Isoperimetry in supercritical bond percolation in dimensions three and higher

TL;DR

This paper proves a shape theorem for isoperimetric subgraphs in supercritical bond percolation on high-dimensional lattices, showing they converge to a deterministic shape and providing sharp asymptotics for a Cheeger constant variant.

Contribution

It establishes a shape theorem for isoperimetric sets in supercritical percolation and confirms a conjecture about the Cheeger constant in this context.

Findings

Almost sure convergence of rescaled isoperimetric subgraphs to a deterministic shape

Construction of a norm defining the limiting shape

Sharp asymptotics for a modified Cheeger constant

Abstract

We study the isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ for supercritical bond percolation on $\mathbb{Z}^d$ with $d\geq 3$. Specifically, we consider the subgraphs of $\textbf{C}_\infty \cap [-n,n]^d$ which have minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs, obtaining that when suitably rescaled, these subgraphs converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of the Cheeger constant for $\textbf{C}_\infty \cap [-n,n]^d$. This settles a conjecture of Benjamini for the version of the Cheeger constant defined here.