Percolation critical probabilities of matching lattice-pairs

TL;DR

This paper establishes a precise condition under which the critical percolation probability of a graph is strictly less than that of its matching graph, extending understanding of percolation thresholds in Euclidean and hyperbolic geometries.

Contribution

It provides a necessary and sufficient condition for the strict inequality of critical probabilities between a graph and its matching graph in non-Euclidean spaces, generalizing previous results.

Findings

Strict inequality holds iff the graph is not a triangulation.

The result applies to quasi-transitive, plane graphs in Euclidean or hyperbolic space.

It complements prior work on the sum of critical probabilities for a graph and its matching graph.

Abstract

A necessary and sufficient condition is established for the strict inequality $p_c(G_*)<p_c(G)$ between the critical probabilities of site percolation on a quasi-transitive, plane graph $G$ and on its matching graph $G_*$. It is assumed that $G$ is properly embedded in either the Euclidean or the hyperbolic plane. When $G$ is transitive, strict inequality holds if and only if $G$ is not a triangulation. The basic approach is the standard method of enhancements, but its implemention has complexity arising from the non-Euclidean (hyperbolic) space, the study of site (rather than bond) percolation, and the generality of the assumption of quasi-transitivity. This result is complementary to the work of the authors ("Hyperbolic site percolation", arXiv:2203.00981) on the equality $p_u(G) + p_c(G_*) = 1$, where $p_u$ is the critical probability for the existence of a unique infinite open cluster. It implies for transitive $G$ that $p_u(G) + p_c(G) \ge 1$, with equality if and only if $G$ is a triangulation.