The critical percolation probability is local
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.