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

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 374-392

We consider the nearest-neighbor simple random walk on ${\mathbb{Z}}^d$, $d\ge 2$, 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 ${\mathbb{Z}}^d$. 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 ${𝖯}_{\omega }^{2n}(0,0)$. We prove that ${𝖯}_{\omega }^{2n}(0,0)$ 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 $O(n^{-2}logn)$ 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 $d=4$.

On considère la marche aléatoire aux plus proches voisins dans ${\mathbb{Z}}^d$, $d\ge 2$, 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 ${\mathbb{Z}}^d$. 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 $2n:{𝖯}_{\omega }^{2n}(0,0)$. On prouve que ${𝖯}_{\omega }^{2n}(0,0)$ est borné par $C{n}^{-d/2}$ pour $d=2,3$ (où $C$ est une constante aléatoire) alors que c’est en $o\left(n^{-2}\right)$ pour $d\ge 5$ 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 $o\left(n^{-2}\right)$ 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 $d=4$.

Classification : 60F05, 60J45, 82C41
Keywords: heat kernel, random conductance model, random walk, percolation, isoperimetry

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