Percolation and isoperimetry on roughly transitive graphs

TL;DR

This paper proves the existence of a percolation phase (p_c < 1) on roughly transitive graphs with polynomial growth and isoperimetric dimension greater than one, using a robust probabilistic renormalization approach.

Contribution

It establishes p_c < 1 for a broad class of graphs, including dependent percolation, without relying on Gromov's theorem, and provides new probabilistic insights even for transitive graphs.

Findings

p_c < 1 for roughly transitive graphs with polynomial growth

Applicable to both dependent and independent percolation

Provides examples of dependent percolation models fitting the results

Abstract

In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent percolation processes, since they are based on a quite robust renormalization technique. When G is transitive, the fact that p_c < 1 was already known before. But even in that case our proof yields some new results and it is entirely probabilistic, not involving the use of Gromov's theorem on groups of polynomial growth. We finish the paper giving some examples of dependent percolation for which our results apply.