Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry

TL;DR

This paper demonstrates that in transitive graphs with polynomial growth, supercritical percolation properties are local and can be inferred from similar graphs with matching local structure, establishing analyticity and sharpness results.

Contribution

It proves locality of supercritical percolation behavior on polynomial growth graphs and introduces new connectivity results for minimal cutsets, answering a longstanding question.

Findings

Percolation quantities are locally determined on similar graphs.

The percolation probability function is analytic in the supercritical regime.

New connectivity results for minimal cutsets are established.

Abstract

We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying $p_c(\mathscr{G})<1$ and take $p>p_c(\mathscr{G})$. Let $\mathscr{H}$ be another such graph and assume that $\mathscr{G}$ and $\mathscr{H}$ have the same ball of radius $r$ for $r$ large. We prove that various quantities regarding percolation of parameter close to $p$ on $\mathscr{H}$ can be well understood from $(\mathscr{G},p)$ alone. This includes uniform versions of supercritical sharpness as well as the Kesten-Zhang bound on the probability of observing a large finite cluster: the constants involved can be chosen to depend only on $(\mathscr{G},p)$. We also prove that $\theta_\mathscr{H}$ is an analytic function of $p$ in the whole supercritical regime and that, for a suitable $\varepsilon=\varepsilon(\mathscr{G},p)>0$, the analytic extension of $\theta_\mathscr{H}$ to the $\varepsilon$-neighbourhood of $p$ in $\mathbb C$ is, uniformly, well approximated by the analytic extension of $\theta_\mathscr{G}$. The proof relies on new results on the connectivity of minimal cutsets; in particular, we answer a question asked by Babson and Benjamini in 1999. We further discuss connections with the conjecture of non-percolation at criticality.