Locality of percolation critical probabilities: uniformly nonamenable case

Abstract

Let $\{G_n{\}}_{n=1}^{\infty }$ be a sequence of transitive infinite connected graphs with $\underset{n\ge 1}{\sup }{p}_c(G_n)<1,$ where each $p_c(G_n)$ is bond percolation critical probability on $G_n.$ Schramm (2008) conjectured that if $G_n$ converges locally to a transitive infinite connected graph $G,$ then $p_c(G_n)\to p_c(G)$ as $n\to \infty .$ We prove the conjecture when $G$ satisfies two rough uniformities, and $\{G_n{\}}_{n=1}^{\infty }$ is uniformly nonamenable.

References in corpus (1)

• Richard Thompson group F is not amenable