Abstract:

Around 2008, Schramm conjectured that the critical probability p_c of a transitive graph is entirely determined by the local geometry of the graph, subject to the global constraint that p_c < 1. In other words, if G_n is a sequence of transitive graphs with p_c(G_n) < 1 for all n converging locally to a transitive graph G then p_c(G_n) converges to p_c(G). Previous works had verified the conjecture in various special cases, including nonamenable graphs of high girth (Benjamini, Nachmias and Peres 2012); Cayley graphs of abelian groups (Martineau and Tassion 2013); nonunimodular graphs (Hutchcroft 2017 and 2018); graphs of uniform exponential growth (Hutchcroft 2018); and graphs of (automatically uniform) polynomial growth (Contreras, Martineau and Tassion 2022). In this talk I will describe our complete resolution of the conjecture in forthcoming joint work with Hutchcroft

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