Counting minimal cutsets and $p_c<1$

TL;DR

This paper proves that for certain graphs, if the percolation critical probability is less than one, then the number of minimal cutsets grows at most exponentially, and it confirms that all uniformly transient graphs have this property.

Contribution

It establishes the converse of the Peierls argument for percolation and proves that all uniformly transient graphs have critical probability less than one.

Findings

Bound on minimal cutsets for p_c<1

p_c<1 for all uniformly transient graphs

New proof for superlinear growth transitive graphs

Abstract

We prove two results concerning percolation on general graphs. - We establish the converse of the classical Peierls argument: if the critical parameter for (uniform) percolation satisfies $p_c<1$, then the number of minimal cutsets of size $n$ separating a given vertex from infinity is bounded above exponentially in $n$. This resolves a conjecture of Babson and Benjamini from 1999. - We prove that $p_c<1$ for every uniformly transient graph. This solves a problem raised by Duminil-Copin, Goswami, Raoufi, Severo and Yadin, and provides a new proof that $p_c<1$ for every transitive graph of superlinear growth.