Spread-out percolation on transitive graphs of polynomial growth

2024/04/26 by Spanos, Panagiotis, Tointon, Matthew · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.2404.17262

Abstract

Let G be a vertex-transitive graph of superlinear polynomial growth. Given r>0, let G_r be the graph on the same vertex set as G, with two vertices joined by an edge if and only if they are at graph distance at most r apart in G. We show that the critical probability p_c(G_r) for Bernoulli bond percolation on G_r satisfies p_c(G_r) ∼ 1/deg(G_r) as r→∞. This extends work of Penrose and Bollobás-Janson-Riordan, who considered the case G=ℤ^d. Our result provides an important ingredient in parallel work of Georgakopoulos in which he introduces a new notion of dimension in groups. It also verifies a special case of a conjecture of Easo and Hutchcroft.

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