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

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

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 2, page 374-392
  • ISSN: 0246-0203

Abstract

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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 𝖯 ω 2 n ( 0 , 0 ) . We prove that 𝖯 ω 2 n ( 0 , 0 ) is bounded by a random constant timesn−d/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.

How to cite

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Berger, N., et al. "Anomalous heat-kernel decay for random walk among bounded random conductances." Annales de l'I.H.P. Probabilités et statistiques 44.2 (2008): 374-392. <http://eudml.org/doc/77975>.

@article{Berger2008,
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&gt;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 timesn−d/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.},
author = {Berger, N., Biskup, M., Hoffman, C. E., Kozma, G.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {heat kernel; random conductance model; random walk; percolation; isoperimetry; simple random walk; random environments},
language = {eng},
number = {2},
pages = {374-392},
publisher = {Gauthier-Villars},
title = {Anomalous heat-kernel decay for random walk among bounded random conductances},
url = {http://eudml.org/doc/77975},
volume = {44},
year = {2008},
}

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
PB - Gauthier-Villars
VL - 44
IS - 2
SP - 374
EP - 392
AB - 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&gt;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 timesn−d/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.
LA - eng
KW - heat kernel; random conductance model; random walk; percolation; isoperimetry; simple random walk; random environments
UR - http://eudml.org/doc/77975
ER -

References

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  1. [1] P. Antal and A. Pisztora. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996) 1036–1048. Zbl0871.60089MR1404543
  2. [2] M. T. Barlow. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004) 3024–3084. Zbl1067.60101MR2094438
  3. [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. Zbl1020.60037MR1967022
  4. [4] N. Berger and M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007) 83–120. Zbl1107.60066MR2278453
  5. [5] M. Biskup and T. Prescott. Functional CLT for random walk among bounded conductances. Electron. J. Probab. 12 (2007) 1323–1348. Zbl1127.60093MR2354160
  6. [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. Zbl0634.60066MR898496
  7. [7] T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 (1999) 181–232. Zbl0922.60060MR1681641
  8. [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. Zbl0571.60044MR814703
  9. [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. Zbl0713.60041MR1003538
  10. [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. Zbl1086.60066MR2214905
  11. [11] S. Goel, R. Montenegro and P. Tetali. Mixing time bounds via the spectral profile. Electron. J. Probab. 11 (2006) 1–26. Zbl1109.60061MR2199053
  12. [12] A. Grigor’yan. Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoamericana 10 (1994) 395–452. Zbl0810.58040MR1286481
  13. [13] G. R. Grimmett. Percolation, 2nd edition. Springer, Berlin, 1999. MR1707339
  14. [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. Zbl0791.60095MR1222363
  15. [15] D. Heicklen and C. Hoffman. Return probabilities of a simple random walk on percolation clusters. Electron. J. Probab. 10 (2005) 250–302 (electronic). Zbl1070.60067MR2120245
  16. [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. Zbl0588.60058MR834478
  17. [17] B. Morris and Y. Peres. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2005) 245–266. Zbl1080.60071MR2198701
  18. [18] P. Mathieu. Quenched invariance principles for random walks with random conductances. J. Statist. Phys. To appear. Zbl1214.82044MR2384074
  19. [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. Zbl1131.82012MR2345229
  20. [20] P. Mathieu and E. Remy. Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004) 100–128. Zbl1078.60085MR2040777
  21. [21] G. Pete. A note on percolation on ℤd: Isoperimetric profile via exponential cluster repulsion. Preprint, 2007. MR2415145
  22. [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. Zbl1156.60074MR2430203
  23. [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. Zbl1070.60090MR2063376
  24. [24] N. T. Varopoulos. Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63 (1985) 215–239. Zbl0573.60059MR803093

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