On considère la marche aléatoire aux plus proches voisins dans , , dont les transitions sont données par un champ de conductances aléatoires bornées . La loi de conductance est iid sur les arêtes, et telle que la probabilité que soit supérieure au seuil de percolation (par arêtes) sur . 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 . On prouve que est borné par pour (où est une constante aléatoire) alors que c’est en pour et pour . En construisant des exemples dont les noyaux de la chaleur décroissent anormalement en avoisinant , on peut prouver que la borne est optimale pour . On parvient également à construire des environnements naturels dépendants de qui présentent le facteur supplémentaire en dimension .
We consider the nearest-neighbor simple random walk on , , driven by a field of bounded random conductances . The conductance law is i.i.d. subject to the condition that the probability of exceeds the threshold for bond percolation on . For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the -step return probability . We prove that is bounded by a random constant times in , while it is in and in . By producing examples with anomalous heat-kernel decay approaching , we prove that the bound in is the best possible. We also construct natural -dependent environments that exhibit the extra factor in .
Mots clés : heat kernel, random conductance model, random walk, percolation, isoperimetry
@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}, pages = {374--392}, publisher = {Gauthier-Villars}, volume = {44}, number = {2}, year = {2008}, doi = {10.1214/07-AIHP126}, mrnumber = {2446329}, zbl = {1187.60034}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP126/} }
TY - JOUR AU - Berger, N. AU - Biskup, M. AU - Hoffman, C. E. AU - Kozma, G. TI - Anomalous heat-kernel decay for random walk among bounded random conductances JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2008 SP - 374 EP - 392 VL - 44 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP126/ DO - 10.1214/07-AIHP126 LA - en ID - AIHPB_2008__44_2_374_0 ER -
%0 Journal Article %A Berger, N. %A Biskup, M. %A Hoffman, C. E. %A Kozma, G. %T Anomalous heat-kernel decay for random walk among bounded random conductances %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 374-392 %V 44 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP126/ %R 10.1214/07-AIHP126 %G en %F 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/articles/10.1214/07-AIHP126/
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