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, p. 374-392
On considère la marche aléatoire aux plus proches voisins dans ${ℤ}^{d}$, $d\ge 2$, dont les transitions sont données par un champ de conductances aléatoires bornées ${\omega }_{xy}\in \left[0,1\right]$. La loi de conductance est iid sur les arêtes, et telle que la probabilité que ${\omega }_{xy}>0$ 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 $2n:{𝖯}_{\omega }^{2n}\left(0,0\right)$. On prouve que ${𝖯}_{\omega }^{2n}\left(0,0\right)$ est borné par $C{n}^{-d/2}$ pour $d=2,3$ (où $C$ est une constante aléatoire) alors que c’est en $o\left({n}^{-2}\right)$ pour $d\ge 5$ et $O\left({n}^{-2}logn\right)$ 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 $o\left({n}^{-2}\right)$ est optimale pour $d\ge 5$. 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 ${ℤ}^{d}$, $d\ge 2$, driven by a field of bounded random conductances ${\omega }_{xy}\in \left[0,1\right]$. The conductance law is i.i.d. subject to the condition that the probability of ${\omega }_{xy}>0$ 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 $2n$-step return probability ${𝖯}_{\omega }^{2n}\left(0,0\right)$. We prove that ${𝖯}_{\omega }^{2n}\left(0,0\right)$ is bounded by a random constant times ${n}^{-d/2}$ in $d=2,3$, while it is $o\left({n}^{-2}\right)$ in $d\ge 5$ and $O\left({n}^{-2}logn\right)$ in $d=4$. By producing examples with anomalous heat-kernel decay approaching $1/{n}^{2}$, we prove that the $o\left({n}^{-2}\right)$ bound in $d\ge 5$ is the best possible. We also construct natural $n$-dependent environments that exhibit the extra $logn$ factor in $d=4$.
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},
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, Tome 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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