Concentration estimates for the isoperimetric constant of the super critical percolation cluster

TL;DR

This paper investigates the variability of the Cheeger constant in supercritical percolation clusters, establishing bounds on its variance that depend on the dimension and percolation parameter.

Contribution

It provides a novel variance bound for the Cheeger constant of the giant component in supercritical percolation, highlighting concentration properties.

Findings

Variance of the Cheeger constant is bounded by a constant over n^d.

The bound depends only on the dimension and percolation parameter.

Results contribute to understanding geometric properties of percolation clusters.

Abstract

We consider the Cheeger constant $\phi(n)$ of the giant component of supercritical bond percolation on $\mathbb{Z}^d/n\mathbb{Z}^d$. We show that the variance of $\phi(n)$ is bounded by $\frac{\xi}{n^d}$, where $\xi$ is a positive constant that depends only on the dimension $d$ and the percolation parameter.