Locality of percolation for graphs with polynomial growth

Abstract

Schramm’s Locality Conjecture asserts that the value of the critical parameter $p_c$ of a graph satisfying $p_c<1$ depends only on its local structure. In this paper, we prove this conjecture in the particular case of transitive graphs with polynomial growth. Our proof relies on two recent works about such graphs, namely supercritical sharpness of percolation by the same authors and a finitary structure theorem by Tessera and Tointon.