing that the matrix
+
+
+
+
+X
+2 + b2
+4b1 · 1
+X
+τ1(t)X
+2
+N−2 Y
+N−4
+N−2
+−τ1(t)X
+2
+N−2 Z
+N−4
+N−2
+τ1(t)X
+N−4
+N−2 Y
+2
+N−2
+γ3(t)Y + b2
+2b1 · 1
+Y
+2γ4(t)Y
+2
+N−2 Z
+N−4
+N−2
+−τ1(t)X
+N−4
+N−2 Z
+2
+N−2
+2γ4(t)Y
+N−4
+N−2 Z
+2
+N−2
+γ3(t)Z + b2
+2b1 · 1
+Z
+
+
+
+
+is nondegenerate. A direct computation, using (6.7), shows that the determinant of the above matrix has the
+same sign as γ3(t), hence is nontrivial, proving Claim 2.
+Theorem 1.6 now follows from
+
+Nodal solutions to problems with Hardy terms
+19
+Claim 3: The function ν2(t) := f5
+�
+λ0(t), λ1(t), λ2(t), t
+�
+has a critical point t1 ∈ (0, t∗
+1).
+Observe that, writing again λi instead of λi(t),
+ν′
+2(t) = ∂f4(λ0(t), λ1(t), λ2(t), t)
+∂t
+= 2b1
+�
+γ′
+3(t)
+�
+λN−2
+1
++ λN−2
+2
+�
++ 2τ ′
+1(t)λ
+N−2
+2
+0
+�
+λ
+N−2
+2
+1
+− λ
+N−2
+2
+2
+�
++ 4γ′
+4(t)λ
+N−2
+2
+1
+λ
+N−2
+2
+2
+�
+,
+where λ0, λ1, λ2 satisfy (6.5), (6.6), (6.7) and (6.8). Therefore, ν′
+2(t) = 0 for t ∈ (0, t∗
+1) is equivalent to
+ι3(t) := γ′
+3(t)
+�
+2γ3(t)(γ3(t) − 2τ 2
+1 (t)) + τ2
+1 (t)(γ3(t) + 2γ4(t))
+�
+− 2τ ′
+1(t)τ1(t)γ3(t)
+�
+γ3(t) + 2γ4(t)
+�
++ 4γ′
+4(t)γ3(t)
+�
+γ3(t) − 2τ 2
+1 (t)
+�
+= 0.
+It is easy to check that ι3(t) → −∞ as t → 0+ and ι3(t∗
+1) > 0 because γ′
+3(t∗
+1) > 0, γ4(t∗
+1) > 0 and γ3(t∗
+1) = 0.
+Hence there exists t1 ∈ (0, t∗
+1) such that ι3(t1) = 0. Claim 3 follows, finishing the proof of Theorem 1.6.
+□
+Remark 6.1. We conjecture that there should also exist t2 ∈ (t∗
+2, 1) such that ι3(t2) = 0. This is not considered
+here because the computations get enormous.
+Acknowledgements: The authors would like to thank Professor Daomin Cao for many helpful discussions
+during the preparation of this paper. This work was carried out while Qianqiao Guo was visiting Justus-
+Liebig-Universit¨at Gießen, to which he would like to express his gratitude for their warm hospitality.
+Funding: Qianqiao Guo was supported by the National Natural Science Foundation of China (Grant No.
+11971385) and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2019JM275).
+References
+[1] Th. Aubin, Probl`emes isop´erim´etriques et espaces de Sobolev, J. Differential Geometry 11 (1976), 573-598.
+[2] A. Bahri, Y. Li, O. Rey, On a variational problem with lack of compactness: the topological effect of the
+critical points at infinity, Calc. Var. Part. Diff. Equ. 3 (1995), 67-93.
+[3] T. Bartsch, T. D’Aprile, A. Pistoia, Multi-bubble nodal solutions for slightly subcritical elliptic problems
+in domains with symmetries, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 30 (2013), 1027-1047.
+[4] T. Bartsch, T. D’Aprile, A. Pistoia, On the profile of sign-changing solutions of an almost critical problem
+in the ball, Bull. London Math. Soc. 45 (2013), 1246-1258.
+[5] T. Bartsch, Q. Guo, Nodal blow-up solutions to slightly subcritical elliptic problems with Hardy-critical
+term, Adv. Nonlinear Stud. 17 (2017), 55-85.
+
+20
+T. Bartsch, Q. Guo
+[6] T. Bartsch, Q. Guo, Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy
+terms, SN Partial Differ. Equ. Appl. (2020) 1:26.
+[7] T. Bartsch, A. Micheletti, A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations
+involving critical growth, Calc. Var. Part. Diff. Equ. 26 (2006), 265-282.
+[8] G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Functional Analysis 100 (1991), 18-24.
+[9] H. Brezis, L.A. Peletier, Asymptotics for elliptic equations involving critical growth, Partial differential
+equations and the calculus of variations, vol. I, Progr. Nonlinear Diff. Equ. Appl. Birkh¨auser, Boston, MA
+1 (1989), 149-192.
+[10] D.M. Cao, P.G. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy poten-
+tial, J. Diff. Equ. 205 (2004), 521-537.
+[11] D.M. Cao, S.J. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and
+Hardy terms, J. Diff. Equ. 193 (2003), 424-434.
+[12] D.M. Cao, S.J. Peng, Asymptotic behavior for elliptic problems with singular coefficient and nearly critical
+Sobolev growth, Ann. Mat. Pura. Appl. 185 (2006), 189-205.
+[13] F. Catrina, Z.Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and
+nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), 229-258.
+[14] M. del Pino, J. Dolbeault, M. Musso, “Bubble-tower” radial solutions in the slightly supercritical Brezis-
+Nirenberg problem, J. Diff. Equ. 193(2) (2003), 280-306.
+[15] I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer.
+Math. Soc. (N.S.) 39(2) (2002), 207-265.
+[16] V. Felli, A. Pistoia, Existence of blowing-up solutions for a nonlinear elliptic equation with Hardy potential
+and critical growth, Comm. Part. Diff. Equ. 31 (2006), 21-56.
+[17] V. Felli, S. Terracini, Fountain-like solutions for nonlinear elliptic equations with critical growth and
+Hardy potential, Comm. Contemp. Math. 7 (2005), 867-904.
+[18] A. Ferrero, F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J.
+Diff. Equ. 177 (2001), 494-522.
+[19] M. Flucher, J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of
+hot spots, Manuscripta Math. 94 (1997), 337-346.
+[20] N. Ghoussoub, F. Robert, The Hardy–Schr¨odinger operator with interior singularity: the remaining cases,
+Calc. Var. Part. Diff. Equ. 56 (20017), paper 149, 54 pp.
+
+Nodal solutions to problems with Hardy terms
+21
+[21] N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy
+exponents, Trans. Amer. Math. Soc. 352 (2000), 5703-5743.
+[22] M. Grossi, F. Takahashi, Nonexistence of multi-bubble solutions to some elliptic equations on convex
+domains, J. Functional Analysis 259 (2010), 904-917.
+[23] Q.Q. Guo, P.C. Niu, Nodal and positive solutions for singular semilinear elliptic equations with critical
+exponents in symmetric domains, J. Diff. Equ. 245 (2008), 3974-3985.
+[24] Z.C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical
+Sobolev exponent, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 8 (1991), 159-174.
+[25] E. Jannelli, The role played by space dimension in elliptic critical problems, J. Diff. Equ. 156 (1999),
+407-426.
+[26] M. Musso, A. Pistoia, Tower of bubbles for almost critical problems in general domains, J. Math. Pures
+Appl. 93 (2010), 1-40.
+[27] M. Musso, J. Wei, Nonradial solutions to critical elliptic equations of Caffarelli-Kohn-Nirenberg type, Int.
+Math. Res. Not. 18 (2012), 4120-4162.
+[28] A. Pistoia, T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet
+problem, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 24 (2007), 325-340.
+[29] O. Rey, Proof of two conjectures of H. Br´ezis and L.A. Peletier, Manuscripta Math. 65 (1989), 19-37.
+[30] O. Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev
+exponent, J. Functional Analysis 89 (1990), 1-52.
+[31] O. Rey, Blow-up points of solutions to elliptic equations with limiting nonlinearity, Differential Integral
+Equations 4 (1991), 1155-1167.
+[32] D. Ruiz, M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Diff. Equ. 190
+(2003), 524-538.
+[33] D. Smets, Nonlinear Schr¨odinger equations with Hardy potential and critical nonlinearities, Trans. Amer.
+Math. Soc. 357 (2005), 2909-2938.
+[34] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353-372.
+[35] S.Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical
+exponent, Adv. Diff. Equations 2 (1996), 241-264.
+E-mail:
+
+22
+T. Bartsch, Q. Guo
+thomas.bartsch@math.uni-giessen.de
+gqianqiao@nwpu.edu.cn
+

KdE2T4oBgHgl3EQfAgZ0/content/2301.03592v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ad28611f3b590fc686abbfabfb9be9d64c3f7a54c69ddc1ffb67c77bab4f92e4
+size 3120024

KdE2T4oBgHgl3EQfAgZ0/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ae690443b8f1237b3062e65bd90794f2fd980f759db0888199b27dd71e22a2d1
+size 7405613

KdE3T4oBgHgl3EQfXwqt/content/tmp_files/2301.04482v1.pdf.txt

@@ -0,0 +1,1531 @@
+Multiple-level Point Embedding for Solving Human Trajectory Imputation
+with Prediction
+KYLE K. QIN, RMIT University, Australia
+YONGLI REN, RMIT University, Australia
+WEI SHAO, RMIT University, Australia
+BRENNAN LAKE, Cuebiq Inc., USA
+FILIPPO PRIVITERA, Cuebiq Inc., USA
+FLORA D. SALIM, University of New South Wales, Australia
+Sparsity is a common issue in many trajectory datasets, including human mobility data. This issue frequently brings more difficulty
+to relevant learning tasks, such as trajectory imputation and prediction. Nowadays, little existing work simultaneously deals with
+imputation and prediction on human trajectories. This work plans to explore whether the learning process of imputation and prediction
+could benefit from each other to achieve better outcomes. And the question will be answered by studying the coexistence patterns
+between missing points and observed ones in incomplete trajectories. More specifically, the proposed model develops an imputation
+component based on the self-attention mechanism to capture the coexistence patterns between observations and missing points among
+encoder-decoder layers. Meanwhile, a recurrent unit is integrated to extract the sequential embeddings from newly imputed sequences
+for predicting the following location. Furthermore, a new implementation called Imputation Cycle is introduced to enable gradual
+imputation with prediction enhancement at multiple levels, which helps to accelerate the speed of convergence. The experimental
+results on three different real-world mobility datasets show that the proposed approach has significant advantages over the competitive
+baselines across both imputation and prediction tasks in terms of accuracy and stability.
+CCS Concepts: • Information systems → Spatial-temporal systems.
+Additional Key Words and Phrases: Location Imputation; Human Trajectory Prediction; Self-attention Network
+ACM Reference Format:
+Kyle K. Qin, Yongli Ren, Wei Shao, Brennan Lake, Filippo Privitera, and Flora D. Salim. 2022. Multiple-level Point Embedding for
+Solving Human Trajectory Imputation with Prediction. ACM Trans. Spatial Algorithms Syst. 1, 1, Article 1 (January 2022), 22 pages.
+https://doi.org/10.1145/1122445.1122333
+1
+INTRODUCTION
+Mining knowledge on datasets such as time series and human mobility data has received much attention from the
+public [4, 6, 17, 24, 28]. However, missing values that commonly exist in this type of dataset can implicitly influence the
+Authors’ addresses: Kyle K. Qin, RMIT University, 124 La Trobe St, Melbourne, VIC, Australia, 3000, kyle.qin@hotmail.com; Yongli Ren, RMIT University,
+124 La Trobe St, Melbourne, VIC, Australia, 3000, yongli.ren@rmit.edu.au; Wei Shao, RMIT University, 124 La Trobe St, Melbourne, VIC, Australia, 3000,
+wei.shao@rmit.edu.au; Brennan Lake, Cuebiq Inc., 15 West 27th Street, New York, NY, USA, 10001, blake@cuebiq.com; Filippo Privitera, Cuebiq Inc., 15
+West 27th Street, New York, NY, USA, 10001, fprivitera@cuebiq.com; Flora D. Salim, University of New South Wales, School of Computer Science and
+Engineering, Engineering Rd, Kensington, NSW, Australia, 2052, flora.salim@unsw.edu.au.
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not
+made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components
+of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to
+redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2022 Association for Computing Machinery.
+Manuscript submitted to ACM
+Manuscript submitted to ACM
+1
+arXiv:2301.04482v1  [cs.LG]  11 Jan 2023
+
+2
+Qin et al.
+quality of the analysis and learning process. As we know, data on human mobility at a fine-grained scale can help with
+understanding people’s movement patterns, activities, and potential intentions, allowing customized recommendations
+and services to be delivered to individuals or groups. However, human trajectory data is frequently incomplete in
+practice for many reasons, such as power or bandwidth limitations on sensor-based devices and varying communication
+frequency of signals between devices and servers. It has been mentioned that the sparsity of data could cause deterioration
+of performance on learning human activities or movements [18, 23]. Therefore, imputing missing values in a trajectory
+becomes a fundamental problem for other learning tasks, such as predictions.
+Human mobility prediction or imputation has become prevalent, with various types of mobility data available for
+collection and processing. Previous works of human trajectory prediction mainly focused on forecasting future positions
+at different levels of granularity, including coordinates [9, 11, 26], specific locations such as Point-of-Interests (POIs)
+[7, 31, 32, 38] and grids or regions on map [8, 22]. Intuitively, the imputation of human trajectories can be formulated
+as the problem of time series imputation, which a number of relevant approaches have solved. Previous algorithms
+proposed for sequence imputation generally leverage the advantages of Generative Adversarial Networks (GANs)
+[6, 20, 21] or Recurrent Neural Networks (RNNs) [3]. However, those generative models that are autoregressive could
+be vulnerable to compounding errors in long-range temporal sequence modeling, and the recurrent models may face
+issues of vanishing gradients and difficult training. Recently, two non-autoregressive models [18, 25] were claimed to
+have more merits for imputing values in sequential data such as traffic flows and pedestrians’ trajectories in a specific
+scene. Nonetheless, these approaches first lack explicit observation and evaluation of the imputation of human mobility
+in the scope of the city. As we know, compared with walking or running trajectories in a small scene, the patterns
+and factors could be dramatically varied when the distribution of movements is spread at the macro city scope (e.g.,
+check-in data). We would encounter more challenges in this kind of dataset which is normally accompanied by arbitrary
+movements, sparsity, and no uniform frequency of points collection. Secondly, the specific dependencies or coexistence
+patterns between visited locations and missing ones in trajectories were not well considered during the training by the
+previous methods. Moreover, little existing work deals with prediction while simultaneously imputing values in the
+data of trajectories.
+This paper plans to tackle the imputation issue by studying the coexistence patterns or dependencies between missing
+points and observed ones while incomplete human trajectories are given. Meanwhile, we argue that while solving the
+data sparsity issue, the imputation and prediction tasks can complement each other. In other words, we develop an
+approach that could achieve mutually beneficial effects between both tasks with interactive optimization. As we know,
+information will be lost when passing messages in a sequential manner of learning. Missing values are often randomly
+distributed in human movement trajectories, potentially inhibiting the efficacy of imputation in a traditional sequential
+manner (e.g., RNNs). Transformer Networks [29] can effectively learn representations of observations for prediction
+purposes by weighting the values of sequences via the attention mechanism. This method is naturally suitable for
+learning in a non-sequential way. We propose a new approach called multIple-level poiNt embeddinG for solving
+tRajectory imputAtion wIth predictioN (INGRAIN). INGRAIN can effectively capture coexistence dependencies among
+observed and missing values via multi-head attention in encoder-decoder layers, which helps ameliorate the imputation
+efficiency. Meanwhile, the model effectively integrates with a recurrent unit to extract sequential embeddings from
+newly imputed sequences to predict the next moving positions. For flexible learning, a new process called the Imputation
+Cycle is designed to enable progressive imputation with prediction at multiple levels by constraining the number of
+missing points for each imputation recurrence. In addition, the model delivers messages from the imputation module to
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+3
+RNN units to generate sequential embeddings for forecasting purposes. In summary, the key contributions of this paper
+are as follows:
+• We propose one new framework that integrates both autoregressive and non-autoregressive components to
+impute missing points in human trajectories and predict future movements.
+• A model is established for trajectory imputation with prediction based on gradual amelioration at multiple levels
+via setting the granularity of imputing points. It is applicable to different human mobility datasets.
+• Comprehensive evaluations are conducted to show the efficiency and effectiveness of our model on three real-
+world human trajectory datasets. The paper provides insights into how the method satisfies accurate estimations
+on missing points and next positions and how trade-offs could be handled in this type of cooperative learning.
+The rest of the paper is organized as follows: Section 2 introduces the related work, and Section 3 provides the
+definitions for the problem of human trajectory imputation with prediction. Section 4 presents the details of the
+proposed solution and its important components. Section 5 shows the experimental results, and the conclusion is given
+in Section 6.
+2
+RELATED WORK
+2.1
+Human Trajectory Prediction
+Many studies have been devoted to human trajectory prediction. They can be categorized according to the granularity of
+targeted locations in prediction. A part of the literature has focused on forecasting grids or regions for next movements
+on maps [8, 22], whereas other parts have explored predicting coordinates [9, 11, 26] or POIs [7, 31, 32] in the future.
+We can further classify the relevant literature from the perspective of traveling scope in human mobility data: some
+studies [7, 26, 31, 32] have concentrated on predicting locations of people across city areas that are often sparsely or
+scattered populated, and others [1, 9, 11] have examined the motions of persons in smaller-scale scenes such as crowds
+on the street or customers on the floor of a building. The patterns and factors of human movement could be particularly
+distinct while the scale of traveling distance is changed. For example, by extracting grid-based stay time statistics of
+users and periodically analyzing their frequency of visiting regions [8], the Inhomogeneous Continuous-Time Markov
+model was used to capture temporal and sequential regularities for predicting the leaving time of objects in regions and
+their next locations. There is a certain positive correlation between people’s morning trajectories and corresponding
+afternoon trajectories [26] in cities, so the integrated similarity metric was developed to estimate similar segments
+of trajectories with temporal segmentation and temporal correlation extraction. In a series of RNNs, LSTM [13] and
+Gated Recurrent Units (GRU) [5] are two popular variants that effectively moderate short-term memory and can help
+to avoid the vanishing gradient problem. Social Long Short-Term Memory [1] was proposed based on LSTM to estimate
+the motions of pedestrians among the crowd in different scenes while taking into account the navigation of all their
+walking neighbors in a shared site.
+2.2
+Missing Data Imputation
+Imputation of missing values on trajectory datasets has become an indispensable work in many applications. Previously,
+a considerable number of methods for trajectory completion focused on inferring missing portion of traffic trajectories
+from sparse GPS samples based on the geometry of road network [16, 19, 33, 36]. For instance, the History-based Route
+Inference System (HRIS) [36] was established with a set of new mapping algorithms that could effectively extract
+and learn the traveling patterns from historical trajectories and incorporate them into the route inference process.
+Manuscript submitted to ACM
+
+4
+Qin et al.
+Along with estimating the traffic flows across junctions in a road network, Li et al. utilized the GPS samples within
+each flow cluster to achieve fine-level completion of individual trajectories [16]. Yin et al. developed a map-matching
+algorithm by evaluating the traveling cost of candidate routes and considering the distance feature and road selection
+behavior of users [33]. However, the imputation methods for traffic trajectories on roads are hard to adapt to outdoor
+human mobility in city areas. Those road network mapping techniques could become ineffective while human beings’
+movements are relatively arbitrary on the map and affected by a variety of factors.
+More recent research related to missing value imputation has mainly relied on techniques to impute sequential data
+including Matrix Factorisation [23], GANs [6, 20, 21, 34] or RNNs [3, 35]. Naghizade et al. [23] proposed a contextual
+model to predict information at missing locations in sparse indoor trajectories via using Graph-regularised Non-
+negative Matrix Factorisation with consideration of implicit social ties among individuals. A model called BRITS [3]
+was built on RNNs to directly learn missing values for time series in a bidirectional recurrent dynamical system without
+strong assumptions. In GAIN [34], the generator imputes the missing values conditioned on observed data, and the
+discriminator then attempts to identify which parts of the conjectured vector are actually observed and which are
+imputed. The former module focuses on the imputation quality, and the latter is forced to learn according to real data
+distribution. In contrast, MASKGAN [6] explicitly trains the generator to produce high-quality samples for infilling text
+on sentences. GRUI [20], an autoregressive model based on GAN, was developed for the imputation of multivariate
+time series, such as electronic medical record datasets or air quality and weather data. Additionally, Liu et al. recently
+presented a non-autoregressive model NAOMI [18] to impute missing values in different sequential datasets, such as
+traffic flows and trajectories of basketball players. The missing values are filled recursively from coarse to fine-grained
+resolutions via a forward and backward RNN-based model. NAOMI considers multiple-resolution imputation, wherein
+new imputations will be performed based on the previous inferences of values.
+2.3
+Attention Mechanism
+In the learning process, recurrent models can make predictions by transiting successive dependencies of observations
+from the beginning of entire sequences. That gives recurrent models the expressive ability to deal with long sequential
+data while maintaining hidden states. Nowadays, autoregressive models such as the Transformer Network are competi-
+tive with or even supersede RNNs on a diverse set of missions. Transformer Network is claimed to effectively weigh and
+learn representations over available observations via the multi-head attention mechanism. The original Transformer
+[29] was established to model and predict sequences in the natural language processing field. Subsequently, GATs [30]
+was developed to operate on graph-structured data, leveraging self-attentional layers to address weighting nodes for
+graph convolutions. Recently, it was adopted by Giuliari et al. [9] to forecast the future motions of people in different
+scenes, which renders better performance than the LSTM-based and Linear approaches.
+2.4
+Main Research Gaps
+In this paper, we attempt to examine whether the imputation of missing points in human historical mobility can bring
+sake to the prediction of future movements, which is rarely investigated by the existing works. As we mentioned,
+our objective focus on inferring missing locations of daily human mobility in city areas. The road network mapping
+techniques for vehicle traffic trajectories can not adapt well to such datasets as the distribution of human beings’
+movements is relatively arbitrary without strictly following the road networks. And human activities could be affected
+heavily by social connections, professions, and weather conditions. In addition, many GAN-based approaches, such as
+MASKGAN, GRUI, and GAIN were not designed or tested for complex human mobility data in terms of constructing
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+5
+Table 1. Symbols and notations.
+Symbols
+Description
+𝑝𝑡
+The point was recorded at time 𝑡
+𝑆𝑢
+The original mobility sequence of user 𝑢
+𝑡𝑟𝑤
+𝑢
+One sub-trajectory of 𝑆𝑢 in 𝑤-𝑡ℎ time window
+𝑈
+The number of users involved
+𝑇
+The maximum number of points considered
+𝐿
+The width of each time window
+𝑆
+′
+𝑢
+A set of sub-trajectories of user 𝑢 from 𝑆𝑢
+𝑝𝑙
+The point at step 𝑙 in a trajectory
+𝑝𝑖
+A missing point in one trajectory
+𝑀
+A masking vector for missing values
+𝐼
+The total number of missing points in trajectory
+𝐷
+The dimension of initial representation
+𝑍
+An initial representation of input points
+𝑧𝑙
+𝑜𝑏𝑠
+The initial representation of point 𝑝𝑙
+𝑧𝑖
+𝑚𝑖𝑠
+The initial representation of missing point 𝑝𝑖
+𝜆1
+The weight of imputation component
+𝜆2
+The weight of prediction component
+𝜆3
+The weight of movement velocity
+coordinates of locations with Spatio-temporal dimensions. Recently, NAOMI and another variant called SingleRes [18]
+were proposed to apply forward and backward RNN on observed points to infer missing values in each trajectory.
+However, those methods still rely on learning sequential dependencies of points in trajectories, and the experiments only
+tested the trajectories of agents in a relatively small scene (e.g., a billiard table or basketball square). The performance
+of daily human movements’ trajectories across city regions remains unknown. This type of trajectory contains more
+arbitrary movements occurring in an ample space with possibly more sparsity. Our approach can provide insights
+for capturing the coexisting dependencies between missing positions and observed ones on different human mobility
+datasets. And the model is established based on the conditional independence assumption, considering both spatial and
+temporal perspectives, which the previous work did not examine sufficiently.
+3
+PROBLEM DEFINITION
+In this section, we define the problem of trajectory prediction and imputation task as follows:
+• The Prediction Task: we denote an original mobility sequence of user 𝑢 as 𝑆𝑢 = {𝑝1, 𝑝2, ..., 𝑝𝑇 }, where 𝑝𝑡 ∈ R2
+is the coordinates of point recorded at time frame 𝑡 and 𝑇 is the maximum number of observed values in
+consideration. A trajectory 𝑡𝑟𝑤
+𝑢 is one sub-sequence of 𝑆𝑢 in 𝑤-𝑡ℎ time window, and the width of the window is
+𝐿. Therefore, 𝑡𝑟𝑤
+𝑢 = {𝑝𝑤
+1 , 𝑝𝑤
+2 , ..., 𝑝𝑤
+𝐿 }, 𝑙-𝑡ℎ is the number of points in the sub-sequence. A set of such trajectories
+can be extracted from the original sequence with a defined time window for user 𝑢: 𝑆
+′
+𝑢 = {𝑡𝑟1𝑢,𝑡𝑟2𝑢, ...𝑡𝑟𝑊
+𝑢 }. The
+whole dataset can be denoted as 𝑆 = {𝑆
+′
+1,𝑆
+′
+2, ...𝑆
+′
+𝑈 } and 𝑈 is the total number of users. The goal of this task is to
+predict the coordinates of the next movement when giving a trajectory.
+Manuscript submitted to ACM
+
+6
+Qin et al.
+• The Imputation Task: we assume that 𝑡𝑟𝑤
+𝑢 denotes one of user 𝑢’s trajectories. In reality, 𝑡𝑟𝑤
+𝑢 may contain a
+portion of missing points for many reasons. Thus, the states of all the points inside the trajectory are represented
+with one masking vector 𝑀 = [𝑚1,𝑚2, ...,𝑚𝐿], where 𝑚𝑙 equals to zero if 𝑝𝑤
+𝑙 is not observed. Otherwise, 𝑚𝑙
+is set to one. The purpose is to infer and substitute the missing values with appropriate alternatives in each
+trajectory.
+Moreover, Table 1 provides more information about the symbols and notations used in this paper.
+4
+PROPOSED APPROACH
+Fig. 1 illustrates the architecture of our proposed method for both human trajectory imputation and prediction. For
+imputation purposes, encoders and decoders based on self-attention are applied to learn coexisting patterns between
+observations and missing points. Then, a recurrent component plays a role in extracting sequential dependencies on
+newly learned embeddings from the Supplement Layer for forwarding prediction. Meanwhile, a mechanism called the
+Imputation Cycle is introduced to achieve progressive learning of both imputation and prediction at multiple levels. The
+model interactively considers learning two main components and improves overall performance. Algorithm 1 shows an
+overview of training conducted using the proposed approach, and the details of the main components are described in
+the following subsections.
+4.1
+Imputation Component
+In the beginning, the representations of each trajectory with missing points are generated via Linear layers and
+Frame-positional Encoding. Then, the imputation component takes the initial representations and captures the spatial-
+temporal dependencies among points through self-attention in either Encoders or Decoders. This process will produce
+embeddings for observations and missing values, respectively. Here, one agile design allows the model to decode 𝑛
+number of missing points at each imputation time, and the process is repeated by the Imputation Cycle. Thus, the final
+impute layer is responsible for inferring 𝑛 number of missing points each time.
+4.1.1
+Feature Initialisation. In the first stage, the framework projects representations of an incomplete trajectory into a
+higher 𝐷-dimensional space via the Linear Embedding Layer. Specifically, the initial representation of one observed
+point 𝑝𝑙 is 𝑧𝑙
+𝑜𝑏𝑠 = 𝑝⊤
+𝑙 𝑊𝑜𝑏𝑠, where ⊤ denotes transpose matrix, 𝑊𝑜𝑏𝑠 is the weight matrix, and 𝑝𝑙 is a zero vector if
+it is missing. We obtain a queue for the incomplete trajectory 𝐸𝑜𝑏𝑠 = {𝑧1
+𝑜𝑏𝑠,𝑧2
+𝑜𝑏𝑠, ...,𝑧𝐿
+𝑜𝑏𝑠}. We extract another queue
+of missing points from 𝐸𝑜𝑏𝑠 as 𝐸𝑚𝑖𝑠 = {𝑧1
+𝑚𝑖𝑠,𝑧2
+𝑚𝑖𝑠, ...,𝑧𝐼
+𝑚𝑖𝑠}. Similarly, the initial representation of a missing point is
+𝑧𝑖
+𝑚𝑖𝑠 = 𝑝⊤
+𝑖 𝑊𝑚𝑖𝑠, where 𝑝𝑖 is also zero vector (0 ≤ 𝑖 ≤ 𝐼) and 𝐼 is the total number of missing points in a trajectory.
+Adding time information to the initial representations of trajectories is essential because the imputation unit relies
+on the attention mechanism for learning without any sequential knowledge. We insert time representation to the points
+as distinctive identifications in trajectory based on the Positional Encoding method [29]. The relevant equations are
+given as follows:
+𝐹 (𝑡,𝑑) =
+
+
+sin(
+𝑡
+10000𝑑/𝐷 ),
+when d is even,
+cos(
+𝑡
+10000𝑑/𝐷 ),
+when d is odd,
+(1)
+where 𝐹 (𝑡,𝑑) outputs the representation vector for time frame 𝑡 recorded for a point in the trajectory, and 𝑑 is the
+𝑑-th dimension in the vector, which is also 𝐷-dimensional. Then, the representation of the time frame is added to
+(e.g., element-wise addition) the initial embedding of the corresponding point in either 𝐸𝑜𝑏𝑠 or 𝐸𝑚𝑖𝑠. Noticeably, the
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+7
+Fig. 1. The proposed framework solves trajectory imputation and prediction jointly. 𝑃1, 𝑃4, 𝑃6, and 𝑃8 are the observed points in the
+trajectory, 𝑃2, 𝑃3, 𝑃5, and 𝑃7 are the missing values. Multiple Encoders and Decoders are employed in the learning process to produce
+high-level embeddings for the observed trajectory and missing values, respectively. First, the Linear embedding layer initializes the
+representations of the trajectory queue and missing point(s), and Time Frame Encoding is responsible for adding observed time points
+to relevant representations. Next, the trajectory embeddings from Encoders are fed to Encoder-Decoder Attention to enhance the
+embedding of missing points. Finally, new embeddings of the trajectory are reconstructed by the Supplement Layer and transferred
+to an RNN-based unit for prediction purposes. In addition, the Imputation Cycle enables the model to conduct gradual imputation on
+multiple levels.
+𝐹 is applied to the missing points in both queues 𝐸𝑜𝑏𝑠 and 𝐸𝑚𝑖𝑠, such that the model can have consistent temporal
+information of missing values from both queues during the imputation.
+The original Positional Encoding encodes the positions of words in a sequence. Here, we encode the time frames of
+points in a trajectory within the observed period. One time frame 𝑡 is a unique numeric index like 0, 1, 2..., which stands
+for the order in which people’s movements occurred in the observation period (day, week, or month). Some existing
+literature [7] has mentioned that the patterns of human mobility tend to occur periodically, and integrating with this
+information can help to learn relevant tasks. Therefore, we add time information using the time positional encoder to
+embed the time frame of points during a considered period. Although the missing points are initially represented with
+Manuscript submitted to ACM
+
+RNN
+learn
+Units
+Predict Layer
+Supplement Layer
+ImputeLayer
+Traj Encoded
+Mssing Point
+Ermbedding
+Embedding
+个
+Add&Normalize
+Add&Normalize
+Feed Forward
+Feed Forward
+Enc-DecAttention
+Nx Encoder
+NxDecoder
+Add&Normalize
+Add&Normalize
+Multi-HeadAttention
+Multi-Head Attention
+Time Frame Encoding
+LinearEmbedding
+LinearEmbedding
+P1
+P2
+P3
+P4
+P5
+P6
+P7
+P8
+P2
+P3
+P5
+P78
+Qin et al.
+zero vectors, the Positional Encoding attaches time features on each missing value which will be further updated by the
+attention-based Encoders and Decoders in the next training stage.
+4.1.2
+Encoder and Decoder. As Fig. 1 shows, multiple Encoders and Decoders are employed to produce embeddings
+by weighting the correlation between observed and missing values, respectively. They share a similar structure:
+Self-attention, Normalisation, and Feed-forward Layers. The self-attention block of the first encoder and decoder
+acquires 𝐸𝑜𝑏𝑠 and 𝐸𝑚𝑖𝑠 as inputs, respectively. And they are responsible for obtaining the internal dependencies
+among the points on each side. Moreover, each decoder contains an Encode-decode Attention to learn further external
+dependencies between an incomplete trajectory and its missing values. More specifically, a 𝑑𝑘-dimensional 𝑞𝑢𝑒𝑟𝑦 vector,
+𝑑𝑘-dimensional 𝑘𝑒𝑦 vector, and 𝑑𝑣-dimensional 𝑣𝑎𝑙𝑢𝑒 vector are built for each point by multiplying its embedding
+with three different weight matrices, respectively. In practice, we deal with a set of points together by wrapping the
+𝑞𝑢𝑒𝑟𝑦 vectors into matrix 𝑄, 𝑘𝑒𝑦𝑠 into matrix 𝐾, and 𝑣𝑎𝑙𝑢𝑒𝑠 into matrix 𝑉 . The final output of the self-attention layer is
+calculated using the following formula:
+𝐴𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(𝑄, 𝐾,𝑉 ) = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥(𝑄𝐾⊤
+√︁
+𝑑𝑘
+)𝑉,
+(2)
+where 𝑑𝑘 is the dimension of 𝐾. Multiple encoders or decoders work sequentially in the imputation component: the
+output of one block will be taken as input by the next block.
+According to [29], multi-head attention that comprises several self-attentions can be applied to jointly synthesize the
+initial representation 𝑍 of points in each input queue from different representation sub-spaces, which helps enhance
+embedding learning. The functions of the operation are given as follows:
+𝑀𝑢𝑙𝑡𝑖𝐻𝑒𝑎𝑑(𝑍) = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1,ℎ𝑒𝑎𝑑2, ...,ℎ𝑒𝑎𝑑ℎ)𝑊 𝑂,
+𝑊 𝑂 ∈ Rℎ𝑑𝑣×𝐷,
+(3)
+ℎ𝑒𝑎𝑑𝑖 = 𝐴𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(𝑍𝑊 𝑄
+𝑖 ,𝑍𝑊 𝐾
+𝑖 ,𝑍𝑊 𝑉
+𝑖 ),
+𝑊 𝑄
+𝑖
+∈ R𝐷×𝑑𝑘,𝑊 𝐾
+𝑖
+∈ R𝐷×𝑑𝑘,𝑊 𝑉
+𝑖
+∈ R𝐷×𝑑𝑣 .
+(4)
+Here, ℎ is the total number of heads in consideration, and each ℎ𝑒𝑎𝑑𝑖 is an individual of self-attention. 𝑄𝑖 = 𝑍𝑊 𝑄
+𝑖 , 𝐾𝑖 =
+𝑍𝑊 𝐾
+𝑖
+and 𝑉𝑖 = 𝑍𝑊 𝑉
+𝑖 . One of the main merits of multi-head attention is that attending computations can be executed
+in parallel to boost the overall runtime. We apply two heads of self-attention in both encoders or decoders in the
+experiments.
+The proposed framework learns the coexistence patterns of the observed and missing points by capturing their
+dependencies between the point embeddings from the encoders and decoders. Fig. 2 illustrates the component of
+an Encoder-Decoder Attention that extracts the dependent relations between the observed trajectory and missing
+points. For the imputation of one incomplete trajectory, the queues are created for missing points and the relevant
+trajectory, respectively. Furthermore, the model imputes 𝑛 missing points each time from the queue of missing points.
+In the example of Fig. 2, when 𝑛 = 2, missing points 𝑃2 and 𝑃3 are first ejected into the Decoder Self-Attention for
+producing embeddings of 𝑃2 and 𝑃3. Simultaneously, the queue of the incomplete trajectory is injected into the Encoder
+Self-Attention for trajectory embedding production. Then, the Encoder-Decoder Attention is responsible for learning
+the dependencies by weighting the relationship of missing points (𝑃2, 𝑃3) and the observed trajectory. Finally, the
+locations of the missing points can be inferred based on their learned embeddings.
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+9
+P1
+P2
+P3
+P4
+P5
+P6
+P7
+P8
+Enc-Dec Attention
+P2
+P3
+P2
+P3
+P5
+P7
+n = 2
+Weighting
+Decoder Self-Attention
+P1
+P2
+P3
+P4
+P5
+P6
+P7
+P8
+Encoder Self-Attention
+Missing Points
+Queue
+Traj Queue
+Fig. 2. The diagram shows that the Encoder-Decoder Attention captures the dependent patterns between the missing points and an
+observed trajectory.
+4.1.3
+Imputation Recurrence. The model is capable of carrying out gradual imputation at multiple levels, which gives
+the model more flexibility by inferring 𝑛 (1 < 𝑛 < 𝐼) missing nodes in each imputation cycle. The encoders yield
+the embeddings of an incomplete trajectory and then transfer them to decoders for weighting with missing values in
+imputation. Before that, the decoder takes the 𝑛 number of missing values as inputs in each cycle. Further, 𝑛 alternatives
+are yielded from the Imputing Linear Layer, and new learning circulation is started based on the previous state. The
+model can train and learn on each trajectory progressively. As Fig. 3 shows, the model takes 𝑛 points from the missing
+points queue in each imputation cycle. In a decoder, the embeddings of 𝑛 missing points are weighted based on observed
+trajectory embedding produced by an encoder. Then, the impute layer outputs the locations of 𝑛 missing points. The
+imputation cycle will be repeated until all the missing points have been learned.
+4.2
+Prediction Component
+The details of the prediction component are provided in this section. At the upper part of the framework in Fig. 1, it is
+the Supplement Layer that fetches the embeddings of observed portion 𝑌𝑜𝑏𝑠 and missing values 𝑌𝑚𝑖𝑠 from the final
+layer of encoder and decoder, respectively. The Supplement layer is responsible for reconstructing a new trajectory
+sequence by replenishing the trajectory representation with the embeddings of the missing points after the encoding
+stage. Subsequently, an RNN-based unit is built upon this module to capture sequential dependency. The formula of a
+Manuscript submitted to ACM
+
+10
+Qin et al.
+P1
+P2
+P3
+P4
+P5
+P6
+P7
+P8
+P2
+P3
+P5
+P7
+Encoder
+Decoder
+Linear &
+Time Enc
+P2
+Impute
+n = 1
+P2
+P3
+P5
+P7
+Encoder
+Decoder
+Linear &
+Time Enc
+P3
+Impute
+...
+P1
+P2
+P3
+P4
+P5
+P6
+P7
+P8
+P2
+P3
+P5
+P7
+Encoder
+Decoder
+Linear &
+Time Enc
+P2
+Impute
+n = 2
+P2
+P3
+P5
+P7
+Encoder
+Decoder
+Linear &
+Time Enc
+P5
+Impute
+P3
+P7
+Missing Points Queue
+Traj Queue
+Fig. 3. The Imputation Cycle enables the model to conduct gradual imputation on multiple levels. In each imputation cycle, 𝑛 missing
+points will be taken into the model for learning with the relevant trajectory information. The diagram shows the imputation process
+when 𝑛 equals 1 or 2.
+supplement operation is given below:
+𝑌𝑠𝑒𝑞(𝑝𝑙) =
+
+
+𝑌𝑚𝑖𝑠 (𝑚𝑖),
+if 𝑝𝑙 = 𝑚𝑖
+𝑌𝑜𝑏𝑠 (𝑜𝑖),
+if 𝑝𝑙 = 𝑜𝑖
+(5)
+where 𝑌𝑠𝑒𝑞 is the embedding matrix of the whole trajectory, 𝑝𝑙 is a point at step 𝑙 in the trajectory, 𝑚𝑖 is a point in
+the missing queue, and 𝑜𝑖 denotes a point in the observation queue. We interpolate the embedding of a point from
+matrix 𝑌𝑚𝑖𝑠 if it is missing; otherwise, the corresponding embedding is extracted from matrix 𝑌𝑜𝑏𝑠. In addition to using
+a ’Replace operation’, we can alternatively consider an ’Add operation’ in the Supplement layer to add the embedding
+of missing values to the corresponding positions in the trajectory representation.
+In the prediction stage, GRU [5] is selected as the recurrent unit because of its efficient computation without
+performance deterioration. This layer is implanted by receiving the embeddings of the newly reconstructed sequence
+from the supplement layer and produces relevant hidden states. The updated formulations of GRU are as follows:
+𝑓𝑡 = 𝜎(𝑊𝑓 𝑥𝑥𝑡 +𝑊𝑓 ℎℎ𝑡−1 + 𝑏𝑓 ),
+(6)
+𝑟𝑡 = 𝜎(𝑊𝑟𝑥𝑥𝑡 +𝑊𝑟ℎℎ𝑡−1 + 𝑏𝑟),
+(7)
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+11
+Algorithm 1 Overview of INGRAIN Training
+Input:
+A set of user trajectory 𝑆
+′
+𝑢 = {𝑡𝑟1𝑢,𝑡𝑟2𝑢, ...𝑡𝑟𝑊
+𝑢 };
+One masking vector 𝑀 = [𝑚1,𝑚2, ...,𝑚𝐿];
+Epochs of training 𝑒𝑝𝑜;
+Number of points for each imputation cycle, 𝑛 < 𝐼.
+1: while training epoch < 𝑒𝑝𝑜 do
+2:
+for each trajectory 𝑡𝑟𝑤
+𝑢 ∈ 𝑆
+′
+𝑢 do
+3:
+Apply mask 𝑀 on 𝑡𝑟𝑤
+𝑢 ;
+4:
+Initialize representations of observed trajectory, 𝐸𝑜𝑏𝑠;
+5:
+Encode time information with 𝐸𝑜𝑏𝑠;
+6:
+while has missing values do
+7:
+Initialize representations of 𝑛 missing points, 𝐸𝑚𝑖𝑠;
+8:
+Apply time encoding on 𝐸𝑚𝑖𝑠;
+9:
+Compute attention for observations, 𝐸
+′
+𝑜𝑏𝑠;
+10:
+Compute attention of 𝑛 missing values in observations, 𝐸′𝑚𝑖𝑠;
+11:
+Reconstruct trajectory based on 𝐸
+′
+𝑜𝑏𝑠 and 𝐸′𝑚𝑖𝑠;
+12:
+Conduct imputation and prediction;
+13:
+Optimisation with fused objective function L𝑙𝑒𝑎𝑟𝑛.
+14:
+end while
+15:
+end for
+16: end while
+𝑐𝑡 = 𝑡𝑎𝑛ℎ(𝑊𝑐𝑥𝑥𝑡 + 𝑟𝑡 ◦𝑊𝑐ℎℎ𝑡−1 + 𝑏𝑐),
+(8)
+ℎ𝑡 = (1 − 𝑓𝑡) ◦ 𝑐𝑡 + 𝑓𝑡 ◦ ℎ𝑡−1,
+(9)
+where 𝑥𝑡 is the embedding of the point in time 𝑡, ℎ𝑡−1 is the output of the last unit, 𝑊 is the weight matrix, 𝑏 is the
+bias vector, ◦ means element-wise multiplication, 𝑓𝑡 is the update gate, 𝑟𝑡 is the reset gate, 𝑐𝑡 is the candidate, and ℎ𝑡
+is the output. Noticeably, the model flexibly subjoins prediction training in each imputation cycle. In other words, it
+can recurrently learn and make forward predictions based on the latest status of the sequence restored by previous
+imputation recurrences. In the testing step, we need to detach the forward prediction from the imputing cycles and
+solely execute it when the imputation of the whole trajectory is completed.
+4.3
+Collaborative Learning Objective
+As we mentioned, there are two main components established in the framework: imputation and prediction units.
+During the training stage of imputation, the purpose is to minimize the mean squared error between imputing values
+and ground truth with the following objective function:
+L𝑖𝑚𝑝 = E𝑋∼𝐶,�
+𝑋∼𝐺𝜃 (𝑋,𝑀)
+��𝐿
+𝑙=1
+��� ˆ𝑥𝑙 − 𝑥𝑙
+���
+2
+�
+,
+(10)
+where 𝐶 = {𝑋 ∗} is the set of original sequences, 𝑀 is a masking vector for missing values, 𝐺𝜃 (𝑋, 𝑀) denotes the
+function to infer missing values for imputation with parameter 𝜃, and ˆ𝑥 is one of the imputed values �𝑋. Subsequently,
+we generate embedding 𝑌 of the sequence from the previous component in the training process of prediction. 𝐽𝜔 (𝑌) is
+the predictor of the next movement with parameter 𝜔, and the objective function of the prediction task is constructed
+Manuscript submitted to ACM
+
+12
+Qin et al.
+as follows:
+L𝑝𝑟𝑒 = E𝑋∼𝐶,𝑌∼𝐺𝜃 (𝑋,𝑀), ˆ𝑦∼𝐽𝜔 (𝑌)
+����ˆ𝑦 − 𝑦
+���
+2
+�
+.
+(11)
+Furthermore, we introduce a third loss function L𝑣𝑒𝑙 to constraint the movement velocity between imputed and
+observed points and comply with the speed observed in trajectories:
+L𝑣𝑒𝑙 = E𝑋∼𝐶,�
+𝑋∼𝐺𝜃 (𝑋,𝑀),𝑣∼𝐻 (𝑋−�
+𝑋),ˆ𝑣∼𝐻 ( �
+𝑋)
+����ˆ𝑣 − 𝑣
+���
+2
+�
+,
+(12)
+where 𝐻 is the function to compute movement speed between each pair of points, 𝑣 is the observed speed in a trajectory,
+and ˆ𝑣 is the speed computed between imputed and observed points.
+To train the entire model, we fuse the optimization of the above objective functions in each execution as shown
+below:
+L𝑙𝑒𝑎𝑟𝑛 = 𝜆1L𝑖𝑚𝑝 + 𝜆2L𝑝𝑟𝑒 + 𝜆3L𝑣𝑒𝑙.
+(13)
+Here, 𝜆1, 𝜆2, and 𝜆3 are the hyperparameters representing the weights of different loss functions correspondingly in
+training. In this way, the model can benefit from optimizing different modules, and this collaborative learning could
+eventually give the model a latent boost of convergence.
+4.4
+Computational Complexity
+In this part, we take into account the main phases of the proposed framework for calculating computational complexity.
+The meanings of symbols used here are independent of the notations in the previous sections. The essential phases
+include initial representation generation, encoder and decoder attention, and RNN-based prediction module.
+The stage of initial representation generation is directly built on Multi-Layer Perceptron (MLP) [15], which approxi-
+mately has time complexity 𝑂(𝑙 ∗ 𝑛 ∗ 𝑑) for one layer of implementation. Here, 𝑙 is the trajectory length, 𝑛 is the input
+dimension (it is regularly a small value), and 𝑑 is the embedding dimension. In the imputation stage, the encoder and
+decoder mainly rely on the self-attention mechanism, which has 𝑂(𝑙2 ∗ 𝑑) [29]. On the other hand, the RNN-based
+module typically contributes to time complexity 𝑂(𝑙 ∗ 𝑑2). In most cases, the trajectory length 𝑙 is smaller than the
+embedding dimension 𝑑. However, self-attention is unnecessary to conduct sequential operations on all the points. It is
+able to consider neighbor locations of size 𝑟 in the input trajectory if it is particularly long [29]. So, the complexity could
+reduce to 𝑂(𝑙 ∗ 𝑟 ∗ 𝑑), which will be considered in future work. Overall, the 𝑂(𝑚𝑎𝑥{𝑙 ∗ 𝑛 ∗ 𝑑,𝑙2 ∗ 𝑑,𝑙 ∗ 𝑑2}) is the total
+complexity of the framework. In the experiments, 𝑛 is two for the input size, and 𝑙 and 𝑑 are configured correspondingly
+to different values.
+5
+EXPERIMENTAL RESULTS
+This section compares the performance of trajectory imputation and prediction for the proposed model and the baselines
+on different human mobility datasets. Then, the primary hyperparameters of the proposed model are assessed intensively.
+A ablation study is provided at the end of the section.
+5.1
+Experimental Setup
+5.1.1
+Datasets. We use real-world human mobility datasets from three different cities worldwide for the experiments.
+The datasets record a broad range of users’ movements among city areas:
+• Geolife Data [37] contains outdoor GPS trajectories of 182 users from April 2007 to August 2012 in Beijing,
+China. The sampling rates vary in trajectories: approximately 91% are logged every 1 to 5 seconds or every 5 to
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+13
+Table 2. The total number of trajectories used in each dataset and 𝐿 denotes the length of each trajectory.
+Geolife
+Cuebiq-AU
+Cuebiq-US
+L = 20
+L = 50
+L = 100
+L = 20
+L = 50
+L = 100
+L = 20
+L = 50
+L = 100
+20,435
+20,435
+20,301
+30,030
+30,030
+30,030
+30,030
+30,030
+30,030
+10 meters per point. Each record contains timestamp, user ID, latitude and longitude. We selected 30 users with
+the most GPS records in January and February 2009 for evaluation.
+• Cuebiq-US Data [14] contains more diverse human movements on a daily basis. The data collection period
+ranged from January 2018 to June 2018, and the location was New York, USA. Sampling frequencies of approxi-
+mately 91% of data range from 1 to 600 seconds per record, and each record has timestamp, device ID, latitude, and
+longitude. Cuebiq’s anonymized and privacy-enhanced data is collected from users who opted for anonymous
+data sharing for research purposes through a GDPR-compliant framework. The trajectories of the 30 most active
+users in May 2018 are extracted for experiments.
+• Cuebiq-AU Data [14] has the same data format as Cuebiq-US. It was collected over two years, from December
+2017 to November 2019, in cities in Australia. The sampling rate of the collected data is similar to that of
+Cuebiq-US for most records. Likewise, The trajectories of 30 users who were most active in October 2019 are
+used for testing.
+Based on the original movement sequences of the users in each dataset, we produce a set of sub-trajectories of the
+users with a defined length (see Section 3). Table 2 gives the total number of trajectories used in each dataset by length.
+Then, the extracted trajectories are randomly separated into a training part (80%) and a testing part (20%). Additionally,
+a varied size of masking vector is randomly created to imitate different degrees of missing values in sub-trajectories for
+imputation. The default probability of generating missing values follows a discrete uniform distribution. Finally, the
+model requires predicting the location of the next movement when an observed trajectory with missing points is given.
+5.1.2
+Metrics. In this paper, the losses of L𝑖𝑚𝑝 and L𝑝𝑟𝑒 are basically average euclidean distances between 2D points.
+In imputation, we infer the coordinates of missing points in trajectories and calculate the average 𝐿2 loss between
+imputed values and ground truth among all users. Moreover, we assess the proposed approach for the prediction
+task, which is to forecast the coordinates of the next location for users if a historical trajectory with missing points is
+given. The average 𝐿2 loss between the predicted values and ground truth is used for the evaluation. In our model, the
+predicted values will be evaluated merely after a trajectory’s imputation is completed during the testing phase.
+5.1.3
+Baselines. For trajectory imputation, two state-of-the-art methods for comparison are NAOMI [18] and Sin-
+gleRes [18]. NAOMI is one of the latest non-autoregressive approaches for sequence imputation. In contrast, SingleRes
+is the autoregressive counterpart, and it can be reduced to BRITS [3] if the adversarial training is discarded. GRUI
+[20] is also an autoregressive model with GAN for time series imputation, which is used to handle the completion of a
+trajectory sequence. Moreover, GAIN [34] is another recent method to impute missing data using GANs. We also test the
+imputation task with a classical approach named KNN + Linear [12], which searches K nearest neighbors from samples
+and then applies Linear regression to impute missing points based on those neighbors. Here, we also implemented
+this approach by inferring a defined number of missing points in a trajectory in different degrees. Simultaneously,
+several RNN variations are used for comparison with the proposed model for prediction purpose, such as stacked LSTM
+Manuscript submitted to ACM
+
+14
+Qin et al.
+Table 3. The results show the L2 loss of imputation and prediction on three different datasets. The percentage of missing points in
+each trajectory is 0.8 for all the tests, and 𝐿 denotes the length of trajectories in different trials.
+L2 Loss for Imputation
+Methods
+Geolife
+Cuebiq - AU
+Cuebiq - US
+L = 20
+L = 50
+L = 100
+L = 20
+L = 50
+L = 100
+L = 20
+L = 50
+L = 100
+KNN + Linear [12]
+4.7095
+4.7014
+8.7260
+1.6466
+1.9836
+1.9567
+0.0238
+0.0254
+0.0323
+GAIN [34]
+2.1597
+2.1903
+2.9478
+0.8619
+0.8842
+1.0788
+0.0125
+0.0126
+0.0137
+GRUI [20]
+0.3884
+0.2584
+0.2443
+0.3150
+0.2062
+0.2088
+0.8407
+0.2871
+0.2189
+NAOMI [18]
+0.0498
+0.1613
+0.0598
+0.2007
+0.0186
+0.0524
+0.0122
+0.0129
+0.0127
+SingleRes [18]
+0.4365
+0.0405
+0.0171
+0.0716
+0.0190
+0.0622
+0.0121
+0.0128
+0.0126
+INGRAIN (ours)
+0.0270
+0.0075
+0.0062
+0.0122
+0.0116
+0.0117
+0.0055
+0.0050
+0.0046
+L2 Loss for Prediction
+B-LSTM [10]
+2.1856
+2.0427
+2.4948
+0.8294
+0.9935
+1.0853
+0.0120
+0.0127
+0.0126
+RNNSearch [2]
+2.2282
+2.0754
+2.5726
+0.8809
+1.0151
+1.0941
+0.0962
+0.0187
+0.0176
+S-GRU [5]
+2.2309
+2.0437
+2.5305
+0.8276
+0.9919
+1.0895
+0.0121
+0.0129
+0.0126
+S-LSTM [27]
+2.2081
+2.0656
+2.5289
+0.8257
+0.9947
+1.0846
+0.0120
+0.0128
+0.0125
+INGRAIN (ours)
+0.0525
+0.0464
+0.0751
+1.0496
+1.0139
+0.9194
+0.0073
+0.0070
+0.0071
+(S-LSTM) [27], bidirectional LSTM (B-LSTM) [10] and stacked GRU (S-GRU) [5]. Another sequence forecasting method
+is RNNSearch [2], which implements the attention mechanism based on RNN to selectively retrieve information from
+the encoder to the decoder for prediction.
+5.1.4
+Implementation. The proposed method (INGRAIN) is implemented with Pytorch, and the Adam algorithm is
+used as the optimizer with a learning rate of 0.001 and batch size of 70. In the imputation component, we use two layers
+of either encoders or decoders. The number of heads (self-attention) in each layer is two, and the dimension of learning
+embedding is 256. In the prediction part, 1-layer GRU is adopted with a hidden size of 256. For training in different
+settings, the number of epochs is 60, and we compute the mean of the best test results for each task in five runs.
+For other imputation methods, NAOMI and SingleRes [18] use the same values of some basic parameters as our model:
+learning rate 0.001, batch size 70, and training epochs 60. The other parameters are default in their implementation.
+GRUI [20], and GAIN [34] are faster for training but harder to achieve convergence. Furthermore, we tried a different
+number of iterations for training to obtain optimal results on different datasets, ranging from 50 to 1000. The baselines
+are all RNN based for the prediction methods, and we can directly compare them with our prediction component by
+applying similar parameters for training, such as learning rate, batch size, training epochs, or size of hidden features.
+5.2
+Performance Analysis
+The evaluations are conducted on Geolife, Cuebiq-AU, and Cuebiq-US, and the sampling rates of points collection vary
+drastically. This section selected the top 20 users of each dataset who were most active within the observed period
+for the learning task evaluation, sensitivity analysis, and ablation study. In addition, to further assess the model’s
+effectiveness, we generate three different groups of users, and each group contains ten persons randomly picked from
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+15
+(a) Geolife - Imputation
+(b) Cuebiq-AU - Imputation
+(c) Cuebiq-US - Imputation
+(d) Geolife - Prediction
+(e) Cuebiq-AU - Prediction
+(f) Cuebiq-US - Prediction
+Fig. 4. We use the 13 most active users of Geolife and the 20 most active users of Cuebiq-AU and Cuebiq-US for this experiment, and
+the length of a trajectory is 20. (a), (b) and (c) demonstrate the imputation loss of the proposed model and baselines on different
+datasets, with varying percentages of missing points in trajectories. (d), (e) and (f) show the results for prediction loss on three
+datasets, respectively.
+the 30 users in each dataset. The data of groups were tested directly on two learning tasks, and the results are shown in
+Fig. 5.
+5.2.1
+Imputation Results. We first evaluate imputation performance for different lengths of trajectories with a certain
+degree of missing values. Table 3 displays the experimental results on three different lengths (20, 50 and 100) of
+trajectories across all the datasets. The missing rate of points is 0.8. The parameters 𝜆1 and 𝜆2 are configured to one for
+our model, which means that the model fully considers the feedback from different components for training. 𝜆3 is not
+considered in this test. Overall, the proposed model has the best imputation performance on these datasets regarding
+average L2 loss. It is also noticeable that the INGRAIN can keep contributing to a minor loss of imputation when the
+length of tested trajectories is increased, whereas no such obvious advantage can be found for the baselines. One reason
+for this could be that longer trajectories contain more sample fragments, and the model can effectively utilize this
+increment of data to infer missing values. Another argument is that a good combination of attention-based imputation
+and prediction components can better enable INGRAIN to overcome imputation in longer trajectories. The proposed
+imputation component is built with an attention mechanism that learns embeddings by weighting the relations between
+the points in trajectories. And the other two baselines rely on learning embeddings in sequential dependencies of the
+trajectories, which could be affected more heavily when more random points are missing.
+Manuscript submitted to ACM
+
+++16
+Qin et al.
+(a) Geolife - Imputation
+(b) Cuebiq-AU - Imputation
+(c) Cuebiq-US - Imputation
+(d) Geolife - Prediction
+(e) Cuebiq-AU - Prediction
+(f) Cuebiq-US - Prediction
+Fig. 5. The experiment was run with three groups of 10 users randomly picked from the 30 most active ones in each dataset. The
+length of a trajectory is 20 in the tests. (a), (b) and (c) demonstrate the imputation loss of the proposed model and baselines on
+different datasets, with varying percentages of missing points in trajectories. (d), (e) and (f) show the results for prediction loss on
+three datasets, respectively.
+Next, we assess the impact of different missing rates of points on the task of imputation by the algorithms. Fig.
+4a and 4b demonstrate the stability of our method in solving imputation on both Geolife and Cuebiq-AU when the
+percentage of missing points varies from 0.2 to 0.8 with a trajectory length of 20. As the missing rate rises, the INGRAIN
+can maintain the loss at a relatively low position while the loss of either NAOMI or SingleRes fluctuates dramatically
+and tends to rise or stay between the missing rate from 0.5 to 0.8. However, the SingleRes performs better on Geolife
+when the rate is smaller than 0.5. As we can see from Fig. 4c, the baselines become steadier on the dataset Cuebiq-US,
+but have (approximately two times) higher loss of imputation than that of our model. In addition, in other tests with
+different groups of random users, Fig. 5 further demonstrates the model’s prominent ability on trajectory imputation
+in terms of accurate estimation and stability. We claim that learning point embeddings based on the mode of a fully
+connected graph (attention mechanism) could better capture the dependencies between missing points and the observed
+trajectories for solving the imputation of daily human mobility in the city regions.
+5.2.2
+Prediction Results. Forward prediction is conducted along with imputation by the proposed model. The results
+of the proposed model again show its advantages in predicting future values after the imputation of the trajectory is
+processed. Table 3 reveals that the INGRAIN is superior to all the other RNN-based baselines (S-GRU, S-LSTM, B-LSTM,
+and RNNSearch) on both Geolife and Cuebiq-US datasets with missing rates of 0.8. For the dataset Cuebiq-AU, the
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+17
+INGRAIN tends to improve in longer trajectories (𝐿=100), although it is worse in shorter ones (𝐿=20 or 50) compared
+with its counterparts. Overall, the results of the baselines deteriorated slightly while the length of trajectories was
+prolonged, with the same degree of missing rate. Our model shows a more reliable capability to overcome the impact of
+missing values in longer trajectories for next-location forecasting.
+Moreover, Fig. 4d, 4e and 4f display the results of prediction with missing rates from 0.2 to 0.8 on three different
+datasets, respectively. And the length of all the trajectories in this test is 20. The performance of the RNN-based baselines
+is stable among the three datasets but weaker in regard to the ability to converge at a smaller loss. In contrast, the
+figure of INGRAIN fluctuates on the Cuebiq-AU but can keep prediction loss at significantly lower values on the other
+two datasets. Fig. 5e and 5f also show the advantages of the proposed model for prediction tasks with different groups
+of random users. In general, the previous results indicate that effectively incorporating the imputation component
+with the prediction unit in INGRAIN would eventually benefit both learning tasks and outperform the counterparts of
+baselines. We claim that the proposed model conducts the prediction based on the status or effect of imputation on
+trajectories, which could potentially enhance the performance.
+5.2.3
+Sensitivity Analysis. In this section, we examine the effect of primary hyperparameters on the performance of
+INGRAIN for both tasks. That gives us more insights into how the proposed method converges in different configurations
+and what trade-offs can be made between two learning tasks. As this paper mentioned, the model is agile to infer a
+defined number of missing points in each imputing cycle. This process iterates until the whole imputation work of
+each trajectory is finished. Thus, we first check how this number potentially acts on the results of two learning tasks
+with the dataset Geolife. In Fig. 6a, the number of missing points from 1 to 5 per imputing cycle is inspected for the
+best average L2 loss of imputation and prediction in each test simultaneously. There is an increase in the loss for both
+criteria when more points are imputed in each cycle. The loss of imputation and prediction is relatively minor when
+the number is one. Although the prediction loss arrives at its lowest position when the number is two, the imputation
+loss value soars. Therefore, using fewer missing points in each imputation operation can offer better learning results
+for both tasks. It is well known that the learning rate is a fundamental factor that affects the convergence of a deep
+learning model. Fig. 6b illustrates that the performance of two tasks becomes gradually worse when we increase the
+value of the learning rate from 0.001 (10−1) to 0.15 (10−1). As we can see, learning rates 0.001 (10−1) and 0.01 (10−1)
+allow the model to produce better results for both imputation and prediction.
+In the assessment of using a different window length for constructing trajectories, Fig. 6c shows that imputation loss
+tends to become smaller in the learning with longer trajectories, and prediction loss fluctuates slightly between around
+0.02 to 0.07. We see that the performance of both imputation and prediction work is relatively advantageous when the
+length is 70. Fig. 6d indicates that using two heads of self-attention can simultaneously perform better for both learning
+tasks. Furthermore, the computational requirement becomes relatively less than applying a bigger number of heads. In
+addition, the evaluation of embedding and hidden size used in the model demonstrates that a smaller value can already
+contribute to a good performance of two tasks, such as 128 or 256. As Fig. 6e and Fig. 6f show, more computation with
+an increased value of such parameters did not provide better results.
+In the following, we examine the functions of imputation and prediction components by changing the values of 𝜆1
+and 𝜆2 without considering the constraint of movement velocity (𝜆3 = 0). As we mentioned in Section 4.3, 𝜆1 and 𝜆2 are
+two hyperparameters that control the feedback of imputation and prediction unit received by the model during training,
+respectively. In Fig. 6g and 6h, higher values indicate that more strength is considered for the corresponding component
+during entire training. We see that the loss of imputation reduces slightly in Fig. 6g when bigger 𝜆1 is configured.
+Manuscript submitted to ACM
+
+18
+Qin et al.
+(a) # of Points per Imp-cycle
+(b) Change of Learning Rate
+(c) Window Length
+(d) Head Number
+(e) Embedding Size
+(f) Hidden Size
+(g) Evaluation of 𝜆1
+(h) Evaluation of 𝜆2
+(i) Evaluation of 𝜆3
+Fig. 6. We evaluate the main hyperparameters of the proposed model for both imputation and prediction on Geolife. (a) shows the
+results for applying different numbers of missing points in each imputing cycle, and (b) illustrates the results with the change in
+learning rate. (c) and (d) show the evaluation of the model for window length and head number, respectively. The figures related
+to embedding and hidden size are given in (e) and (f). In addition, (g), (h) and (i) are the results of investigation for 𝜆1, 𝜆2 and 𝜆3,
+respectively.
+Meanwhile, the prediction loss simultaneously undergoes a reversed change (rise). However, the combination 𝜆1 = 1
+and 𝜆2 = 1 could offer a better trade-off for the performance of both tasks. Similarly, in a different run, the loss of
+prediction decreases drastically while the value of 𝜆2 is raised from 0.2 to 1 along with a fixed 𝜆1 in Fig. 6h, and both
+tasks are beneficial when 𝜆2 is round 0.3. In addition, Fig. 6i presents the results of varied 𝜆3 when 𝜆1 = 1 and 𝜆2 = 1. It
+is apparent that the performance of both task benefit at most when 𝜆3 = 0.8.
+In Fig. 7, we compare the performance of the proposed model and the baselines with different distributions of missing
+values generation, such as Uniform and Poisson distribution. Fig. 7a, 7b and 7c demonstrate the imputation loss of the
+proposed model and baselines on different datasets, respectively. Fig. 7d, 7e and 7f are the results for the prediction
+Manuscript submitted to ACM
+
+O-Loss-lmp
+Loss-Pred
+0.038
+0.3
+0.036
+0.25
+Loss
+LOSS
+0.034
+0.2
+Imputation
+0.032
+0.15
+0.03
+0.1
+0.028
+0.05
+0.026
+0
+2
+cCi
+4
+5
+Number of PointsO-Loss-Imp
+Loss-Pred
+0.8
+0.6
+1.5
+Imputation Loss
+LOsS
+0.4
+0.2
+0.5
+0.001
+0.005
+0.01
+0.05
+0.1
+Learning Rate (10-1)O- Loss-Imp
+Loss-Pred
+0.04
+0.08
+0.03
+0.06
+Loss
+Loss
+Imputation
+Prediction 
+0.02
+0.04
+0.01
+0.02
+0
+20
+35
+50
+70
+100
+Window LengthO- Loss-lmp
+Loss-Pred
+0.03
+0.07
+0.027
+0.06
+Loss
+LOss
+Imputation L
+0.024
+0.05
+iction
+0.021
+0.04
+redi
+P
+0.018
+0.03
+0.015
+0.02
+2
+4
+8
+16
+32
+Number of HeadsO-Loss-Imp
+Loss-Pred
+0.05
+1.5
+0.04
+1.2
+Loss
+LOSS
+Imputationl
+0.03
+0.9
+0.02
+0.6
+0.01
+0.3
+0
+128
+256
+512
+768
+1024
+Embedding SizeO-Loss-lmp
+Loss-Pred
+0.05
+0.8
+0.04
+0.6
+Imputation Loss
+Prediction Loss
+0.03
+0.4
+0.02
+0.2
+0.01
+0
+128
+256
+512
+768
+1024
+Hidden SizeO-Loss-Imp
+Loss-Pred
+0.036
+0.15
+0.034
+0.13
+Loss
+Loss 
+Imputation L
+0.032
+0.11
+0.03
+0.09
+0.028
+0.07
+0.026
+0.05
+0.2
+0.4
+0.6
+0.8
+入1 (入2= 1)O- Loss-lmp
+Loss-Pred
+0.032
+1.2
+0.03
+1
+Loss
+LOss
+0.028
+0.8
+Imputation I
+Prediction 
+0.026
+0.6
+0.024
+0.4
+0.022
+0.2
+0.02
+0.2
+0.4
+0.6
+0.8
+入2 (入1= 1)O- Loss-Imp
+Loss-Pred
+0.01
+0.1
+0.0098
+0.08
+Loss
+Loss
+Imputation l
+Prediction 
+0.0096
+0.06
+0.0094
+0.04
+0.0092
+0.02
+0.2
+0.4
+0.6
+0.8
+1
+入3 (入1=入2= 1)Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+19
+(a) Geolife - Imputation
+(b) Cuebiq-AU - Imputation
+(c) Cuebiq-US - Imputation
+(d) Geolife - Prediction
+(e) Cuebiq-AU - Prediction
+(f) Cuebiq-US - Prediction
+Fig. 7. This experiment was run with a group of 10 users randomly picked from the 30 most active ones in each dataset, and the
+trajectory length is 20. (a), (b) and (c) demonstrate the imputation loss of the proposed model and baselines on three datasets,
+with different distributions of missing values generation. (d), (e) and (f) show the results for the prediction task on three datasets,
+respectively.
+task on three datasets, respectively. As we can see, when using various distributions for the experiment, the proposed
+approach exhibits significant advantages on the imputation task. Meanwhile, our model can still keep a competitive
+performance on the prediction task compared with the other algorithms.
+5.2.4
+Ablation Study. We conduct an ablation study of 𝜆1, 𝜆2, and 𝜆3 to check the model’s performance when feedback
+of the imputation component, prediction component, or speed constraint is totally discarded during optimization. The
+proposed model will fully consider the feedback of imputation in the optimization process when 𝜆1 = 1 and 𝜆2 = 1
+indicates that the optimizer will receive the feedback of the prediction component without any abandon. In contrast,
+zero value means that the feedback from one component is totally left out during training. It is found that when the
+feedback from one task is totally discarded during the training of the whole model, the outputs of that task will be
+meaningless and random floats because no specific optimization arises based on the ground truth. Thus, we omit those
+relevant results on the Table 4. We temporarily neglect the movement speed constraint by setting 𝜆3 to zero in the
+beginning. Furthermore, the table illustrates that the setting of 𝜆1 = 1 and 𝜆2 = 0 could achieve better imputation
+results in some cases, accompanied by the sacrifice of optimization for the prediction. However, considering the full
+feedback of both components (𝜆1 = 1, 𝜆2 = 1) could also provide competitive prediction performance in most cases
+(the length of trajectory in 20 or 50). This experiment also indicates that switching one of two main components could
+give the model the flexibility to concentrate better on optimizing a single task. In addition, adding movement speed
+Manuscript submitted to ACM
+
+GRA
+NAO
+onolekGRA
+AO
+SngleRIGRA
+NAO
+onoerNGRA
+-GFGRA
+S-GRIGRA
+-GF20
+Qin et al.
+Table 4. The model will not consider the feedback from the imputation unit during training when 𝜆1=0. Otherwise, 𝜆1=1. 𝜆2 and 𝜆3
+are responsible for controlling the feedback from the prediction unit and the weight of speed constraint, respectively. Loss-I and
+Loss-P represent the L2 loss of imputation and prediction, respectively. Further, the portion of missing points is 0.8 for all the tests,
+and L denotes the length of trajectories in different trials. Basically, the learning of a task is infeasible if its designated optimization is
+totally ignored. However, another task has the chance of getting a slight improvement than considering more optimization units in
+the same iterations of training.
+Weights
+Tasks
+Geolife
+Cuebiq - AU
+Cuebiq - US
+𝜆1
+𝜆2
+𝜆3
+L = 20
+L = 50
+L = 20
+L = 50
+L = 20
+L = 50
+1
+1
+0
+Loss-I
+0.0270
+0.0075
+0.0122
+0.0117
+0.0055
+0.0050
+Loss-P
+0.0525
+0.0464
+1.0497
+1.0140
+0.0073
+0.0070
+1
+0
+0
+Loss-I
+0.0275
+0.0098
+0.0113
+0.0109
+0.0054
+0.0055
+Loss-P
+∼
+∼
+∼
+∼
+∼
+∼
+0
+1
+0
+Loss-I
+∼
+∼
+∼
+∼
+∼
+∼
+Loss-P
+0.0959
+0.0393
+0.8611
+1.1230
+0.0114
+0.0082
+1
+1
+1
+Loss-I
+0.0107
+0.0062
+0.0120
+0.0110
+0.0055
+0.0048
+Loss-P
+0.0648
+0.0499
+0.9350
+1.2609
+0.0070
+0.0076
+Table 5. The RNN-based unit and the Supplement layer are two modules that support the imputation and prediction learning
+process. The table shows the impact of these two modules on the model’s performance. ’Add operation’ or ’Replace operation’ is used
+individually in the Supplement layer with or without the RNN unit. The tests were conducted on Geolife with a trajectory length of
+20.
+Components
+Geolife
+RNN Unit
+Add Operation
+Replace Operation
+Loss-I
+Loss-P
+-
+-
+-
+0.0105
+0.0143
+✓
+-
+-
+0.0097
+0.0480
+-
+-
+✓
+0.0159
+0.0695
+✓
+✓
+-
+0.0094
+0.0732
+✓
+-
+✓
+0.0090
+0.0886
+constraint (𝜆3 = 1) could lead to a slight improvement of imputation on Geolife, which has a more stable sampling rate
+of points collection than the other two datasets.
+As mentioned before, the RNN-based unit and the Supplement layer are two modules that support the imputation and
+prediction learning process. An evaluation is given of their effects on the overall performance of the proposed model in
+Table 5. ’Add operation’ or ’Replace operation’ is used individually in the Supplement layer to incorporate missing
+values’ embedding into the trajectory representation. Five tests were conducted for each combination on Geolife with a
+trajectory length of 20, and the mean values were reported. We can see that combining a supplement operation and
+an RNN unit tends to contribute to a minor loss of imputation. In contrast, the prediction can not benefit too much
+from the joint use of two components. However, we can find that integration of RNN unit for learning can improve
+imputation performance to some extent.
+Manuscript submitted to ACM
+
+Multiple-level Point Embedding for Solving Human Trajectory Imputation with Prediction
+21
+6
+CONCLUSION
+Usually, human mobility data are incomplete in practice, leading to bias or difficulties in learning tasks, such as imputation
+and prediction. We propose a new approach incorporating non-autoregressive and autoregressive components to help
+trajectory imputation and prediction. The model effectively learns the dependence between observations and missing
+values on multiple levels with the advantage of self-attention. Meanwhile, one RNN-based unit is applied to extract
+potential features recurrently from the newly learned sequences. Intensive experiments are conducted on three datasets:
+Geolife, Cuebiq-AU, and Cuebiq-US. The results show that the proposed model can achieve advanced performance
+in both learning tasks compared to the baselines. Additionally, the analysis of primary hyperparameters reveals how
+trade-offs could be made between different tasks with proper settings. Moreover, the flexible configuration of switching
+the acceptance of additional feedback enables us to pay more attention to individual units to attain better results for a
+specific task. In the future, we plan to conduct more experiments on more diverse types of mobility datasets (e.g., POIs
+and grid-based datasets) and analyze the potential factors that crucially influence the learning of different imputation
+algorithms.
+REFERENCES
+[1] Alexandre Alahi, Kratarth Goel, Vignesh Ramanathan, Alexandre Robicquet, Li Fei-Fei, and Silvio Savarese. 2016. Social lstm: Human trajectory
+prediction in crowded spaces. In IEEE conference on computer vision and pattern recognition. 961–971.
+[2] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 2014. Neural machine translation by jointly learning to align and translate. arXiv preprint
+arXiv:1409.0473 (2014).
+[3] Wei Cao, Dong Wang, Jian Li, Hao Zhou, Lei Li, and Yitan Li. 2018. Brits: Bidirectional recurrent imputation for time series. In Advances in Neural
+Information Processing Systems. 6775–6785.
+[4] Cynthia Chen, Jingtao Ma, Yusak Susilo, Yu Liu, and Menglin Wang. 2016. The promises of big data and small data for travel behavior (aka human
+mobility) analysis. Transportation research part C: emerging technologies 68 (2016), 285–299.
+[5] Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Yoshua Bengio. 2014. Empirical evaluation of gated recurrent neural networks on sequence
+modeling. arXiv preprint arXiv:1412.3555 (2014).
+[6] William Fedus, Ian Goodfellow, and Andrew M Dai. 2018. MaskGAN: Better text generation via filling in the_. arXiv preprint arXiv:1801.07736
+(2018).
+[7] Jie Feng, Yong Li, Chao Zhang, Funing Sun, Fanchao Meng, Ang Guo, and Depeng Jin. 2018. Deepmove: Predicting human mobility with attentional
+recurrent networks. In 2018 world wide web conference. 1459–1468.
+[8] Győző Gidófalvi and Fang Dong. 2012. When and where next: individual mobility prediction. In SIGSPATIAL. 57–64.
+[9] Francesco Giuliari, Irtiza Hasan, Marco Cristani, and Fabio Galasso. 2020. Transformer Networks for Trajectory Forecasting. arXiv preprint
+arXiv:2003.08111 (2020).
+[10] Alex Graves, Santiago Fernández, and Jürgen Schmidhuber. 2005. Bidirectional LSTM networks for improved phoneme classification and recognition.
+In International conference on artificial neural networks. Springer, 799–804.
+[11] Agrim Gupta, Justin Johnson, Li Fei-Fei, Silvio Savarese, and Alexandre Alahi. 2018. Social gan: Socially acceptable trajectories with generative
+adversarial networks. In IEEE Conference on Computer Vision and Pattern Recognition. 2255–2264.
+[12] Trevor Hastie, Robert Tibshirani, and Jerome Friedman. 2009. The elements of statistical learning: data mining, inference, and prediction. Springer
+Science & Business Media.
+[13] Sepp Hochreiter and Jürgen Schmidhuber. 1997. Long short-term memory. Neural computation 9, 8 (1997), 1735–1780.
+[14] Cuebiq Inc. 2021. Data for Good - Cuebiq. https://www.cuebiq.com/about/data-for-good/
+[15] Tarun Khanna. 1990. Foundations of neural networks. Addison-Wesley Longman Publishing Co., Inc.
+[16] Yang Li, Yangyan Li, Dimitrios Gunopulos, and Leonidas Guibas. 2016. Knowledge-based trajectory completion from sparse GPS samples. In
+Proceedings of the 24th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems. 1–10.
+[17] Biwei Liang, Tengjiao Wang, Shun Li, Wei Chen, Hongyan Li, and Kai Lei. 2016. Online learning for accurate real-time map matching. In PAKDD.
+Springer, 67–78.
+[18] Yukai Liu, Rose Yu, Stephan Zheng, Eric Zhan, and Yisong Yue. 2019. NAOMI: Non-autoregressive multiresolution sequence imputation. In Advances
+in Neural Information Processing Systems. 11238–11248.
+[19] Yin Lou, Chengyang Zhang, Yu Zheng, Xing Xie, Wei Wang, and Yan Huang. 2009. Map-matching for low-sampling-rate GPS trajectories. In
+Proceedings of the 17th ACM SIGSPATIAL international conference on advances in geographic information systems. 352–361.
+Manuscript submitted to ACM
+
+22
+Qin et al.
+[20] Yonghong Luo, Xiangrui Cai, Ying Zhang, Jun Xu, et al. 2018. Multivariate time series imputation with generative adversarial networks. In Advances
+in Neural Information Processing Systems. 1596–1607.
+[21] Yonghong Luo, Ying Zhang, Xiangrui Cai, and Xiaojie Yuan. 2019. E2GAN: End-to-End Generative Adversarial Network for Multivariate Time
+Series Imputation. In 28th International Joint Conference on Artificial Intelligence. AAAI Press, 3094–3100.
+[22] Anna Monreale, Fabio Pinelli, Roberto Trasarti, and Fosca Giannotti. 2009. Wherenext: a location predictor on trajectory pattern mining. In
+Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining. 637–646.
+[23] Elham Naghizade, Jeffrey Chan, Yongli Ren, and Martin Tomko. 2018. Contextual Location Imputation for Confined WiFi Trajectories. In Pacific-Asia
+Conference on Knowledge Discovery and Data Mining. Springer, 444–457.
+[24] Elham Naghizade, Lars Kulik, Egemen Tanin, and James Bailey. 2020. Privacy-and context-aware release of trajectory data. ACM Transactions on
+Spatial Algorithms and Systems (TSAS) 6, 1 (2020), 1–25.
+[25] Mengshi Qi, Jie Qin, Yu Wu, and Yi Yang. 2020. Imitative Non-Autoregressive Modeling for Trajectory Forecasting and Imputation. In IEEE/CVF.
+12736–12745.
+[26] Amin Sadri, Flora D Salim, Yongli Ren, Wei Shao, John C Krumm, and Cecilia Mascolo. 2018. What will you do for the rest of the day? an approach
+to continuous trajectory prediction. Proc. of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies 2, 4 (2018), 1–26.
+[27] Martin Sundermeyer, Ralf Schlüter, and Hermann Ney. 2012. LSTM neural networks for language modeling. In Thirteenth annual conference of the
+international speech communication association.
+[28] Douglas Do Couto Teixeira, Aline Carneiro Viana, Jussara M Almeida, and Mrio S Alvim. 2021. The impact of stationarity, regularity, and context
+on the predictability of individual human mobility. ACM Transactions on Spatial Algorithms and Systems 7, 4 (2021), 1–24.
+[29] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is
+all you need. In Advances in neural information processing systems. 5998–6008.
+[30] Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. 2017. Graph attention networks. arXiv
+preprint arXiv:1710.10903 (2017).
+[31] Hao Wang, Huawei Shen, Wentao Ouyang, and Xueqi Cheng. 2018. Exploiting POI-Specific Geographical Influence for Point-of-Interest Recommen-
+dation.. In IJCAI. 3877–3883.
+[32] Xianjing Wang, Flora D Salim, Yongli Ren, and Piotr Koniusz. 2020. Relation Embedding for Personalised Translation-Based POI Recommendation.
+In Pacific-Asia Conference on Knowledge Discovery and Data Mining. Springer, 53–64.
+[33] Yifang Yin, Rajiv Ratn Shah, Guanfeng Wang, and Roger Zimmermann. 2018. Feature-based map matching for low-sampling-rate GPS trajectories.
+ACM Transactions on Spatial Algorithms and Systems (TSAS) 4, 2 (2018), 1–24.
+[34] Jinsung Yoon, James Jordon, and Mihaela Schaar. 2018. Gain: Missing data imputation using generative adversarial nets. In International Conference
+on Machine Learning. PMLR, 5689–5698.
+[35] Jinsung Yoon, William R Zame, and Mihaela van der Schaar. 2018. Estimating missing data in temporal data streams using multi-directional recurrent
+neural networks. IEEE Transactions on Biomedical Engineering 66, 5 (2018), 1477–1490.
+[36] Kai Zheng, Yu Zheng, Xing Xie, and Xiaofang Zhou. 2012. Reducing uncertainty of low-sampling-rate trajectories. In 2012 IEEE 28th international
+conference on data engineering. IEEE, 1144–1155.
+[37] Yu Zheng, Lizhu Zhang, Xing Xie, and Wei-Ying Ma. 2009. Mining interesting locations and travel sequences from GPS trajectories. In 18th
+International conference on World Wide Web. 791–800.
+[38] Fan Zhou, Hantao Wu, Goce Trajcevski, Ashfaq Khokhar, and Kunpeng Zhang. 2020. Semi-supervised trajectory understanding with poi attention
+for end-to-end trip recommendation. ACM Transactions on Spatial Algorithms and Systems (TSAS) 6, 2 (2020), 1–25.
+Manuscript submitted to ACM
+

LdE3T4oBgHgl3EQfvwu2/content/tmp_files/2301.04697v1.pdf.txt

@@ -0,0 +1,2052 @@
+Prepared for submission to JHEP
+Regular Black Holes and Horizonless Ultra-Compact
+Objects in Lorentz-Violating Gravity
+Jacopo Mazza and Stefano Liberati
+SISSA, International School for Advanced Studies,
+via Bonomea 265, 34136 Trieste, Italy
+IFPU, Institute for Fundamental Physics of the Universe,
+via Beirut 2, 34014 Trieste, Italy
+INFN, Sezione di Trieste,
+via Valerio 2, 34127 Trieste, Italy
+E-mail: jmazza@sissa.it, liberati@sissa.it
+Abstract: There is growing evidence that Hoˇrava gravity may be a viable quantum
+theory of gravity. It is thus legitimate to expect that gravitational collapse in the full,
+non-projectable version of the theory should result in geometries that are free of space-
+time singularities. Previous analyses have shown that such geometries must belong to one
+of the following classes: simply connected regular black holes with inner horizons; non-
+connected black holes “hiding” a wormhole mouth (black bounces); simply connected or
+non-connected horizonless compact objects.
+Here, we consider a singular black hole in
+the low-energy limit of non-projectable Hoˇrava gravity, i.e. khronometric theory, and de-
+scribe examples of its possible “regularisations”, covering all of the viable classes. To our
+knowledge, these examples constitute the first instances of black holes with inner universal
+horizons, of black bounces and of stars with a de Sitter core in the context of Lorentz-
+violating theories of gravity.
+Keywords: Regular black hole, wormhole, Hoˇrava gravity, khronometric theory
+arXiv:2301.04697v1  [gr-qc]  11 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+A (singular) black hole solution
+4
+3
+Regularisations of the singularity
+7
+3.1
+Connected regularisation
+7
+3.2
+Non-connected regularisation
+8
+4
+Horizons
+10
+4.1
+Connected regularisation
+10
+4.2
+Non-connected regularisation
+12
+5
+Causal structure
+13
+5.1
+Connected regularisation
+13
+5.2
+Non-connected regularisation
+14
+6
+Deviations from vacuum, or The effective SET
+15
+6.1
+Hayward’s effective sources
+17
+6.2
+Black bounce’s effective sources
+20
+7
+Conclusions
+22
+A Optical scalars
+24
+B 2D expansions
+25
+1
+Introduction
+General relativity (GR) is notoriously non-renormalisable at the perturbative level [1, 2].
+Though this is not the only reason why a quantum theory of gravity remains elusive, it
+still represents an important technical impediment. To circumvent the issue, Hoˇrava [3]
+proposed a field theory of gravity in which power-counting renormalisability is manifest
+thanks to the addition, to the action of gravity, of terms that are higher-order in spatial
+derivatives. This choice improves the ultra-violet (UV) behaviour of propagators while
+ensuring that no ghosts are introduced; but it comes at the cost of breaking diffeomorphism
+invariance and, locally, Lorentz invariance. In the following years, several improvements
+to the original idea have been proposed, including a “healthy” extension known as non-
+projectable Hoˇrava gravity [4–6], in which diffeomorphism invariance is restored via the
+introduction of an irrotational, unit-norm, timelike vector field — the so-called æther.
+– 1 –
+
+There are now strong indications — [7–12] and references therein — that this version of
+the theory is renormalisable beyond power-counting: if this property is confirmed, non-
+projectable Hoˇrava gravity will represent an example of a consistent quantum theory of
+gravity in four spacetime dimensions.
+At low energies, one can neglect all but the lowest-dimensional operators. The theory
+thereby obtained has an action of the form
+S = −
+1
+16πG
+�
+d4x √−g
+�
+R + λ(∇aua)2 + β∇aub∇bua + αaaaa
+�
+,
+(1.1)
+where aa = ub∇bua is the æther’s acceleration and α, β, λ are three dimensionless cou-
+plings. This action coincides with that of Einstein–æther theory [13, 14], when the æther is
+constrained to be hypersurface-orthogonal [15, 16] — this justifies the choice of terminology
+for ua. However, since hypersurface orthogonality restricts the number of physical degrees
+of freedom, the theory specified by eq. (1.1) goes by its own name: khronometric theory.
+The parameters α, β, λ are tightly constrained by observations [17]: |β| ≲ 10−15
+and either |α| ≲ 10−7 with λ unconstrained or |α| ≲ 0.25 × 10−4 with λ ≈ α/(1 − 2α).
+Moreover, λ > 0 to avoid ghosts. Since α and β seem both very small, one may at times
+consider a “minimal theory” in which they are set to zero exactly, while λ remains free.
+Due to hypersurface orthogonality, the æther can be expressed as
+ua = ∇aT
+N
+with
+N =
+�
+∇bT∇bT ,
+(1.2)
+and T a scalar field called khronon.
+The khronon’s level sets are three-dimensional,
+everywhere-spacelike hypersurfaces that provide a time foliation with a preferred status.
+The existence of a preferred foliation — or equivalently of a preferred reference frame, in
+this case the one provided by the æther — is a direct consequence of the Lorentz-violating
+nature of Hoˇrava gravity, and it opens the door to the existence of modified dispersion
+relations. Indeed, khronometric theory allows for superluminal propagation of signals and
+even contains an instantaneous mode that travels at infinite speed.
+In such a context, it would be natural to expect that the notion of black hole had
+no meaning. Surprisingly, the theory does admit black hole solutions [18–27]. The moral
+equivalent of the event horizon is the so-called universal horizon (UH): a compact constant-
+khronon surface that traps modes of any speed — even the instantaneous one. When the
+spacetime is stationary, meaning that there exists a Killing vector field χa that is timelike
+at infinity, the UH is characterised by the conditions [28]
+ua χa
+����
+UH
+= 0
+and
+aa χa
+����
+UH
+̸= 0 .
+(1.3)
+Known black hole solutions harbour a spacetime singularity at their centre, exactly as
+their general-relativistic counterparts do. One can conjecture, however, that these singular-
+ities might be “cured” by properly taking into account all the higher-dimensional operators
+that define the full theory. In this scenario, the end state of gravitational collapse would
+be more appropriately described by a non-singular (or regular) configuration, consisting of
+– 2 –
+
+a non-singular metric and a non-singular æther flow. Solutions of this kind are however
+still lacking, and their derivation is probably going to be challenging.
+On the other hand, a lot is known about singularities (and how to avoid their formation)
+in the context of purely metric theories of gravity: it is thus natural to wonder whether
+this know-how could guide the search for non-singular configurations in Hoˇrava gravity. In
+particular, if one assumes that pseudo-Riemannian geometry provides a good description of
+the spacetimes resulting from quantum gravitational regularisation at late times, Penrose’s
+singularity theorem [29] can be used to compile a classification of all the non-singular
+geometries that gravitational collapse could result in [30, 31].
+This approach is purely
+geometric and therefore largely theory-agnostic.
+Surprisingly, the resulting list of viable spacetimes is remarkably short. In essence,
+once a trapping horizon is formed somewhere in the spacetime, there are only two options
+for avoiding an inner singularity: either the geometry is endowed with an “inner” trapping
+horizon; or the singularity is replaced by a wormhole mouth, which is “hidden” behind the
+trapping horizon. Note that, while in the former case the spacetime is simply connected,
+the latter case is characterised by the existence of a minimum radius that renders the
+spacetime topologically different. For this reason, we will refer to these alternatives as
+connected and non-connected regularisation, respectively. The only way to avoid either of
+them is to prevent the formation of the trapping horizon altogether. In principle, one can
+still consider both connected and non-connected configurations without horizons: while a
+connected horizonless object is a star (though possibly of an exotic kind), a non-connected
+one is a traversable wormhole.
+The analysis of [30] was performed under the assumption of Lorentz invariance but
+it can be extended to the framework of Lorentz-breaking theories [32].
+Despite some
+technical subtleties, the main result carries over: non-singular configurations are either
+simply connected or non-connected; and they might display a universal trapping horizon
+(the moral equivalent of the trapping horizon) or not — but if they are connected and with
+horizon, a second universal inner horizon must exist.
+Such classification is best suited for describing dynamical situations, namely the col-
+lapse of some energy density leading to the formation of a compact object. In particular,
+[30, 32] assume global hyperbolicity and therefore do not admit, for instance, stationary
+simply connected regular black holes. Yet, if the evolution of the geometry is characterised
+by a timescale that is much longer than any other relevant timescale, a stationary non-
+singular spacetime might provide an approximate description that is sufficiently accurate
+for phenomenological purposes.
+This line of reasoning has given rise to a thriving industry, fuelled by recent obser-
+vational achievements, aimed at constructing models of non-singular geometries. Models
+of simply connected regular black holes have a longer history, dating back to the work of
+Bardeen in 1968 [33], and are therefore more abundant in the literature — see e.g. [34–36]
+end references therein. Several (though fewer) instances of wormholes whose mouths are
+hidden behind an horizon exist as well [37–42].
+Only very recently [43], it was realised that the same models can describe horizonless
+objects too: typically, horizons only exist when the parameters that specify the model fall
+– 3 –
+
+within a given range; but when they do not, the corresponding geometries are still perfectly
+viable. In particular, the connected regular black holes are counterparts of compact stars
+that usually have a de Sitter core and are therefore instances of gravastar-like objects
+[44, 45].
+These models, which are usually constructed ad hoc and without referring to physically
+well-motivated theories, implicitly assume that all the relevant information concerning the
+geometry is encoded in the metric. In theories like Hoˇrava gravity, in contrast, the preferred
+foliation has a crucial, genuinely physical role that is not entirely played by the metric.
+The goal of this paper, therefore, is to construct explicit examples of non-singular
+geometries, connected and non-connected, in the context of low-energy Hoˇrava gravity.
+Such examples will consist of a metric and an æther flow, both of which will be free of
+singularities. They will not constitute solutions of khronometric theory in vacuum, nor
+with any simple form matter; yet, they will display all the key features that are expected
+for exact non-singular solutions of both khronometric theory and the full Hoˇrava gravity.
+In particular, these configurations will exhibit pairs of inner/outer UHs, hidden wormholes
+and (gravastar-like) compact stars with de Sitter core — all features that, to our knowledge,
+have never been described before in this context.
+Non-singular black holes have been searched for, but not found, in (2 + 1) projectable
+Hoˇrava gravity in [46]. Spherical stars in Einstein-æther and khronometric theory have
+been investigated in [47], while examples of wormholes are given in [48, 49]; however, the
+æther flow considered in these references is often assumed to be parallel to the Killing
+vector: our analysis is therefore substantially different.
+The paper is structured as follows. Section 2 is an introduction to the singular solution
+we take as a starting point for constructing our non-singular geometries. Such geometries
+are introduced in section 3 and characterised in the following sections. In particular, sec-
+tion 4 investigates horizons (Killing and universal); section 5 describes the causal structure;
+section 6 quantifies the deviations away from the vacuum of the khronometric theory. Fi-
+nally, section 7 reports our conclusions. We also include appendices A and B, which provide
+further details on the non-singular configurations.
+2
+A (singular) black hole solution
+The equations of motion obtained by varying eq. (1.1) with respect to δgab and δT can be
+written as:
+Gab = 0 ,
+(2.1)
+∇a
+�Aa
+N
+�
+= 0 .
+(2.2)
+Eq. (2.1) is the equivalent of the Einstein’s equation, indeed one can write Gab = Gab −T æ
+ab,
+with Gab the Einstein’s tensor and T æ
+ab the stress-energy tensor (SET) of the æther. Eq. (2.2)
+is the equation of motion of the khronon: the vector Aa is built out of ua and its derivatives
+and is orthogonal to the æther uaAa = 0. To include matter, one can add the matter action
+– 4 –
+
+Smat to eq. (1.1). This yields source terms that appear on the right-hand side of eqs. (2.1)
+and (2.2).
+In this paper, we will investigate spherically symmetric and static spacetimes, and
+assume that the same symmetries extend to the æther field. This implies the existence of a
+Killing vector χa, timelike at spatial infinity, along which both the metric and æther are Lie-
+dragged. Adopting in-going Eddington–Finkelstein coordinates, we can generically write
+the metric and the æther as
+ds2 = F(r) dv2 − 2 dv dr − R2(r) dΩ2 ,
+(2.3)
+ua∂a = A(r)∂v + y(r)∂r ,
+(2.4)
+with
+y = −1 − A2F
+2A
+(2.5)
+so that uaua = +1. With these notations, the projection of the æther along the Killing
+vector is
+ua χa = 1 + A2F
+2A
+.
+(2.6)
+An exact solution is known for α = 0 [26].1 In our coordinates, it is given by2
+F = 1 − r0
+r − β r4
+æ
+r4 ,
+R(r) = r ,
+(2.7)
+A(r) =
+1
+F(r)
+�
+−r2
+æ
+r2 ±
+�
+F(r) + r4æ
+r4
+�
+,
+(2.8)
+where r0 is twice the Arnowitt–Deser–Misner (ADM) mass of the spacetime (measured in
+appropirate units) and ræ is another, a priori independent, integration constant. Depend-
+ing on the relative magnitude of r0 and ræ, the quantity
+F(r) + r4
+æ
+r4
+(2.9)
+may become negative, thus rendering A(r) ill-defined. One can further check that this
+quantity coincides with (ua χa)2, hence its zeroes correspond to UHs.
+Only when the
+parameters satisfy
+ræ = r0
+4
+� 27
+1 − β
+�1/4
+(2.10)
+1Another exact solution can be found for β + λ = 0.
+2Reference [26] only reports the plus sign, although the equations of motion actually allow for both.
+However, the square root, with its sign, coincides with ua χa, which must become negative upon crossing
+the universal horizon [50]. Hence, the choice of [26] is in fact ill-behaved at the universal horizon; all their
+conclusions still stand, despite this clarification.
+– 5 –
+
+does the solution describe a black hole with one universal horizon located at
+rUH = 3
+4r0 .
+(2.11)
+When this fine-tuned choice is assumed — as we will do for the rest of this paper —, one
+can write
+ua ��a = 1
+r2
+�
+r − 3
+4r0
+� �
+r2 + r0
+2 r + 3r2
+0
+16 .
+(2.12)
+The signs have been chosen so that this quantity tends to one at spatial infinity but changes
+sign upon crossing the universal horizon — as it must [50]. This corresponds to choosing
+the plus sign in eq. (2.8) outside of the UH and the minus inside.
+The UH has an associated surface gravity that seems to provide some thermal prop-
+erties, in a way similar to the surface gravity of horizons in GR. It is defined as
+κUH = −1
+2aa χa ,
+(2.13)
+which on the singular solution evaluates to
+κsing.
+UH =
+2
+√
+2
+3
+√
+3r0
+√1 − β .
+(2.14)
+The metric also exhibits a Killing horizon (KH), associated with the zero of F(r).
+Clearly, when β = 0 the metric reduced to that of Schwarzschild and the KH is located at
+rKH = r0. More generally, one can write F(r) = 0 as
+r3(r − r0) −
+� 27
+256
+β
+1 − β r4
+0
+�
+= 0 ;
+(2.15)
+hence, one can deduce that the KH moves towards larger values of r as β increases (we
+always assume β < 1), i.e. rKH ≥ r0. Thus, the KH always encloses the UH. The equation
+does not have any more roots. Note that A(r) is well-behaved at the KH, as can be verified
+by expanding close to r = rKH:
+A(r) =
+1
+F ′ (rKH) (r − rKH)
+�r2
+KHF ′ (rKH)
+2r2æ
+(r − rKH)
+�
++ O
+�
+(r − rKH)2�
+(2.16)
+= r2
+KH
+2r2æ
++ O
+�
+(r − rKH)2�
+.
+(2.17)
+This metric is singular at r = 0, as one can check e.g. by evaluating the Kretschmann
+scalar:
+RabcdRabcd = 12r2
+0r6 + 10βr0r4
+ær3 + 39β2r8
+æ
+r12
+.
+(2.18)
+The components of the æther also seem ill-defined at that point, although this statement
+relies on the choice of coordinates. To check that the æther flow is in fact singular at
+r = 0 one should characterise it in terms of scalar quantities. Since the æther constitutes
+a timelike non-geodesic congruence, a rather natural choice is to describe it in terms of its
+optical scalars: 3 the expansion, the square of the symmetric shear and the square of the
+antisymmetric twist. Further details can be found in appendix A.
+3The term “optical scalars” is usually reserved for null geodesic congruences.
+We are abusing this
+terminology, hopefully without confusion.
+– 6 –
+
+3
+Regularisations of the singularity
+As mentioned in section 1, there are only two qualitatively different ways of avoiding,
+i.e. “regularising”, the central singularity: we have referred to these alternatives as con-
+nected and non-connected regularisation. The connected regularisation corresponds to a
+physical scenario in which gravity effectively becomes weaker and “turns off” at r = 0.
+Metrics that exhibit such behaviour can be built by modifying a singular solution (typi-
+cally Schwarzschild) in a simple way. The non-connected regularisation instead does not
+correspond to a weakening of gravity. On the contrary, gravity becomes so strong that it
+induces a change in the topology of spacetime. As a consequence, a finite region containing
+r = 0 gets excised from the spacetime: this alternative is therefore characterised by the
+existence of a minimal length scale corresponding, roughly speaking, to the radius of the
+smallest sphere centred at r = 0. Despite the different topology, metrics portraying this
+scenario can be built by modifying a singular solution too.
+To be explicit and as clear as possible, in the following we will explore two specific
+examples, one connected and one not. Most of the qualitative considerations, however,
+hold true in general.
+Appendix B reports further details, using a characterisation in terms of two-dimensional
+congreunces in order to make contact with the language of [32].
+3.1
+Connected regularisation
+A simple way to construct a simply connected non-singular metric, starting from a singular
+one, is to upgrade the parameter r0 to a function of the radius r0(r). Remarkably, when this
+replacement is performed on eq. (2.7) and eq. (2.8), the resulting æther becomes regular
+too. Explicitly, the metric components of eq. (2.3) and the æther one of eq. (2.4) become
+F(r) = 1 − r0(r)
+r
+− β r4
+æ(r)
+r4
+,
+R(r) = r ,
+(3.1)
+A(r) =
+1
+F(r)
+�
+−r2
+æ(r)
+r2
++ 1
+r2
+�
+r − 3
+4r0(r)
+� �
+r2 + r0(r)
+2
+r + 3r2
+0(r)
+16
+�
+,
+(3.2)
+ræ(r) = r0(r)
+4
+� 27
+1 − β
+�1/4
+.
+The function r0(r) is arbitrary, except for a minimum set of requirements [34–36, 51]: it
+should be “well-behaved”, in the sense that it should not introduce new singularities (this is
+guaranteed if e.g. r0(r) > 0); it should not spoil asymptotic flatness, i.e. limr→∞ r0(r) = 2M
+with M the ADM mass; and, crucially, it must be r0(r) = O
+�
+r3�
+close to r = 0. This last
+requirement makes the components of the metric manifestly regular at the origin and
+prevents divergences in any scalar polynomial built out of the Riemann tensor and the
+metric. Expanding close to r = 0, one has
+F(r) = 1 − cr2 + O
+�
+r3�
+,
+(3.3)
+showing that the geometry of the inner core is asymptotically de Sitter (anti-de Sitter) if
+c > 0 (c < 0) or Minkowski if c = 0.
+– 7 –
+
+The components of the æther are now manifestly regular, too. In particular,
+A(r) = 1 + c
+2r2 + O
+�
+r3�
+and
+y(r) = O
+�
+r3�
+,
+(3.4)
+i.e. in the limit r → 0 the æther coincides with the Killing vector, up to corrections of
+order O
+�
+r2�
+. This is precisely the trivial æther flow that one would expect in a maximally
+symmetric space. The first derivatives of F(r) and A(r) are similarly well-behaved close
+to r = 0, which ensures that all the optical scalars characterising the æther congruence are
+regular too — details can be found in appendix A.
+Note, incidentally, that the geometry described by eq. (3.2) is certainly non-singular,
+in general, only for r ≥ 0. If one allows the coordinate r to become negative, one might still
+encounter spacetime singularities [52, 53] — in the form of divergences in the curvatures
+or in the sense of geodesic incompleteness. In order to interpret eq. (3.2) as a non-singular
+black hole, therefore, one must limit the domain of r to [0, +∞). Clearly, this is coherent
+with the interpretation of r as a radius, and with the fact that r = 0 is, at any given v, a
+point (i.e. a degenerate, zero-radius sphere).
+Many explicit forms of r0(r) have been proposed and the corresponding metrics have
+been extensively studied: all of them are characterised by at least one additional parameter,
+usually carrying the dimensions of a length, upon which r0(r) depends continuously. Often,
+r0(r) reduces to a constant for some particular values of the parameters (typically zero).
+In this limit, the regularisation is undone.
+In what follows, we will present calculations for a particular choice of r0(r) introduced
+by Hayward [54],
+r0(r) = 2M
+r3
+r3 + 2Mℓ2
+(3.5)
+(we have called ℓ the additional parameter; note that r0 = 2M for ℓ = 0), but the fea-
+tures we will describe are generic: other well-studied examples, e.g. the Bardeen [33] or
+Dymnikova [55] metrics, yield very similar results.
+3.2
+Non-connected regularisation
+Many instances of wormhole exist in the literature (see e.g. [56]). In the past, the attention
+has mostly focused on “traversable” wormholes, i.e. wormholes with a timelike mouth
+that can be traversed in both directions.
+However, there has been recently a growing
+interest towards “hidden” wormholes, i.e. wormholes whose mouths are shielded by trapping
+horizons. In the presence of a single outer horizon such mouth is not traversable, being
+spacelike, and indeed the metric can be more precisely characterised as a “black-bounce”
+being endowed with a minimum radius.
+The example that we investigate here is a very simple black-bounce geometry proposed
+by Simpson and Visser [37]. The original metric is a slight modification of the Schwarzschild
+one, formally obtained by replacing any instance of r with
+√
+r2 + ℓ2. When this trick is
+applied to the singular solution eqs. (2.7) and (2.8), the metric components of eq. (2.3) and
+– 8 –
+
+the æther one of eq. (2.4) take the form
+F(r) = 1 −
+r0
+√
+r2 + ℓ2 − β
+r4
+æ
+(r2 + ℓ2)2 ,
+R(r) =
+�
+r2 + ℓ2 ,
+(3.6)
+A(r) =
+1
+F(r)
+�
+�−
+r2
+æ
+r2 + ℓ2 +
+1
+r2 + ℓ2
+��
+r2 + ℓ2 − 3
+4r0
+� �
+(r2 + ℓ2) + r0
+√
+r2 + ℓ2
+2
++ 3r2
+0
+16
+�
+� .
+(3.7)
+We are still assuming ræ = 271/4(1−β)−1/4(r0/4), without r-dependence, as in the singular
+solution.
+In this example, regularity is manifest, since all components of both the metric and
+the æther approach a finite non-zero limit as r → 0. (Details on the æther’s optical scalars
+can be found in appendix A.) Notably, R(r) = ℓ + O
+�
+r2�
+, meaning that, at any given time
+v, r = 0 is not a point; rather, it is a sphere of surface area 4πℓ2. It is therefore clear how
+this example could be generalised: any other R(r) that attains a finite value in r = 0 works
+as fine — provided one writes F (R(r)) and A (R(r)).
+Since the metric and the æther are invariant under r → −r, one can extend the
+domain of the coordinate r to (−∞, +∞). Hence, the geometry should be interpreted as
+representing a wormhole whose asymptotically flat ends are at r → +∞ and r → −∞ and
+whose mouth, a sphere of surface area 4πℓ2, is located at r = 0. We will often call “our
+universe” the (r > 0)-patch and “other universe” the (r < 0)-patch.
+It is useful to express eqs. (3.6) and (3.7) in terms of a new coordinate ϱ =
+√
+r2 + ℓ2.
+We have
+ds2 = F(ϱ) dv2 − 2δ(ϱ) dv dϱ − R2(ϱ) dΩ2 ,
+(3.8)
+ua∂a = A(ϱ)∂v + y(ϱ)
+δ(ϱ)∂ϱ ,
+(3.9)
+where
+δ(ϱ) = dr
+dϱ =
+ϱ
+�
+ϱ2 − ℓ2
+(3.10)
+and
+F(ϱ) = 1 − r0
+ϱ − β r4
+æ
+ϱ4 ,
+R(ϱ) = ϱ ,
+(3.11)
+A(ϱ) =
+1
+F(ϱ)
+�
+−r2
+æ
+ϱ2 + 1
+ϱ2
+�
+ϱ − 3
+4r0
+� �
+ϱ2 + r0
+2 ϱ + 3
+16r2
+0
+�
+.
+(3.12)
+The new coordinate ϱ has a simple physical interpretation: it is the aerial radius, i.e. sur-
+faces of constant ϱ (and v) are spheres with area 4πϱ2. In this coordinate, the components
+of the metric and of the æther look identical to those of the singular case, except for δ;
+however, ϱ can never reach zero, since min(ϱ) = ϱ (r = 0) = ℓ > 0, and the geometry thus
+remains free of spacetime singularities. There is now a coordinate singularity at the throat
+ϱ = ℓ, hence this coordinate system can only describe one of the two universes at a time.
+– 9 –
+
+4
+Horizons
+The regularised metrics introduced above, similarly to the singular solution, exhibit KHs
+located at the solutions of F(r) = 0.
+These horizons are still surfaces of infinite red-
+shift/blueshift for matter that is minimally coupled to the metric and uncoupled to the
+æther. However, because of the breaking of local Lorentz invariance, they are not causal
+horizons since the presence of superluminal signals affects the causal structure [57]. As
+mentioned above, in khronometric gravity the role of causal horizons is instead played by
+UHs (see e.g. the relative discussion in [57] and references therein).
+Nonetheless, it is easy to see that in the configurations we are exploring these structures
+are tightly related. Indeed, for both kinds of regularisation (as for the singular case) one
+can write
+(ua χa)2 = F(r) + f(r)
+with
+f(r) > 0 ,
+(4.1)
+so, since both ua χa and F are positive at infinity, ua χa can reach zero only in a region in
+which F(r) is negative. This means that UHs must necessarily lie in a “trapped region”
+— as one would call it in GR. Hence, the presence of a UH implies the existence of a
+KH that encloses it. Moreover, F(r) must be positive at r = 0 in the connected case and
+at r = −∞ in the disconnected case. This entails that KHs have to come in pairs and
+therefore, generically, UHs have too.
+Thus, non-singular black hole geometries typically exhibit a nested structures of Killing
+and universal horizons. In the following subsections we will describe this structure in detail
+for the specific examples that we are exploring, but the above considerations can be proven
+to be general by exploiting the notion of the degree of a map. Moreover, in appendix B
+we present an alternative, local characterisation of UHs in terms of the expansions of two
+congruences, in a language that makes contact with [32].
+4.1
+Connected regularisation
+We start by setting β = 0, for simplicity.
+KHs are given by r = r0(r) and UHs by
+4r/3 = r0(r).
+Whether these equations admit solutions or not is a model-dependent
+question. When r0(r) is that of eq. (3.5), for example, the answer depends on the value of
+the parameter ℓ.
+As an illustration, in figure 1 we plot Hayward’s r0(r) for three values of ℓ; the plot
+also reports two straight lines, with slope equal to one (dashed line) and 4/3 (solid line)
+respectively: the intersections of r0(r) with these lines determine the horizons. When ℓ
+is small, we can count two intersections with the dashed line and two with the solid one.
+Hence, this configuration presents two KHs and two UHs; coming from infinity, they are
+met in the following order: outer KH, outer UH, inner UH, inner KH. As ℓ is increased,
+the curve relative to r0(r) moves towards the bottom-left corner of the picture: inner and
+outer horizons thus approach each other. They keep approaching until the two UHs merge
+into a single, degenerate UH; this happens at a threshold value of ℓ = M/2 above which
+no UH exist. Similarly, the KHs keep approaching until they merge into a degenerate KH
+and then disappear: this second threshold corresponds to a higher value of ℓ = 4M/(3
+√
+3).
+– 10 –
+
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.5
+1.0
+1.5
+2.0
+Figure 1: Plot of Hayward’s choice of r0(r) for three values of the parameter ℓ. Intersec-
+tions with the straight (dashed) black line correspond to UHs (KHs) in the minimal theory
+α = β = 0.
+Therefore, we can distinguish three qualitatively different regimes: a non-singular black
+hole regime, characterised by an inner/outer UH pair (as well as an inner/outer KH pair);
+an intermediate regime in which there are two KHs but no UHs; and a star-like regime
+with no horizons. Although technically a black hole is present only in the first regime, an
+object in the intermediate regime would still appear “almost black”, given that low energy
+modes would linger for an extremely long time at the KH before being able to escape to
+infinity.
+Reinstating the parameter β does not greatly distort this picture, since its only effect
+is that of displacing the KHs. Eq. (2.15) remains valid upon replacing r0 with r0(r), so
+increasing the value of β shifts the outer KH outwards. The inner KH, instead, moves
+inwards. That is, increasing β has the effect of pushing KHs further apart; this is the
+opposite effect one has by increasing the regularisation parameter ℓ, which instead pushes
+KHs closer together. The location of UHs is unaffected by β.
+In the black hole regime, the universal horizons each have a surface gravity. Plugging
+eq. (3.3) in the definition eq. (2.13), we get
+κUH =
+4 − 3r′
+0(r)
+3
+√
+6r0
+√1 − β
+����
+UH
+,
+(4.2)
+which should be evaluated at each of the UHs. Note that when r′
+0 = 0 we recover the result
+for the singular solution eq. (2.14).
+In the black hole and in the intermediate regime, the horizon radii provide an intuitive
+way of telling the “size” of the compact object. It would be useful to extend this notion
+to the star-like regime by defining an appropriate effective radius. A particularly simple
+choice is to pick the unique r⋆ for which F ′(r⋆) = 0. This is the radius of maximum (metric)
+– 11 –
+
+redshift and thus quantifies the compactness of the star. Moreover, in the limit in which
+ℓ approaches (from above) the threshold value for the KH’s formation, r⋆ approaches the
+(degenerate) horizon radius.
+Explicitly, we have
+F ′ = −
+�r0
+r
+�′ �
+1 + 4 27
+256
+β
+1 − β
+�r0
+r
+��
+,
+(4.3)
+hence F ′ = 0 reduces to
+r0(r) − rr′
+0(r) = 0 ,
+(4.4)
+independently on β. For Hayward’s choice, we find
+r⋆ =
+�
+4Mℓ2�1/3 .
+(4.5)
+4.2
+Non-connected regularisation
+As in the previous case, the existence and location of horizons is, strictly speaking, a
+model-dependent question. In the simple example that we are considering, the answer is
+determined by the only free parameter ℓ. The discussion becomes particularly simple if
+one resorts to the coordinate ϱ.
+KHs are solutions of
+ϱ3(ϱ − r0) −
+�27
+56
+β
+1 − β r4
+0
+�
+= 0 ,
+(4.6)
+which is formally the same as eq. (2.15). Call ϱKH the (unique) solution; in the r coordi-
+nates, this corresponds to
+r2
+KH = ϱ2
+KH − ℓ2 .
+(4.7)
+Hence, for ℓ < ϱKH the spacetime has one KH per each universe, located at r = ±rKH
+with rKH =
+�
+ϱ2
+KH − ℓ2; when instead ℓ > ϱKH the spacetime has no KHs; the limiting
+case ℓ = ϱKH corresponds to the two horizons coinciding with the wormhole mouth, which
+in this case is null. Similarly to the simply connected configuration, one can easily check
+that increasing ℓ makes the KH shrink, while increasing β makes it larger.
+For what concerns the UHs, instead, they are located at
+ϱUH = 3
+4r0 .
+(4.8)
+That is, when ℓ < 3r0/4 there is one UH per each universe, located at r = ±rUH with
+rUH =
+�
+ϱ2
+UH − ℓ2; when instead ℓ > 3r0/4 there are no UHs. As before, the equality
+corresponds to a degenerate case for which the mouth of the wormhole coincides with the
+UH. Analogously to the previous case, increasing ℓ makes the UH shrink while β has no
+effect at all; note that rUH < rKH.
+– 12 –
+
+The surface gravity of the UH has a particularly simple form:
+κUH = κsing.
+UH
+�
+1 − ℓ2
+ϱ2
+UH
+,
+(4.9)
+where κsing.
+UH is the surface gravity for the singular solution written in eq. (2.14). Thus, for
+ℓ = ϱUH = 3r0/4 the UH is “degenerate” and the black hole is extremal, in the sense that
+its UH’s surface gravity vanishes.
+5
+Causal structure
+In a Lorentz-violating theory of gravity like khronometric theory, the causal structure is
+not determined by null rays. Rather, the theory exhibits a preferred foliation, specified
+by constant-khronon surfaces: it is the embedding of the leafs of the foliation into the
+four-dimensional spacetime, therefore, that defines the causal structure.
+Due to spherical symmetry, the khronon does not depend on the angles in either of the
+spacetimes we constructed. Hence, we can visualise the causal structure by simply plotting,
+in an appropriate (time–radius) plane, the surfaces of constant khronon. The most natural
+definition of time is given in terms of the null coordinate v as
+dt∗ = dv − dr ;
+(5.1)
+this is a (Killing-)“horizon-penetrating” time. The more familiar time t, given by dt =
+dv − dr/F, would not be appropriate, since the components of the metric and of the
+æther are singular at the KHs when expressed in terms of it.
+In addition, we plot the flow of the æther, written in its covariant form. Since the
+æther is by definition orthogonal to constant-khronon hypersurfaces, the information pro-
+vided by these plots and by those representing constant-khronon surfaces is not independent
+but complementary. We report both, hoping this will benefit the reader.
+5.1
+Connected regularisation
+The causal structure corresponding to the Hayward-like regularisation is summarised in
+figure 2. Each row corresponds to a different value of ℓ and therefore to a different regime:
+the top row represents a non-singular black hole, the second row the intermediate regime
+and the third row the star-like regime.
+The left-hand panel displays the flow of ua, written in the (t∗, r) coordinates: the
+horizontal component of the arrows is ur = y while the vertical one is ut∗ = ua χa. The
+right-hand panel, instead, presents constant-khronon lines.
+At large r, the æther is almost vertical, i.e. aligned with the Killing vector, and
+constant-khronon lines are also lines of constant t∗.
+As one approaches to smaller r,
+however, the æther tilts inwards. Nothing remarkable happens at the outer KH, whose
+location is depicted for reference only. At the outer UH, instead, the æther is horizontal
+while the constant-khronon lines exponentially recede away from the UH — i.e. they peal
+off, in an amount that is controlled by the surface gravity, exactly as null rays would do at
+– 13 –
+
+0
+1
+2
+3
+4
+5
+6
+7
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.0
+0.1
+0.2
+0.3
+0.4
+0.00
+0.05
+0.10
+0.15
+(a) ℓ = 0.25M.
+0
+1
+2
+3
+4
+5
+6
+7
+-3
+-2
+-1
+0
+1
+2
+3
+0.0
+0.1
+0.2
+0.3
+0.4
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+(b) ℓ = 0.25M.
+0
+1
+2
+3
+4
+5
+6
+7
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+(c) ℓ = 0.6M.
+0
+1
+2
+3
+4
+5
+6
+7
+-3
+-2
+-1
+0
+1
+2
+3
+(d) ℓ = 0.6M.
+0
+1
+2
+3
+4
+5
+6
+7
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+(e) ℓ = 1.3M.
+0
+1
+2
+3
+4
+5
+6
+7
+-3
+-2
+-1
+0
+1
+2
+3
+(f) ℓ = 1.3M.
+Figure 2: Hayward non-singular black hole: æther flow (left) and constant-khronon sur-
+faces (right). Black solid lines mark universal horizons, dashed lines Killing horizons; the
+dotted line signals the star’s effective radius. The first row depicts the case with outer and
+inner KHs and UHs (with inlets magnifying the inner KH-UH region), the middle row the
+case with only two KHs, the bottom row the case of an ultracompact, horizonless, object.
+a trapping horizon. Note that behind the outer UH the æther points downwards, meaning
+that it flows in the opposite direction with respect to the Killing vector.
+Moving to yet smaller r, the æther becomes horizontal again at the inner UH. The
+constant-khronon lines instead pile up exponentially at the UH — as null rays would do
+at an inner trapping horizon. Inside the inner UH the æther points again in the same
+direction as the Killing vector. Note that nothing remarkable takes place at the inner KH.
+Close to r = 0, the flow becomes identical to that of large r, i.e. to that of flat space.
+5.2
+Non-connected regularisation
+The æther flow and constant-khronon lines for the black-bounce-like regularisation are
+displayed in figure 3. As before, each row corresponds to a different regime: the first to
+a black bounce with UHs, the second to one with KHs but no UHs and the third to one
+without horizons.
+– 14 –
+
+-4
+-2
+0
+2
+4
+0.0
+0.5
+1.0
+1.5
+2.0
+(a) ℓ = 0.25M.
+-4
+-2
+0
+2
+4
+-3
+-2
+-1
+0
+1
+2
+3
+(b) ℓ = 0.25M.
+-4
+-2
+0
+2
+4
+0.0
+0.5
+1.0
+1.5
+2.0
+(c) ℓ = 1.8M.
+-4
+-2
+0
+2
+4
+-3
+-2
+-1
+0
+1
+2
+3
+(d) ℓ = 1.8M.
+-4
+-2
+0
+2
+4
+0.0
+0.5
+1.0
+1.5
+2.0
+(e) ℓ = 2.5M.
+-4
+-2
+0
+2
+4
+-3
+-2
+-1
+0
+1
+2
+3
+(f) ℓ = 2.5M.
+Figure 3: Black bounce: æther flow (left) and constant-khronon surfaces (right). Black
+solid lines mark universal horizons, dashed lines Killing horizons; the dotted line signals
+the mouth. the first row depicts a black bounce with a UH and a KH per side, the second
+row represents a configuration with just one with KHs per side but no UHs, the third row
+portraits a naked (without horizons) traversable wormhole
+The æther, which is aligned with the Killing vector at infinity, tilts inwards as one
+moves closer to r = 0. At the UH it becomes horizontal, it flows in the opposite direction
+until it becomes horizontal again at the UH in the other universe and then returns aligned
+with the Killing vector at r = −∞. The constant-khronon curves pile off the UH in our
+universe and pile up at the UH in the other universe. Note that the KHs and the throat
+look like any other location to the æther.
+6
+Deviations from vacuum, or The effective SET
+As already mentioned in the Introduction, section 1, the non-singular configurations we are
+describing are supposed to be solutions of the full Hoˇrava action, and in this sense cannot
+be expected to be vacuum solutions of its low-energy version — khronometric theory.
+Nevertheless, it is still instructive to study the extent to which our configurations fail
+to be (vacuum) solutions. Our strategy is inspired by the analysis of non-vacuum solutions,
+– 15 –
+
+since any metric and æther flow can be seen as solutions of the equations of motion with an
+appropriate matter content (in this case, one that couples to the metric and to the æther).
+For this reason, we will speak of “effective sources” and use terms such as “energy density”
+and “pressures”. It should be clear, however, that this is merely a choice of terminology
+and we are not positing the existence of any physical form matter. Indeed, such density
+and pressures can be seen as components of an effective SET induced by the higher-order
+terms of the Hoˇrava action. Understanding whether this is actually the case is a crucial
+but open question, given the daunting task of manipulating such extra terms.
+Thus, in the following we will characterise the form and distribution of such effective
+sources, in order to better understand the regular geometries we are proposing.
+Let us start by noticing that the equation of motion for the khronon eq. (2.2) can be
+written as
+1
+N J = 0
+with
+J = [∇aAa − aaAa] .
+(6.1)
+The lapse N can be chosen (almost) arbitrarily, since it depends on how the khronon is
+parametrised; the scalar quantity J instead only depends on the æther and can be computed
+unequivocally. Thus, J will be the khronon’s effective source.
+Similar considerations hold for the Einstein’s equations eq. (2.1). Since the components
+of Gab clearly depend on the choice of coordinates, one first needs to find a coordinate-
+independent way of characterising the source. One way to achieve this goal would be to
+compute its eigenvalues, which are scalars under general coordinate transformations. When
+the eigenvalues are real, they can be interpreted as energy density and principal pressures of
+some non-perfect fluid. Unfortunately, while this characterisation works for the Einstein’s
+tensor, it fails for the SET of the æther, since there are regions in the spacetime where
+the eigenvalues are complex (i.e. the æther’s SET is of Type IV in the Hawking–Ellis
+classification [58]).
+However, since in the framework of Hoˇrava gravity there exists a preferred foliation and
+therefore a preferred observer, it makes sense to characterise the deviations from vacuum
+as measured by such observer. Hence, we compute the projection
+ρ(u) = Gabuaub ,
+(6.2)
+which we could interpret as the energy density measured by an observer that is comoving
+with the æther. We then pick another vector sa that is spacelike, outward-pointing, of unit
+norm and orthogonal to ua and use it to define a radial pressure as
+p(s) = Gabsasb .
+(6.3)
+We use
+sa∂a = A∂v + w∂r
+with
+w = +1 + A2F
+2A
+.
+(6.4)
+Finally, we could define a tangential pressure in an analogous way, or simply as
+p⊥ = −Gθ
+θ = −Gφ
+φ .
+(6.5)
+– 16 –
+
+Since the parameters α, β, λ enter the action as coupling constants, all the scalars
+that we have just introduced share the same simple structure. Consider ρ(u) as an example:
+it can be written as
+ρ(u) = ρ(u)
+G + αρ(u)
+α
++ βρ(u)
+β
++ λρ(u)
+λ
+.
+(6.6)
+Here, ρ(u)
+G
+derives from the Einstein’s tensor while each of ρ(u)
+α , ρ(u)
+β , ρ(u)
+λ
+derives from the
+operators that appear in the action multiplied respectively by α, β and λ. Clearly, each
+of them still depends on β (and on ℓ), since the explicit form of the metric and of the
+æther does; but not on α nor λ. We will use analogous notations for the decompositions
+of p(s), p⊥ and J, with the only difference that J has no “JG” part. One can check that
+p(s)
+α = −p⊥
+α in all the cases that we consider.
+A remark is in order, at this point. The singular geometry of eqs. (2.7) and (2.8) is
+a solution of the equations of motion only for α = 0. Indeed, as will be made explicit in
+the next two subsections, the effective sources proportional to α generically do not vanish
+— not even in the limit ℓ → 0, in which the singular geometry is retrieved. Hence, one
+might worry that allowing α ̸= 0 in the analysis of the non-singular configurations is not
+consistent.
+Here, however, we choose to keep α ̸= 0. The reason is that, in the absence of some
+custodial symmetry that protects it against running, the higher-order operators in the
+action of Hoˇrava gravity will generically affect the value of α (as well as that of β and λ) at
+the level of the effective field theory. Hence, we cannot presume it to be zero at this stage.
+Still, at low energies, the effect of higher-order operators is negligible and the value of
+α is the one set by (low-energy) observations: α ≲ O
+�
+10−4�
+— cf. section 2. One should
+bear in mind, therefore, that at large distances the effective sources proportional to α are
+highly suppressed.
+6.1
+Hayward’s effective sources
+Here, we remain agnostic on the specific choice of r0(r) for as long as possible; however, the
+explicit results are often cumbersome and not particularly enlightening. For this reason,
+we specialise to Hayward’s choice and discuss, in particular, the asymptotic behaviour of
+the effective sources.
+Khronon’s equation
+One finds that Jβ = Jλ; clearly, these are zero when ℓ = 0. Jα
+instead is non-zero even in the limit in which the regularisation parameter vanishes, since
+the singular solution we started with is an exact (vacuum) solution only for α = 0. The
+explicit expressions are not particularly enlightening and we hence omit them here. We
+can however get useful insights by looking at their asymptotic behaviour.
+At infinity, we find
+J = αJα + (β + λ)Jβ,λ
+= α
+�
+−6
+√
+3
+�
+1
+1 − β
+M4
+r5 + O
+�
+r−6��
++ (β + λ)
+�
+540
+√
+3
+�
+1
+1 − β
+M3ℓ2
+r6
++ O
+�
+r−7��
+. (6.7)
+– 17 –
+
+The different scaling between Jα and Jβ,λ is not surprising, since the former does not vanish
+in the ℓ → 0 limit — as previously argued. In any case, it is easy to see from the above
+expression that the effective source vanishes rapidly as one moves away from the object.
+The sources may be large at intermediate radii, but become very small in the opposite
+limit, small r, and vanish exactly at r = 0. Indeed, expanding around this point, we find
+J = α
+�
+27
+√
+3
+4
+�
+1
+1 − β
+r5
+ℓ6 + O
+�
+r6�
+�
++ (β + λ)
+�
+27
+√
+3
+�
+1
+1 − β
+r3
+ℓ4 + O
+�
+r4��
+.
+(6.8)
+Hence, the connected non-singular configuration is almost a solution of khronometric
+theory at very large and very small distances from the centre.
+Einstein’s equations
+Plugging in the Ans¨atze eqs. (3.2) and (3.3), we find a series of
+additional identities:
+ρ (u)
+G = −p(s)
+G ,
+p⊥
+λ = p(s)
+λ
+(6.9)
+ρ(u)
+G + βρ(u)
+β
+= r′
+0
+r2 + βρ(u)
+λ
+,
+p(s)
+G + βp(s)
+β
+= −r′
+0
+r2 + βp(s)
+λ ,
+p⊥
+G + βp⊥
+β = −r′′
+0
+2r + βp⊥
+λ
+(6.10)
+Hence, we can write
+ρ(u) = r′
+0
+r2 + (β + λ)
+27r2
+0
+128(1 − β)
+�r′
+0
+r2
+�2
++ αρ(u)
+α ,
+(6.11)
+p(s) = −r′
+0
+r2 − (β + λ)
+27r2
+0
+128(1 − β)r5
+�
+2r(r′
+0)2 + r0(rr′′
+0 − 2r′
+0)
+�
++ αp(s)
+α ,
+(6.12)
+p⊥ = −r′′
+0
+2r − (β + λ)
+27r2
+0
+128(1 − β)r5
+�
+2r(r′
+0)2 + r0(rr′′
+0 − 2r′
+0)
+�
+− αp(s)
+α
+(6.13)
+The expressions of ρ(u)
+α
+and p(s)
+α
+are slightly more involved and we omit them here.
+Focusing on Hayward’s choice, we can read off the asymptotic behaviours. At large r
+we have
+ρ(u) =
+�12M2ℓ2
+r6
++ O
+�
+r−9��
++ (β + λ)
+� 243M6ℓ4
+2(1 − β)r12 + O
+�
+r−15��
++ α
+�M2
+2r4 + O
+�
+r−5��
+,
+(6.14)
+p(s) = −
+�12M2ℓ2
+r6
++ O
+�
+r−9��
++ (β + λ)
+� 243M5ℓ2
+2(1 − β)r9 + O
+�
+r−12��
++ α
+�M2
+2r4 + O
+�
+r−5��
+,
+(6.15)
+p⊥ =
+�24M2ℓ2
+r6
++ O
+�
+r−9��
++ (β + λ)
+� 243M5ℓ2
+2(1 − β)r9 + O
+�
+r−12��
+− α
+�M2
+2r4 + O
+�
+r−5��
+.
+(6.16)
+Clearly, these effective sources display “tails” that extend, in principle, up to infinity;
+the tails however decrease very rapidly, hence for all practical purposes the spacetime
+surrounding the object can be considered empty. Further note that, as anticipated, these
+– 18 –
+
+0
+1
+2
+3
+4
+5
+6
+7
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+(a) Energy density and principal pressures de-
+riving from the Einstein’s tensor.
+0
+1
+2
+3
+4
+5
+6
+7
+-0.2
+-0.1
+0.0
+0.1
+(b) Energy density and principal pressures de-
+rived from the æther’s SEMT. These curves
+should be multiplied by λ.
+Figure 4: Components of the energy density and the principal pressures, as measured
+by an observer comoving with the æther, for the Hayward non-singular configuration in
+the minimal theory α = β = 0. The position of the horizons in this coordinate strongly
+depends on ℓ: the vertical black lines mark UH (solid) and KH (dashed) for ℓ = 0.25M; the
+dotted line corresponds to the star’s effective radius (which coincides with the degenerate
+KH in the extremal case).
+sources do not vanish in the limit ℓ → 0 because of the terms ∝ α. This is simply due to
+the fact that the singular configuration is a solution only for α = 0.
+At r = 0, the following expansions hold:
+ρ(u) =
+� 3
+ℓ2 + O
+�
+r3��
++ (β + λ)
+�
+243r6
+128(1 − β)ℓ8 + O
+�
+r7��
++ α
+� 3
+ℓ2 + O(r)
+�
+,
+(6.17)
+p(s) = −
+� 3
+ℓ2 + O
+�
+r3��
+− (β + λ)
+�
+243r6
+64(1 − β)ℓ8 + O
+�
+r7��
++ α
+� r2
+2ℓ4 + O
+�
+r3��
+,
+(6.18)
+p⊥ = −
+� 3
+ℓ2 + O
+�
+r3��
+− (β + λ)
+�
+243r6
+64(1 − β)ℓ8 + O
+�
+r7��
+− α
+� r2
+2ℓ4 + O
+�
+r3��
+.
+(6.19)
+Note that the Einstein’s tensor presents a de Sitter form.
+Focusing on the minimal theory (α = β = 0), the analytic expressions become more
+tractable. We report them for completeness:
+ρ(u)
+G = −p(s)
+G =
+12M2ℓ2
+(r3 + 12Mℓ2)2 ,
+p⊥
+G = −24M2ℓ2(Mℓ2 − r3)
+(r3 + 2Mℓ2)3
+,
+(6.20)
+ρ(u)
+λ
+=
+243M6ℓ4r6
+2(r3 + 2Mℓ2)6 ,
+p(s)
+λ
+= p⊥
+λ = 243M5ℓ2r6(r3 − 2Mℓ2)
+2(r3 + 2Mℓ2)6
+;
+(6.21)
+and provide their plots in figure 4.
+In order to produce the figures, we exploit a self-similarity property that these functions
+enjoy: when written in terms of the variable x = r
+�
+Mℓ2�−1/3, they only depend on ℓ
+through a multiplicative factor, which we can remove. Thus, the curves in figure 4 are ℓ-
+independent: the ℓ-dependence can be reinstated by simply rescaling the axes appropriately.
+– 19 –
+
+The location of the horizons in the x coordinate depends markedly on ℓ: for reference,
+figure 4 reports an illustrative example. It also reports the location of the star’s effective
+radius, which is defined only for ℓ > 4M/(3
+√
+3) but has an otherwise ℓ-independent x-
+coordinate:
+r⋆ =
+�
+4Mℓ2�1/3 �→ x⋆ = 41/3 ≃ 1.59 .
+(6.22)
+The fact that x⋆ does not depend on the regularisation parameter might sound suspicious,
+as it seems to suggest that the value of the effective sources at the star’s radius is always
+the same, irrespective of the value of ℓ. Physical intuition would suggest the opposite:
+the sources corresponding to large stars should be “dilute” with respect to those of more
+compact stars. Physical intuition does indeed paint the correct picture: the value of each of
+the effective sources at the star’s radius is given by a constant (of order one in appropriate
+units) divided by ℓ2. So increasing ℓ does suppress the deviations away from vacuum.
+Inspecting the figure, we realise that the effective sources are typically negligible at
+the scale of the outer horizon. They are still small, though less so, at the scale of the star’s
+radius, and become almost zero very rapidly as one moves outwards. We deduce that most
+of the phenomenologically relevant phenomena involving these non-singular objects take
+place, for all practical purposes, in vacuum.
+6.2
+Black bounce’s effective sources
+In the non-connected case, we are forced to analyse the specific example provided by the
+black bounce spacetime. The analytic expressions are often reasonably compact: when this
+is the case, we report them in full. However, as in the previous section, we will put the
+emphasis on the more informative asymptotic behaviours of the effective sources.
+Khronon’s equation
+We find that Jλ = 0 identically. This means that, remarkably, the
+khronon’s equation of motion is satisfied in the minimal theory α = β = 0. For the more
+general cases, Jα and Jβ can be written as
+Jα = r
+�
+Mϱ2P5(ϱ) + ℓ2P6(ϱ)
+�
+jα(ϱ)
+and
+Jβ = rℓ2jβ(ϱ) ,
+(6.23)
+where jα and jβ do not depend on ℓ while P5(ϱ) and P6(ϱ) are polynomials of degree five
+and six, respectively, in ϱ.
+At infinity, one finds the following expansions:
+J = α
+�
+−6
+√
+3
+�
+β
+1 − β
+M4
+ϱ5 + O
+�
+ϱ−6�
+�
++ β
+�
+12
+√
+3
+�
+β
+1 − β
+M2ℓ2
+ϱ5
++ O
+�
+ϱ−6�
+�
+,
+(6.24)
+so even in this case the effective sources go to zero very rapidly.
+As the expressions in eq. (6.23) make explicit, these functions are O(r) close to r = 0,
+for all nonzero values of ℓ. Note that the limit ℓ → 0 is, as expected, singular.
+– 20 –
+
+Einstein’s equations
+We find that the term proportional to λ in Gab vanishes identically:
+this means that, in the minimal theory α = β = 0, the only deviations from vacuum come
+from the Einstein tensor. In other words
+ρ(u)
+λ
+= p(s)
+λ
+= p⊥
+λ = 0 .
+(6.25)
+The analytic expression of the effective energy density is
+ρ(u) = − ℓ2
+8ϱ8
+�
+8ϱ4 − 32ϱ3M + 27M4�
++ α
+�
+M
+ϱ2MP4(ϱ) + ℓ2P5(ϱ)
+8ϱ8 (4ϱ2 + 4ϱM + 3M2)
+�
+,
+(6.26)
+where the Pn are polynomials of degree n in ϱ; note in particular that nothing depends on
+β — ρ(u)
+G
+and ρ(u)
+β
+separately do, but the sum ρ(u)
+G + βρ(u)
+β
+does not. For the pressures, we
+find
+p(s) = − ℓ2
+8ϱ8
+�
+8ϱ4 + 27M4�
++ α
+�
+M2r2 �
+4ϱ2 + 6ϱM + 9M2�2
+8ϱ8 (4ϱ2 + 4ϱM + 3M2)
+�
+,
+(6.27)
+p⊥ = ℓ2(ϱ − M)
+ϱ5
+− α
+�
+M2r2 �
+4ϱ2 + 6ϱM + 9M2�2
+8ϱ8 (4ϱ2 + 4ϱM + 3M2)
+�
+;
+(6.28)
+as before, the β-dependence cancels out.
+Once again, we focus on the asymptotic behaviour. At infinity
+ρ(u) =
+�
+− ℓ2
+ϱ4 + O
+�
+ϱ−5��
++ α
+�
+−M2
+2ϱ4 + O
+�
+ϱ−5��
+,
+(6.29)
+p(s) = −p⊥ + O
+�
+ϱ−5�
+= ρ(u) + O
+�
+ϱ−5�
+.
+(6.30)
+Note that the fall-off rate of the tails is still rather fast. Again, it is easy to see that even
+for ℓ → 0 one does not recover vacuum if α ̸= 0. As in the previous case this is simply due
+to the fact that only for α = 0 the considered singular BH spacetime is an exact solution
+of the field equations in vacuum.
+At r = 0, instead, the values of the sources are nonzero and controlled by the regular-
+isation parameter ℓ:
+ρ(u) = −8ℓ4 + 27M4 − 32ℓ3M
+8ℓ6
++ α
+�27M4 − 8Mℓ3
+8ℓ6
+�
+,
+(6.31)
+p(s) = −8ℓ4 + 27M4
+ℓ6
+,
+(6.32)
+p⊥ = ℓ − M
+ℓ3
+.
+(6.33)
+For symmetry reasons, the throat is an extremal point (either a local minimum or maxi-
+mum) for these functions.
+Finally, we again focus on the minimal theory and provide plots of the nonzero sources.
+As the analytic expressions make clear, once the coordinate ϱ is employed the dependence
+on ℓ is trivial, since this parameter only enters as a multiplicative factor. For this reason,
+– 21 –
+
+0
+1
+2
+3
+4
+5
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+Figure 5: Energy density and principal pressures, as measured by an observer comoving
+with the æther, for the black-bounce non-singular metric in the minimal theory α = β = 0.
+The black vertical lines mark the UH (solid) and KH (dashed), which may be present or not
+depending on the value of ℓ. Dotted lines signal the position of the mouth for two choices
+of ℓ, corresponding to a hidden (ℓ = 0.25M) and a traversable (ℓ = 2.5M) wormhole
+respectively. Recall that, since ϱ =
+√
+r2 + ℓ2, the region ϱ < ℓ is unphysical and should be
+removed; for this reason it is shaded.
+we decide to plot, in figure 5, the ℓ-independent part only, as a function of ϱ. For refer-
+ence, dotted lines mark the location of the wormhole mouth for two specific choices of ℓ,
+corresponding to a hidden and a traversable wormhole respectively.
+We remind the reader that, although the plot extends to ϱ = 0, min(ϱ) = ℓ. Hence, for
+any given choice ℓ, the region ϱ < ℓ does not belong to the spacetime and should therefore
+be removed: in figure 5 this is rendered by shading. The curves should thus be cut off at
+ϱ = ℓ and joined smoothly with a mirror copy of themselves; moreover, they should be
+multiplied by ℓ2.
+The upshot of this analysis is that the deviations away from vacuum are sizeable only
+in a region close to the mouth, but decay very fast as one moves away from the object.
+Therefore, the physics in the surrounding of the black bounce is well described by the
+equations of vacuum khronometric theory.
+7
+Conclusions
+In this paper, we have presented explicit examples of simply connected and non-connected
+regularisations of an exact static and spherically symmetric black-hole solution in low-
+energy Hoˇrava gravity.
+These configurations, consisting of a non-singular metric and a non-singular æther flow,
+may or may not exhibit universal and/or Killing horizons. To our knowledge, our connected
+– 22 –
+
+configurations represent the first instances of regular black holes and of stars with (anti-)de
+Sitter core in the context of khronometric theory. Similarly, the non-connected configura-
+tions are the first instances, in this context, of hidden and traversable wormholes whose
+æther is not aligned with the Killing vector associated to staticity.
+Our proposals are summarised in eqs. (3.2) and (3.3) (connected) and eqs. (3.6)
+and (3.7) (non-connected). They are not solutions of khronometric theory; however, they
+instantiate all of the few physically viable classes of non-singular end states of gravitational
+collapse: connected regular black holes and horizonless objects, non-connected hidden and
+traversable wormholes. Therefore, if non-projectable Hoˇrava gravity is indeed a UV com-
+pletion of khronometric theory, gravitational collapse in the full theory should produce
+solutions that are qualitatively akin to the configurations that we have hereby described.
+In this frame of mind, the deviations of our configurations from the vacuum of khrono-
+metric theory should be interpreted as arising from the contributions of higher-order oper-
+ators. Checking directly whether this is the case is probably close to impossible. It seems
+important, therefore, to seek indirect ways of validating this conjecture.
+Even within the low-energy theory, however, our proposals present several intriguing
+features. For instance, when a UH is present both the connected and non-connected con-
+figurations represent one-parameter generalisations of the exact solution.
+Notably, this
+additional parameter can be used to trim the surface gravity of the UH, thereby affect-
+ing the thermal properties of the black hole. In particular, it seems possible to construct
+configurations in which the UH is extremal, i.e. its surface gravity vanishes.
+Moreover, connected non-singular black holes necessarily have multiple UHs. Since
+the peeling properties of the inner UH are highly reminiscent of those of inner KHs in GR,
+which are believed to be unstable (see e.g. [59–61] and references therein), it is legitimate
+to wonder whether inner UHs will be subject to a similar (mass inflation) instability in
+spite of the modified dispersion relations associated to Lorentz-breaking matter. We think
+this question deserves further investigation in the future.
+Finally, we note that, both on theoretical as well as phenomenological grounds, the
+Lorentz-violating effects in matter should be suppressed by some energy scale higher than
+the one associated with Lorentz breaking in Hoˇrava gravity (see e.g. [62] or [63] and ref-
+erences therein). This implies that for all practical purposes light rays and test particles,
+typically used for BH phenomenology nowadays, essentially move along geodesics of the
+metric — regardless of the specifics of the modified dispersion relation.
+Previous studies have shown that the geodesics of the Hayward (as well as Bardeen,
+Dymnikova et similia) and black bounce spacetimes are parametrically close to those of the
+Schwarzschild metric. In particular, these metrics typically admit an unstable light ring
+— whose properties are connected, for instance, to the shape and size of electromagnetic
+shadows, or to the frequencies of the longest-lived quasinormal modes — that lies close to
+the one of Schwarzschild. (Moreover, in the horizonless regime they can admit another,
+stable light ring, located inside the unstable light ring or at the wormhole mouth, which
+might be associated to a non-linear instability [64].)
+Therefore, when probed at low energies — e.g. employing very long-baseline interfer-
+ometry, accretion disk spectroscopy, star dynamics or early inspiral gravitational waves
+– 23 –
+
+— these objects represent phenomenologically viable “mimickers” of Schwarzschild black
+holes, similar but not identical to their singular counterparts. The search for signatures of
+these subtle deviations could then mark the dawn of a new channel for quantum gravity
+phenomenology.
+Acknowledgements
+We wish to thank Edgrado Franzin, Enrico Barausse and Mario Herrero-Valea for the
+precious discussions and their comments on an early draft of the manuscript. The au-
+thors acknowledge funding from the Italian Ministry of Education and Scientific Research
+(MIUR) under the grant PRIN MIUR 2017-MB8AEZ.
+A
+Optical scalars
+A coordinate-independent way to characterise the æther congruence is through the optical
+scalars: 4
+expansion
+θ = ∇aua ,
+(A.1)
+shear squared
+σ2
+with
+σab = ∇(aub) − u(aab) ,
+(A.2)
+twist squared
+ω2
+with
+ωab = ∇[buc] − u[aab] .
+(A.3)
+Other interesting scalars are ua χa and aa χa, as they are associated with properties of the
+UHs.
+Since the æther is hypersurface-orthogonal, Frobenius’ theorem implies that the twist
+vanishes. (Note that ω2 ∝ (u[a∇buc])2.) Moreover
+aa χa =
+�
+−a2 = y (ua χa)′ .
+(A.4)
+When evaluated on the Ansatz of eqs. (2.3) and (2.4), one finds
+θ = y′ + 2yR′
+R
+(A.5)
+and a similar, though lengthier, expression for σ2.
+All this quantities thus depend al-
+gebraically on the functions F(r), R(r), A(r) and their first derivatives, in a way that
+renders the following statement manifestly true: when F(r), R(r), A(r) are of class C1
+and bounded, all the scalars introduced above are C1 and bounded. We have computed
+them explicitly for the singular solution and found that they are ill-behaved at the origin;
+and then on the connected and on the non-connected non-singular configurations, checking
+that they are indeed well-behaved everywhere — in particular, at the origin, at the UHs
+and at the KHs.
+4The shear is usually taken to be traceless; our definition is good enough for our purposes.
+– 24 –
+
+B
+2D expansions
+In order to make contact with the arguments of [32], we complement our analysis with a
+discussion on the local characterisation of horizons.
+We start by considering a closed, spacelike 2-surface S 2. The subspace of the tangent
+space that is orthogonal to the tangent space of S 2 is spanned by two vectors that can
+be taken timelike, future-pointing and spacelike, outward-pointing — respectively. In our
+case, a simple choice for S 2 is any sphere centred at the origin. The two vectors are then
+the æther and the vector sa of eq. (6.4) used to define the tangential pressure.
+The induced metric on S 2 is
+hab = gab − uaub + sasb
+(B.1)
+and can be used to define the scalars
+θ(X) = hab∇aXb
+with
+X = {u, s} .
+(B.2)
+These are expansions, but should not be confused with the optical scalar θ, which is defined
+in terms of a three-dimensional transverse metric. θ(u) and θ(s), and in particular their
+signs, determine whether S 2 is a universal (marginally) trapped surface.
+With our Ans¨atze of eqs. (2.3) and (2.4), we have
+θ(u) = 2yR′
+R
+and
+θ(s) = 2(ua χa)R′
+R .
+(B.3)
+Recall that y < 0. Hence, on the singular solution eqs. (2.7) and (2.8), θ(u) is always
+negative, i.e. the future-directed congruence is always converging, while θ(s) has the sign
+of ua χa. Thus, ua χa = 0 marks a universal trapping horizon. Note that both expansions
+diverge as r → 0, meaning that r = 0 is a caustic. Penrose’s theorem then implies that
+this is in fact a singularity, in the sense that the spacetime is not geodesically complete.
+On the simply connected configurations of eqs. (3.2) and (3.3) we still have that θ(u) < 0
+and that θ(s) has the sign of ua χa, but we know that in this case ua χa = 0 has multiple
+roots and in particular it is positive in a neighbourhood of r = 0. Hence there exist multiple
+universal trapping horizons. Further note that in this case r = 0 is not a caustic anymore,
+since θ(u) → 0 and θ(s) → 1 as r → 0.
+In the non-connected configuration the sign of the two expansions depends also on
+R′/R, which is positive in our universe but negative in the other. I.e. both congruences
+vanish and change sign at the wormhole mouth r = 0.
+References
+[1] G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Ann.
+Inst. H. Poincare Phys. Theor. A 20 (1974) 69.
+[2] M.H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B
+266 (1986) 709.
+– 25 –
+
+[3] P. Hoˇrava, Quantum Gravity at a Lifshitz Point, Physical Review D 79 (2009) 084008
+[0901.3775].
+[4] D. Blas, O. Pujolas and S. Sibiryakov, On the Extra Mode and Inconsistency of Hoˇrava
+Gravity, Journal of High Energy Physics 2009 (2009) 029 [0906.3046].
+[5] D. Blas, O. Pujolas and S. Sibiryakov, Consistent extension of Hoˇrava gravity, Physical
+Review Letters 104 (2010) 181302 [0909.3525].
+[6] D. Blas, O. Pujolas and S. Sibiryakov, Models of non-relativistic quantum gravity: The good,
+the bad and the healthy, Journal of High Energy Physics 2011 (2011) 18 [1007.3503].
+[7] J. Bellorin, C. Borquez and B. Droguett, BRST symmetry and unitarity of the Hoˇrava
+theory, 2212.14079.
+[8] J. Bellorin, C. Borquez and B. Droguett, Cancellation of divergences in the nonprojectable
+Hoˇrava theory, Physical Review D 106 (2022) 044055 [2207.08938].
+[9] A.O. Barvinsky, A.V. Kurov and S.M. Sibiryakov, Beta functions of (3 + 1)-dimensional
+projectable Hoˇrava gravity, Physical Review D 105 (2022) 044009 [2110.14688].
+[10] A.O. Barvinsky, M. Herrero-Valea and S.M. Sibiryakov, Towards the renormalization group
+flow of Hoˇrava gravity in (3 + 1) dimensions, Physical Review D 100 (2019) 026012
+[1905.03798].
+[11] A.O. Barvinsky, D. Blas, M. Herrero-Valea, S.M. Sibiryakov and C.F. Steinwachs, Hoˇrava
+gravity is asymptotically free (in 2 + 1 dimensions), Physical Review Letters 119 (2017)
+211301 [1706.06809].
+[12] A.O. Barvinsky, D. Blas, M. Herrero-Valea, S.M. Sibiryakov and C.F. Steinwachs,
+Renormalization of Hoˇrava Gravity, Physical Review D 93 (2016) 064022 [1512.02250].
+[13] T. Jacobson and D. Mattingly, Gravity with a dynamical preferred frame, Physical Review D
+64 (2001) 024028 [gr-qc/0007031].
+[14] T. Jacobson, Einstein-æther gravity: A status report, in Proceedings of From Quantum to
+Emergent Gravity: Theory and Phenomenology — PoS(QG-Ph), vol. 43, p. 020, SISSA
+Medialab, Oct., 2008, DOI.
+[15] T. Jacobson, Extended hoˇrava gravity and einstein-aether theory, Physical Review D 81
+(2010) 101502 [1001.4823].
+[16] T. Jacobson, Undoing the twist: The Hoˇrava limit of Einstein-aether, Physical Review D 89
+(2014) 081501 [1310.5115].
+[17] N. Franchini, M. Herrero-Valea and E. Barausse, The relation between general relativity and
+a class of hoˇrava gravity theories, Physical Review D 103 (2021) 084012 [2103.00929].
+[18] C. Eling and T. Jacobson, Black Holes in Einstein-Aether Theory, Classical and Quantum
+Gravity 23 (2006) 5643 [gr-qc/0604088].
+[19] E. Barausse, T. Jacobson and T.P. Sotiriou, Black holes in Einstein-aether and
+Hoˇrava-Lifshitz gravity, Physical Review D 83 (2011) 124043 [1104.2889].
+[20] D. Blas and S. Sibiryakov, Hoˇrava gravity vs. thermodynamics: The black hole case, Physical
+Review D 84 (2011) 124043 [1110.2195].
+[21] E. Barausse and T.P. Sotiriou, Slowly rotating black holes in Hoˇrava-Lifshitz gravity,
+Physical Review D 87 (2013) 087504 [1212.1334].
+– 26 –
+
+[22] E. Barausse and T.P. Sotiriou, A no-go theorem for slowly rotating black holes in
+Hoˇrava-Lifshitz gravity, Physical Review Letters 109 (2012) 181101 [1207.6370].
+[23] E. Barausse, T.P. Sotiriou and I. Vega, Slowly rotating black holes in Einstein-æther theory,
+Physical Review D 93 (2016) 044044 [1512.05894].
+[24] J. Oost, S. Mukohyama and A. Wang, Spherically symmetric exact vacuum solutions in
+Einstein-aether theory, Universe 7 (2021) 272 [2106.09044].
+[25] M. Bhattacharjee, S. Mukohyama, M.-B. Wan and A. Wang, Gravitational collapse and
+formation of universal horizons in Einstein-æther theory, Phys. Rev. D 98 (2018) 064010
+[1806.00142].
+[26] P. Berglund, J. Bhattacharyya and D. Mattingly, Mechanics of universal horizons, Physical
+Review D 85 (2012) 124019 [1202.4497].
+[27] S. Janiszewski, A. Karch, B. Robinson and D. Sommer, Charged black holes in Hoˇrava
+gravity, Journal of High Energy Physics 2014 (2014) 163 [1401.6479].
+[28] J. Bhattacharyya, M. Colombo and T.P. Sotiriou, Causality and black holes in spacetimes
+with a preferred foliation, Classical and Quantum Gravity 33 (2016) 235003 [1509.01558].
+[29] R. Penrose, Gravitational Collapse and Space-Time Singularities, Physical Review Letters 14
+(1965) 57.
+[30] R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, Geodesically complete black
+holes, Physical Review D 101 (2020) 084047 [1911.11200].
+[31] R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, Opening the Pandora’s box at
+the core of black holes, Classical and Quantum Gravity 37 (2020) 145005 [1908.03261].
+[32] R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, Geodesically complete black holes
+in Lorentz-violating gravity, Journal of High Energy Physics 2022 (2022) 122 [2111.03113].
+[33] J.M. Bardeen, Non-singular general relativistic gravitational collapse, in Proceedings of the
+International Conference GR5, (Tblisi, Georgia), Tbilisi University Press, 1968.
+[34] S. Ansoldi, Spherical black holes with regular center: A review of existing models including a
+recent realization with Gaussian sources, 0802.0330.
+[35] H. Maeda, Quest for realistic non-singular black-hole geometries: Regular-center type,
+Journal of High Energy Physics 2022 (2022) 108 [2107.04791].
+[36] L. Sebastiani and S. Zerbini, Some remarks on non-singular spherically symmetric
+space-times, 2206.03814.
+[37] A. Simpson and M. Visser, Black-bounce to traversable wormhole, Journal of Cosmology and
+Astroparticle Physics 2019 (2019) 042 [1812.07114].
+[38] A. Simpson, P. Martin-Moruno and M. Visser, Vaidya spacetimes, black-bounces, and
+traversable wormholes, Classical and Quantum Gravity 36 (2019) 145007 [1902.04232].
+[39] F.S.N. Lobo, M.E. Rodrigues, M.V.d.S. Silva, A. Simpson and M. Visser, Novel black-bounce
+spacetimes: Wormholes, regularity, energy conditions, and causal structure, Physical Review
+D 103 (2021) 084052 [2009.12057].
+[40] E. Franzin, S. Liberati, J. Mazza, A. Simpson and M. Visser, Charged black-bounce
+spacetimes, Journal of Cosmology and Astroparticle Physics 2021 (2021) 036 [2104.11376].
+– 27 –
+
+[41] K.A. Bronnikov and J.C. Fabris, Regular phantom black holes, Physical Review Letters 96
+(2006) 251101 [gr-qc/0511109].
+[42] K.A. Bronnikov, H. Dehnen and V.N. Melnikov, Regular black holes and black universes,
+General Relativity and Gravitation 39 (2007) 973 [gr-qc/0611022].
+[43] R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, A connection between regular
+black holes and horizonless ultracompact stars, 2211.05817.
+[44] P.O. Mazur and E. Mottola, Gravitational Vacuum Condensate Stars, Proceedings of the
+National Academy of Sciences 101 (2004) 9545 [gr-qc/0407075].
+[45] E. Mottola, The Effective Theory of Gravity and Dynamical Vacuum Energy, Journal of High
+Energy Physics 2022 (2022) 37 [2205.04703].
+[46] G. Lara, M. Herrero-Valea, E. Barausse and S.M. Sibiryakov, Black Holes in
+Ultraviolet-Complete Hoˇrava Gravity, Physical Review D 103 (2021) 104007 [2103.01975].
+[47] C. Eling and T. Jacobson, Spherical Solutions in Einstein-Aether Theory: Static Aether and
+Stars, Classical and Quantum Gravity 23 (2006) 5625 [gr-qc/0603058].
+[48] J. Chojnacki and J. Kwapisz, Finite action principle and Hoˇrava-Lifshitz gravity: Early
+universe, black holes, and wormholes, Physical Review D 104 (2021) 103504 [2102.13556].
+[49] K. Lin and W.-L. Qian, Ellis drainhole solution in Einstein-æther gravity and the axial
+gravitational quasinormal modes, The European Physical Journal C 82 (2022) 529
+[2203.03081].
+[50] F. Del Porro, M. Herrero-Valea, S. Liberati and M. Schneider, Time Orientability and
+Particle Production from Universal Horizons, Physical Review D 105 (2022) 104009
+[2201.03584].
+[51] V.P. Frolov, Notes on non-singular models of black holes, Physical Review D 94 (2016)
+104056 [1609.01758].
+[52] T. Zhou and L. Modesto, Geodesic incompleteness of some popular regular black holes,
+2208.02557.
+[53] C. Lan and Y.-F. Wang, Singularities of regular black holes and the monodromy method for
+asymptotic quasinormal modes, Chinese Physics C (2022) [2205.05935].
+[54] S.A. Hayward, Formation and evaporation of non-singular black holes, Physical Review
+Letters 96 (2006) 031103 [gr-qc/0506126].
+[55] I. Dymnikova, Vacuum nonsingular black hole, General Relativity and Gravitation 24 (1992)
+235.
+[56] M. Visser, Lorentzian Wormholes: From Einstein to Hawking, AIP-Press (1996).
+[57] R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, Causal hierarchy in modified
+gravity, Journal of High Energy Physics 2020 (2020) 55 [2005.08533].
+[58] S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge
+Monographs on Mathematical Physics, Cambridge University Press, Cambridge (1973),
+10.1017/CBO9780511524646.
+[59] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio and M. Visser, On the viability of
+regular black holes, Journal of High Energy Physics 2018 (2018) 23 [1805.02675].
+– 28 –
+
+[60] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio and M. Visser, Inner horizon
+instability and the unstable cores of regular black holes, Journal of High Energy Physics 2021
+(2021) 132 [2101.05006].
+[61] F. Di Filippo, R. Carballo-Rubio, S. Liberati, C. Pacilio and M. Visser, On the Inner
+Horizon Instability of Non-Singular Black Holes, Universe 8 (2022) 204.
+[62] S. Liberati, L. Maccione and T.P. Sotiriou, Scale hierarchy in Hoˇrava-Lifshitz gravity: A
+strong constraint from synchrotron radiation in the Crab nebula, Physical Review Letters 109
+(2012) 151602 [1207.0670].
+[63] S. Liberati, Tests of Lorentz invariance: A 2013 update, Classical and Quantum Gravity 30
+(2013) 133001 [1304.5795].
+[64] P.V.P. Cunha, E. Berti and C.A.R. Herdeiro, Light ring stability in ultra-compact objects,
+Physical Review Letters 119 (2017) 251102 [1708.04211].
+– 29 –
+