Isoperimetry in two-dimensional percolation

TL;DR

This paper investigates the geometric properties of large isoperimetric sets in the infinite cluster of supercritical bond percolation on the square lattice, characterizing their asymptotic shape and related isoperimetric quantities.

Contribution

It proves the asymptotic shape of large isoperimetric sets in the infinite percolation cluster and confirms a conjecture about their isoperimetric profile and Cheeger constant in the plane.

Findings

Asymptotic shape characterized by a plane isoperimetric problem with a specific norm.

Anchored isoperimetric profile scales to a deterministic limit.

Cheeger constant of the giant component converges to a deterministic value.

Abstract

We consider the unique infinite connected component of supercritical bond percolation on the square lattice and study the geometric properties of isoperimetric sets, i.e., sets with minimal boundary for a given volume. For almost every realization of the infinite connected component we prove that, as the volume of the isoperimetric set tends to infinity, its asymptotic shape can be characterized by an isoperimetric problem in the plane with respect to a particular norm. As an application we then show that the anchored isoperimetric profile with respect to a given point as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the plane.