Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

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

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Abstract

We consider the nearest-neighbor simple random walk on ℤd, d≥2, driven by a field of bounded random conductances ωxy∈[0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy>0 exceeds the threshold for bond percolation on ℤd. 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 $\mathsf{P}_{\omega}^{2n}(0,0)$. We prove that $\mathsf{P}_{\omega}^{2n}(0,0)$ is bounded by a random constant times nd/2 in d=2, 3, while it is o(n−2) in d≥5 and O(n−2log n) in d=4. By producing examples with anomalous heat-kernel decay approaching 1/n2, we prove that the o(n−2) bound in d≥5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d=4.

Résumé

On considère la marche aléatoire aux plus proches voisins dans ℤd, d≥2, dont les transitions sont données par un champ de conductances aléatoires bornées ωxy∈[0, 1]. La loi de conductance est iid sur les arêtes, et telle que la probabilité que ωxy>0 soit supérieure au seuil de percolation (par arêtes) sur ℤd. 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 : $\mathsf{P}_{\omega}^{2n}(0,0)$. On prouve que $\mathsf{P}_{\omega}^{2n}(0,0)$ est borné par Cnd/2 pour d=2, 3 (où C est une constante aléatoire) alors que c’est en o(n−2) pour d≥5 et O(n−2log n) pour d=4. En construisant des exemples dont les noyaux de la chaleur décroissent anormalement en avoisinant 1/n2, on peut prouver que la borne o(n−2) est optimale pour d≥5. On parvient également à construire des environnements naturels dépendants de n qui présentent le facteur log n supplémentaire en dimension d=4.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 44, Number 2 (2008), 374-392.

Dates
First available in Project Euclid: 11 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1207948225

Digital Object Identifier
doi:10.1214/07-AIHP126

Mathematical Reviews number (MathSciNet)
MR2446329

Zentralblatt MATH identifier
1187.60034

Subjects
Primary: 60F05: Central limit and other weak theorems 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Heat kernel Random conductance model Random walk Percolation Isoperimetry

Citation

Berger, N.; Biskup, M.; Hoffman, C. E.; Kozma, G. Anomalous heat-kernel decay for random walk among bounded random conductances. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 2, 374--392. doi:10.1214/07-AIHP126. https://projecteuclid.org/euclid.aihp/1207948225


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