On percolation critical probabilities and unimodular random graphs

TL;DR

This paper explores percolation critical probabilities on unimodular random graphs, establishing new relationships, counterexamples, and conditions for convergence, thereby advancing understanding of phase transitions in complex graph structures.

Contribution

It generalizes classical percolation thresholds to unimodular graphs, proves equality of certain critical probabilities, and provides counterexamples and convergence conditions.

Findings

$p_c = ilde{p_c}$ for bounded degree unimodular graphs

Existence of unimodular graphs with $p_T < p_c$ and sub-exponential growth

Counterexamples where $p_c(G_n)$ does not converge to $p_c(G)$

Abstract

We investigate generalisations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\tilde{p_c}$ defined by Duminil-Copin and Tassion (2015) to bounded degree unimodular random graphs. We further examine Schramm's conjecture in the case of unimodular random graphs: does $p_c(G_n)$ converge to $p_c(G)$ if $G_n\to G$ in the local weak sense? Among our results are the following: 1. $p_c=\tilde{p_c}$ holds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and $p_T < p_c$; i.e., the classical sharpness of phase transition does not hold. 2. We give conditions which imply $\lim p_c(G_n) = p_c(\lim G_n)$. 3. There are sequences of unimodular graphs such that $G_n\to G$ but $p_c(G)>\lim p_c(G_n)$ or $p_c(G)<\lim p_c(G_n)<1$. As a corollary to our positive results, we show that for any transitive graph with sub-exponential volume growth there is a sequence $T_n$ of large girth bi-Lipschitz invariant subgraphs such that $p_c(T_n)\to 1$. It remains open whether this holds whenever the transitive graph has cost 1.