Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two

TL;DR

This paper investigates the minimal boundary-to-volume ratio subgraphs of the giant component in supercritical bond percolation on a 2D lattice, showing convergence to continuum optimizers and confirming a conjecture about the Cheeger constant.

Contribution

It establishes the almost sure convergence of isoperimetric subgraphs to continuum optimizers and proves the Cheeger constant scales to a deterministic isoperimetric ratio in 2D.

Findings

Convergence of rescaled isoperimetric subgraphs to continuum optimizers.

The Cheeger constant converges to a deterministic isoperimetric ratio.

Settles a conjecture of Benjamini in dimension two.

Abstract

We study the isoperimetric subgraphs of the giant component $\textbf{C}_n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\textbf{C}_n$ having minimal edge boundary to volume ratio. In contrast to the work of Biskup, Louidor, Procaccia and Rosenthal, the edge boundary is taken only within $\textbf{C}_n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constant of $\textbf{C}_n$ scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.