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The critical percolation probability is local (2310.10983v1)
Published 17 Oct 2023 in math.PR
Abstract: We prove Schramm's locality conjecture for Bernoulli bond percolation on transitive graphs: If $(G_n){n\geq 1}$ is a sequence of infinite vertex-transitive graphs converging locally to a vertex-transitive graph $G$ and $p_c(G_n) \neq 1$ for every $n \geq 1$ then $\lim{n\to\infty} p_c(G_n)=p_c(G)$. Equivalently, the critical probability $p_c$ defines a continuous function on the space $\mathcal{G}*$ of infinite vertex-transitive graphs that are not one-dimensional. As a corollary of the proof, we obtain a new proof that $p_c(G)<1$ for every infinite vertex-transitive graph that is not one-dimensional.