Effective resistances for supercritical percolation clusters in boxes

TL;DR

This paper provides a precise estimate of the effective resistance in supercritical percolation clusters within boxes and demonstrates how this influences the cover time of random walks, revealing differences from standard lattice behavior.

Contribution

It offers a sharp estimate for the effective resistance in supercritical percolation clusters and analyzes its impact on the cover time of simple random walks.

Findings

Effective resistance estimate for supercritical clusters

Cover time on clusters is comparable to $n^d ( ext{log } n)^2$

Difference in cover time behavior compared to full lattice for $d \,\geq\, 3$

Abstract

Let $\mathcal{C}^n$ be the largest open cluster for supercritical Bernoulli bond percolation in $[-n, n]^d \cap \mathbb{Z}^d$ with $d \ge 2$. We obtain a sharp estimate for the effective resistance on $\mathcal{C}^n$. As an application we show that the cover time for the simple random walk on $\mathcal{C}^n$ is comparable to $n^d (\log n)^2$. Noting that the cover time for the simple random walk on $[-n, n]^d \cap \mathbb{Z}^d$ is of order $n^d \log n$ for $d \ge 3$ (and of order $n^2 (\log n)^2$ for $d = 2$), this gives a quantitative difference between the two random walks for $d \ge 3$.