Functional CLT for Random Walk Among Bounded Random Conductances

Marek Biskup (UCLA)
Timothy M Prescott (UCLA)


We consider the nearest-neighbor simple random walk on $Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $\omega$'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in $d\ge5$ due to anomalously slow decay of the probability that the walk returns to the starting point at a given time.

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Pages: 1323-1348

Publication Date: October 25, 2007

DOI: 10.1214/EJP.v12-456


  1. Antal, Peter; Pisztora, Agoston. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996), no. 2, 1036--1048. MR1404543 (98b:60168)
  2. Barlow, Martin T. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004), no. 4, 3024--3084. MR2094438 (2006e:60146)
  3. M.T. Barlow and J.-D. Deuschel (2007).Quenched invariance principle for the random conductance model with unbounded conductances.In preparation.
  4. Bass, Richard F. On Aronson's upper bounds for heat kernels. Bull. London Math. Soc. 34 (2002), no. 4, 415--419. MR1897420 (2003c:35054)
  5. Benjamini, Itai; Mossel, Elchanan. On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Related Fields 125 (2003), no. 3, 408--420. MR1967022 (2004c:60139)
  6. Berger, Noam; Biskup, Marek. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007), no. 1-2, 83--120. MR2278453 (Review)
  7. N. Berger, M. Biskup, C.E. Hoffman and G. Kozma (2006). Anomalous heat-kernel decay for random walk among bounded random conductances. Ann. Inst. Henri Poincar\'e (to appear).
  8. M. Biskup and J.-D. Deuschel, in preparation.
  9. M. Biskup and H. Spohn (2007). Scaling limit for a class of gradient fields with non-convex potentials. Preprint arxiv:0704.3086.
  10. Coulhon, T.; Grigor'yan, A.; Pittet, C. A geometric approach to on-diagonal heat kernel lower bounds on groups. Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1763--1827. MR1871289 (2002m:20067)
  11. De Masi, A.; Ferrari, P. A.; Goldstein, S.; Wick, W. D. Invariance principle for reversible Markov processes with application to diffusion in the percolation regime. Particle systems, random media and large deviations (Brunswick, Maine, 1984), 71--85, Contemp. Math., 41, Amer. Math. Soc., Providence, RI, 1985. MR0814703 (87a:60077)
  12. De Masi, A.; Ferrari, P. A.; Goldstein, S.; Wick, W. D. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 (1989), no. 3-4, 787--855. MR1003538 (91e:60107)
  13. R. Durrett (1996). Probability Theory and Examples (Third edition),Duxbury Press, Belmont, CA.
  14. Fontes, L. R. G.; Mathieu, P. On symmetric random walks with random conductances on ${\Bbb Z}\sp d$. Probab. Theory Related Fields 134 (2006), no. 4, 565--602. MR2214905 (2006m:60142)
  15. Giacomin, Giambattista; Olla, Stefano; Spohn, Herbert. Equilibrium fluctuations for $\nabla\phi$ interface model. Ann. Probab. 29 (2001), no. 3, 1138--1172. MR1872740 (2003c:60161)
  16. Goel, Sharad; Montenegro, Ravi; Tetali, Prasad. Mixing time bounds via the spectral profile. Electron. J. Probab. 11 (2006), no. 1, 1--26 (electronic). MR2199053 (2007e:60075)
  17. Grimmett, Geoffrey. Percolation.Second edition.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6 MR1707339 (2001a:60114)
  18. Grimmett, G. R.; Marstrand, J. M. The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 (1990), no. 1879, 439--457. MR1068308 (91m:60186)
  19. Kesten, Harry; Zhang, Yu. The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 (1990), no. 2, 537--555. MR1055419 (91i:60278)
  20. Kipnis, C.; Varadhan, S. R. S. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986), no. 1, 1--19. MR0834478 (87i:60038)
  21. Liggett, T. M.; Schonmann, R. H.; Stacey, A. M. Domination by product measures. Ann. Probab. 25 (1997), no. 1, 71--95. MR1428500 (98f:60095)
  22. Lovász, László; Kannan, Ravi. Faster mixing via average conductance. Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999), 282--287 (electronic), ACM, New York, 1999. MR1798047 (2001i:68179)
  23. Morris, B.; Peres, Yuval. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2005), no. 2, 245--266. MR2198701 (2007a:60042)
  24. P. Mathieu (2006). Quenched invariance principles for random walks with random conductances, J.~Statist. Phys. (to appear).
  25. P. Mathieu and A.L. Piatnitski (2007).Quenched invariance principles for random walks on percolation clusters. Proc. Roy. Soc. A463, 2287--2307.
  26. Nash, J. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 1958 931--954. MR0100158 (20 #6592)
  27. C. Rau (2006). Sur le nombre de points visit\'es par une marche al\'eatoire sur un amas infini de percolation, Bull. Soc. Math. France (to appear).
  28. Sidoravicius, Vladas; Sznitman, Alain-Sol. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2004), no. 2, 219--244. MR2063376 (2005d:60155)

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