Anomalous heat-kernel decay for random walk among bounded random conductances

Berger, N.; Biskup, M.; Hoffman, C. E.; Kozma, G.

Berger, N. ; Biskup, M. ; Hoffman, C. E. ; Kozma, G.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 374-392 / Harvested from Numdam

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Résumé
Abstract

On considère la marche aléatoire aux plus proches voisins dans $ℤ^{d}$ , $d \geq 2$ , dont les transitions sont données par un champ de conductances aléatoires bornées $ω_{x y} \in [0, 1]$ . La loi de conductance est iid sur les arêtes, et telle que la probabilité que $ω_{x y} > 0$ soit supérieure au seuil de percolation (par arêtes) sur $ℤ^{d}$ . 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 $2 n : 𝖯_{ω}^{2 n} (0, 0)$ . On prouve que $𝖯_{ω}^{2 n} (0, 0)$ est borné par $C n^{- d / 2}$ pour $d = 2, 3$ (où $C$ est une constante aléatoire) alors que c’est en $o (n^{- 2})$ pour $d \geq 5$ et $O (n^{- 2} log n)$ 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 $d \geq 5$ . On parvient également à construire des environnements naturels dépendants de $n$ qui présentent le facteur $log n$ supplémentaire en dimension $d = 4$ .

We consider the nearest-neighbor simple random walk on $ℤ^{d}$ , $d \geq 2$ , driven by a field of bounded random conductances $ω_{x y} \in [0, 1]$ . The conductance law is i.i.d. subject to the condition that the probability of $ω_{x y} > 0$ exceeds the threshold for bond percolation on $ℤ^{d}$ . For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the $2 n$ -step return probability $𝖯_{ω}^{2 n} (0, 0)$ . We prove that $𝖯_{ω}^{2 n} (0, 0)$ is bounded by a random constant times $n^{- d / 2}$ in $d = 2, 3$ , while it is $o (n^{- 2})$ in $d \geq 5$ and $O (n^{- 2} log n)$ in $d = 4$ . By producing examples with anomalous heat-kernel decay approaching $1 / n^{2}$ , we prove that the $o (n^{- 2})$ bound in $d \geq 5$ is the best possible. We also construct natural $n$ -dependent environments that exhibit the extra $log n$ factor in $d = 4$ .

Publié le : 2008-01-01

MR 2446329 | Zbl 1187.60034 | 3 citations dans Numdam

DOI : https://doi.org/10.1214/07-AIHP126
Classification: 60F05, 60J45, 82C41

@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},
     volume = {44},
     year = {2008},
     pages = {374-392},
     doi = {10.1214/07-AIHP126},
     mrnumber = {2446329},
     zbl = {1187.60034},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 44 (2008) pp. 374-392. doi : 10.1214/07-AIHP126. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_2_374_0/

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