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.

We consider the nearest-neighbor simple random walk on d , d2, driven by a field of bounded random conductances ω xy 0,1. The conductance law is i.i.d. subject to the condition that the probability of ω xy >0 exceeds the threshold for bond percolation on 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 ω 2n (0,0). We prove that 𝖯 ω 2n (0,0) is bounded by a random constant times n -d/2 in d=2,3, while it is o(n -2 ) in d5 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(n -2 ) bound in d5 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 d , d2, dont les transitions sont données par un champ de conductances aléatoires bornées ω xy 0,1. La loi de conductance est iid sur les arêtes, et telle que la probabilité que ω xy >0 soit supérieure au seuil de percolation (par arêtes) sur 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 ω 2n (0,0). On prouve que 𝖯 ω 2n (0,0) est borné par Cn -d/2 pour d=2,3 (où C est une constante aléatoire) alors que c’est en o(n -2 ) pour d5 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(n -2 ) est optimale pour d5. On parvient également à construire des environnements naturels dépendants de n qui présentent le facteur logn supplémentaire en dimension d=4.

DOI: 10.1214/07-AIHP126
Classification: 60F05, 60J45, 82C41
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] P. Antal and A. Pisztora. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996) 1036-1048. | MR | Zbl

[2] M. T. Barlow. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004) 3024-3084. | MR | Zbl

[3] I. Benjamini and E. Mossel. 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

[4] N. Berger and M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007) 83-120. | MR | Zbl

[5] M. Biskup and T. Prescott. Functional CLT for random walk among bounded conductances. Electron. J. Probab. 12 (2007) 1323-1348. | MR | Zbl

[6] E. A. Carlen, S. Kusuoka and D. W. Stroock. Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 245-287. | Numdam | MR | Zbl

[7] T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 (1999) 181-232. | MR | Zbl

[8] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. 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

[9] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 (1989) 787-855. | MR | Zbl

[10] L. R. G. Fontes and P. Mathieu. On symmetric random walks with random conductances on ℤd. Probab. Theory Related Fields 134 (2006) 565-602. | MR | Zbl

[11] S. Goel, R. Montenegro and P. Tetali. Mixing time bounds via the spectral profile. Electron. J. Probab. 11 (2006) 1-26. | MR | Zbl

[12] A. Grigor'Yan. Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoamericana 10 (1994) 395-452. | MR | Zbl

[13] G. R. Grimmett. Percolation, 2nd edition. Springer, Berlin, 1999. | MR

[14] G. R. Grimmett, H. Kesten and Y. Zhang. Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 (1993) 33-44. | MR | Zbl

[15] D. Heicklen and C. Hoffman. Return probabilities of a simple random walk on percolation clusters. Electron. J. Probab. 10 (2005) 250-302 (electronic). | MR | Zbl

[16] C. Kipnis and S. R. S. Varadhan. 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

[17] B. Morris and Y. Peres. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2005) 245-266. | MR | Zbl

[18] P. Mathieu. Quenched invariance principles for random walks with random conductances. J. Statist. Phys. To appear. | MR

[19] P. Mathieu and A. L. Piatnitski. 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

[20] P. Mathieu and E. Remy. Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004) 100-128. | MR | Zbl

[21] G. Pete. A note on percolation on ℤd: Isoperimetric profile via exponential cluster repulsion. Preprint, 2007. | MR

[22] C. Rau. 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] V. Sidoravicius and A.-S. Sznitman. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2004) 219-244. | MR | Zbl

[24] N. T. Varopoulos. Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63 (1985) 215-239. | MR | Zbl

Cited by Sources: