Effective resistances for supercritical percolation clusters in boxes

Abe, Yoshihiro

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 935-946.

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

Let ${𝒞}^n$ 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 ${𝒞}^n$. As an application we show that the cover time for the simple random walk on ${𝒞}^n$ is comparable to $n^d{(logn)}^2$. Noting that the cover time for the simple random walk on ${[-n,n]}^d\cap {\mathbb{Z}}^d$ is of order $n^{d}logn$ for $d\ge 3$ (and of order $n^2{(logn)}^2$ for $d=2$), this gives a quantitative difference between the two random walks for $d\ge 3$.

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