Isoperimetry in supercritical bond percolation in dimensions three and higher

Abstract

We study the isoperimetric subgraphs of the infinite cluster ${\text{C}}_{\infty }$ for supercritical bond percolation on ${\mathbb{Z}}^d$ with $d\ge 3$. Specifically, we consider the subgraphs of ${\text{C}}_{\infty }\cap [-n,n{]}^d$ 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 ${\text{C}}_{\infty }\cap [-n,n{]}^d$. This settles a conjecture of Benjamini for the version of the Cheeger constant defined here.

arXiv comment73 pages, exposition greatly improved, to appear Ann. Inst. H. Poincaré Probab. Statist