arxiv:2304.01431

Planar site percolation on semi-transitive graphs

Published on Apr 4, 2023

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Abstract

For semi-transitive graphs with specific embedding properties, the sum of the site percolation threshold and that of their matching graphs equals one.

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Semi-transitive graphs, defined in hps98 as examples where ``uniform percolation" holds whenever p>p_c, are a large class of graphs more general than quasi-transitive graphs. Let G be a semi-transitive graph with one end which can be properly embedded into the plane with uniformly bounded face degree for finite faces and minimal vertex degree at least 7. We show that p_u^{site}(G) +p_c^{site}(G_*)=1, where G_* denotes the matching graph of G. This fulfils and extends an observation of Sykes and Essam in 1964 (SE64) to semi-transitive graphs.

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