Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs

TL;DR

This paper investigates percolation on nonunimodular transitive graphs, proving the existence of multiple phases with infinite clusters, verifying a longstanding conjecture, and establishing mean-field critical behavior at the phase transition.

Contribution

It proves the existence of multiple infinite cluster phases and confirms the mean-field critical exponents for percolation on nonunimodular graphs, extending previous results to broader classes.

Findings

Existence of a non-empty phase with infinite light clusters

Verification of the nonunimodular case of a conjecture by Benjamini and Schramm

Triangle condition holds at criticality, implying mean-field critical exponents

Abstract

We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a non-empty phase in which there are infinite light clusters, which implies the existence of a non-empty phase in which there are infinitely many infinite clusters. That is, we show that $p_c<p_h \leq p_u$ for any such graph. This answers a question of Haggstrom, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which implies that various critical exponents exist and take their mean-field values. All our results apply, for example, to the product $T_k\times \mathbb{Z}^d$ of a $k$-regular tree with $\mathbb{Z}^d$ for $k\geq 3$ and $d \geq 1$, for which these results were previously known only for large $k$. Furthermore, our methods also enable us to establish the basic topological features of the phase diagram for anisotropic percolation on such products, in which tree edges and $\mathbb{Z}^d$ edges are given different retention probabilities. These features had only previously been established for $d=1$, $k$ large.