Effective resistances for supercritical percolation clusters in boxes
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 d avec d2. 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 d est de l’ordre de n d logn quand d3 (et de n 2 (logn) 2 quand d=2), ceci montre une différence quantitative entre les deux marches si d3.

Let 𝒞 n be the largest open cluster for supercritical Bernoulli bond percolation in [-n,n] d d with d2. 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 d is of order n d logn for d3 (and of order n 2 (logn) 2 for d=2), this gives a quantitative difference between the two random walks for d3.

DOI : https://doi.org/10.1214/14-AIHP604
Mots clés : effective resistances, simple random walks, cover times, gaussian free fields, supercritical percolation
@article{AIHPB_2015__51_3_935_0,
     author = {Abe, Yoshihiro},
     title = {Effective resistances for supercritical percolation clusters in boxes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {935--946},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {3},
     year = {2015},
     doi = {10.1214/14-AIHP604},
     mrnumber = {3365968},
     language = {en},
     url = {http://archive.numdam.org/item/AIHPB_2015__51_3_935_0/}
}
Abe, Yoshihiro. Effective resistances for supercritical percolation clusters in boxes. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 935-946. doi : 10.1214/14-AIHP604. http://archive.numdam.org/item/AIHPB_2015__51_3_935_0/

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