Effective resistances for supercritical percolation clusters in boxes

Abe, Yoshihiro

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 935-946 / Harvested from Numdam

Résumé

On considère la percolation de Bernoulli par arêtes dans le régime surcritique. Soit $\mathcal {C}^{n}$ le plus grand amas de percolation dans $[-n,n]^{d}\cap \mathbb {Z}^{d}$ avec $d\ge 2$. Nous obtenons une estimation précise de la résistance effective sur $\mathcal {C}^{n}$. Comme application, nous montrons que le temps de recouvrement d’une marche simple sur $\mathcal {C}^{n}$ est de l’ordre de $n^{d}(\log n)^{2}$. En remarquant que le temps de recouvrement d’une marche simple sur $[-n,n]^{d}\cap \mathbb {Z}^{d}$ est de l’ordre de $n^{d}\log n$ quand $d\ge 3$ (et de $n^{2}(\log n)^{2}$ quand $d=2$), ceci montre une différence quantitative entre les deux marches si $d\ge 3$.

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 $d\ge 3$.

Détail

Publié le : 2015-01-01

MR 3365968

DOI : https://doi.org/10.1214/14-AIHP604

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