Is the critical percolation probability local?

TL;DR

This paper investigates whether the critical percolation probability is a local property, demonstrating its proximity to the infinite tree case on certain non-amenable graphs and proving a finite analogue for expanders.

Contribution

It provides evidence supporting Schramm's conjecture on the locality of the critical percolation probability for specific classes of graphs.

Findings

Critical probability on large girth non-amenable graphs is close to that on infinite regular trees.

Finite analogue of the conjecture proved for expander graphs.

Supports the idea that p_c is a local property in these graph classes.

Abstract

We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. This is a special case of a conjecture due to O. Schramm on the locality of p_c. We also prove a finite analogue of the conjecture for expander graphs.