On percolation critical probabilities and unimodular random graphs

Abstract

We investigate generalizations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\tilde{p} _c$ defined by Duminil-Copin and Tassion [11] 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:

• ${p_c}={\tilde{p} _c}$ 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.

• We give conditions which imply $\lim{p_c} (G_n)= {p_c}(\lim G_n)$.

• 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 $\mathcal{T} _n$ of large girth bi-Lipschitz invariant subgraphs such that ${p_c}(\mathcal{T} _n)\to 1$. It remains open whether this holds whenever the transitive graph has cost 1.