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 .
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 .
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://archive.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://archive.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://archive.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, Volume 44 (2008) no. 2, pp. 374-392. doi : 10.1214/07-AIHP126. http://archive.numdam.org/articles/10.1214/07-AIHP126/
[1] On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996) 1036-1048. | MR | Zbl
and .[2] Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004) 3024-3084. | MR | Zbl
.[3] On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Related Fields 125 (2003) 408-420. | MR | Zbl
and .[4] Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007) 83-120. | MR | Zbl
and .[5] Functional CLT for random walk among bounded conductances. Electron. J. Probab. 12 (2007) 1323-1348. | MR | Zbl
and .[6] Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 245-287. | Numdam | MR | Zbl
, and .[7] Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 (1999) 181-232. | MR | Zbl
.[8] Invariance principle for reversible Markov processes with application to diffusion in the percolation regime. In Particle Systems, Random Media and Large Deviations (Brunswick, Maine) 71-85. Contemp. Math. 41. Amer. Math. Soc., Providence, RI, 1985. | MR | Zbl
, , and .[9] An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 (1989) 787-855. | MR | Zbl
, , and .[10] On symmetric random walks with random conductances on ℤd. Probab. Theory Related Fields 134 (2006) 565-602. | MR | Zbl
and .[11] Mixing time bounds via the spectral profile. Electron. J. Probab. 11 (2006) 1-26. | MR | Zbl
, and .[12] Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoamericana 10 (1994) 395-452. | MR | Zbl
.[13] Percolation, 2nd edition. Springer, Berlin, 1999. | MR
.[14] Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 (1993) 33-44. | MR | Zbl
, and .[15] Return probabilities of a simple random walk on percolation clusters. Electron. J. Probab. 10 (2005) 250-302 (electronic). | MR | Zbl
and .[16] A central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104 (1986) 1-19. | MR | Zbl
and .[17] Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2005) 245-266. | MR | Zbl
and .[18] Quenched invariance principles for random walks with random conductances. J. Statist. Phys. To appear. | MR
.[19] Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007) 2287-2307. | MR | Zbl
and .[20] Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004) 100-128. | MR | Zbl
and .[21] A note on percolation on ℤd: Isoperimetric profile via exponential cluster repulsion. Preprint, 2007. | MR
.[22] Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation, Bull. Soc. Math. France. To appear. | Numdam | MR | Zbl
.[23] Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2004) 219-244. | MR | Zbl
and .[24] Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63 (1985) 215-239. | MR | Zbl
.Cited by Sources: