Counting minimal cutsets and $p_c<1$

TL;DR

The paper characterizes when percolation on an infinite graph has a uniformly percolating phase by linking it to the growth rate of minimal cutsets: it proves ${p_c^*(G)<1}$ if and only if ${\kappa(G)<\infty}$, where ${\kappa(G)=\sup_{n\ge1} q_n^{1/n}}$ and ${q_n}$ counts minimal cutsets of size ${n}$ from a vertex to infinity. It also shows that every uniformly transient graph satisfies ${\kappa(G)<\infty}$, yielding ${p_c^*(G)<1}$ for such graphs, and extends these ideas to transitive graphs with superlinear growth, anchored isoperimetric inequalities, and related percolation bounds. The authors develop a Peierls-type framework combined with positive association, a novel Markov-chain covering lemma, and a Gaussian free field (GFF) approach to bound the occurrence of prescribed cutsets, providing both combinatorial and probabilistic routes to the main results. These findings unify geometric, probabilistic, and analytic perspectives on ${p_c<1}$ for broad graph classes and offer new tools for analyzing giant clusters and percolation phases, with potential extensions to related models like the Ising model.

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.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 3
• proof
• Lemma 3
• proof