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

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.

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.

DOI : 10.1214/07-AIHP126
Classification : 60F05, 60J45, 82C41
Mots clés : heat kernel, random conductance model, random walk, percolation, isoperimetry
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     title = {Anomalous heat-kernel decay for random walk among bounded random conductances},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {374--392},
     publisher = {Gauthier-Villars},
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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, Tome 44 (2008) no. 2, pp. 374-392. doi : 10.1214/07-AIHP126. http://www.numdam.org/articles/10.1214/07-AIHP126/

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