Anomalous heat-kernel decay for random walk among bounded random conductances
Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 2, p. 374-392

We consider the nearest-neighbor simple random walk on d , d2, driven by a field of bounded random conductances ω xy 0,1. The conductance law is i.i.d. subject to the condition that the probability of ω xy >0 exceeds the threshold for bond percolation on d . For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the 2n-step return probability 𝖯 ω 2n (0,0). We prove that 𝖯 ω 2n (0,0) is bounded by a random constant times n -d/2 in d=2,3, while it is o(n -2 ) in d5 and O(n -2 logn) in d=4. By producing examples with anomalous heat-kernel decay approaching 1/n 2 , we prove that the o(n -2 ) bound in d5 is the best possible. We also construct natural n-dependent environments that exhibit the extra logn factor in d=4.

On considère la marche aléatoire aux plus proches voisins dans d , d2, dont les transitions sont données par un champ de conductances aléatoires bornées ω xy 0,1. La loi de conductance est iid sur les arêtes, et telle que la probabilité que ω xy >0 soit supérieure au seuil de percolation (par arêtes) sur d . Pour les environnements dont l’origine est connectée à l’infini à l’aide d’arêtes à conductances positives, on étudie l’asymptotique de la probabilité de retour à l’instant 2n:𝖯 ω 2n (0,0). On prouve que 𝖯 ω 2n (0,0) est borné par Cn -d/2 pour d=2,3 (où C est une constante aléatoire) alors que c’est en o(n -2 ) pour d5 et O(n -2 logn) pour d=4. En construisant des exemples dont les noyaux de la chaleur décroissent anormalement en avoisinant 1/n 2 , on peut prouver que la borne o(n -2 ) est optimale pour d5. On parvient également à construire des environnements naturels dépendants de n qui présentent le facteur logn supplémentaire en dimension d=4.

DOI : https://doi.org/10.1214/07-AIHP126
Classification:  60F05,  60J45,  82C41
Keywords: heat kernel, random conductance model, random walk, percolation, isoperimetry
@article{AIHPB_2008__44_2_374_0,
     author = {Berger, N. and Biskup, M. and Hoffman, C. E. and Kozma, G.},
     title = {Anomalous heat-kernel decay for random walk among bounded random conductances},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {2},
     year = {2008},
     pages = {374-392},
     doi = {10.1214/07-AIHP126},
     zbl = {1187.60034},
     mrnumber = {2446329},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2008__44_2_374_0}
}
Berger, N.; Biskup, M.; Hoffman, C. E.; Kozma, G. Anomalous heat-kernel decay for random walk among bounded random conductances. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 2, pp. 374-392. doi : 10.1214/07-AIHP126. http://www.numdam.org/item/AIHPB_2008__44_2_374_0/

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