Locality of percolation for Abelian Cayley graphs

Abstract

We prove that the value of the critical probability for percolation on an Abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function $\mathrm{p}_{\mathrm{c}}$ defined on the set of Cayley graphs of Abelian groups of rank at least $2$ is continuous for the Benjamini–Schramm topology. The proof involves group-theoretic tools and a new block argument.