Effective resistances for supercritical percolation clusters in boxes

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 $d\ge 3$.

17 pages, 4 figures

References in corpus (3)

• Random walks on supercritical percolation clusters
• Quenched invariance principles for random walks and elliptic diffusions in random media with boundary
• Cover times for sequences of reversible Markov chains on random graphs