Locality of percolation for graphs with polynomial growth

TL;DR

This paper proves Schramm's Locality Conjecture for transitive graphs with polynomial growth, showing that the critical percolation threshold depends solely on local graph structure.

Contribution

It confirms the conjecture for a broad class of graphs by leveraging recent results on supercritical sharpness and a finitary structure theorem.

Findings

Proves locality of percolation threshold for polynomial growth graphs

Utilizes recent advances in supercritical percolation analysis

Establishes a connection between local structure and critical percolation value

Abstract

Schramm's Locality Conjecture asserts that the value of the critical percolation parameter $p_c$ of a graph satisfying $p_c<1$ depends only on its local structure. In this note, 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.