Abstract.
In this work we principally study random walk on the supercritical infinite cluster for bond percolation on ℤd. We prove a quenched functional central limit theorem for the walk when d≥4. We also prove a similar result for random walk among i.i.d. random conductances along nearest neighbor edges of ℤd, when d≥1.
Article PDF
Similar content being viewed by others
References
Adams, R.A.: Sobolev spaces. Academic Press, New York, 1975
Anshelevich, V.V., Khanin, K.M., Ya., Sinai, G.: Symmetric random walks in random environments. Commun. Math. Phys. 85, 449–470 (1982)
Barlow, M.T.: Random walks on supercritical percolation clusters. To appear in Ann. Probab.
Barlow, M.T., Bass, R.F.: Stability of parabolic Harnack inequalities. To appear in Trans. Amer. Math. Soc.
Boivin, D.: Weak convergence for reversible random walks in random environment. Ann. Probab. 21 (3), 1427–1440 (1993)
Boivin, D., Depauw, J.: Spectral homogeneization of reversible random walks on ℤd in a random environment. Stochastic Process. Appl. 104, 29–56 (2003)
Bolthausen, E., Sznitman, A.S.: Ten lectures on random media. DMV Seminar, Band 32, Birkhäuser, Basel, 2002
Bolthausen, E., Sznitman, A.S.: On the static and dynamic points of views for certain random walks in random environment. Meth. Appl. Anal. 9 (3), 345–376 (2002) Also accessible on www.math.ethz.ch/∼sznitman/preprint.shtml
Caputo, P., Ioffe, D.: Finite volume approximation of the effective diffusion matrix: the case of independent bond disorder. Ann. Inst. H. Poincaré, PR 39, 505–525 (2003)
Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Revista Matematica Iberoamericana 15 (1), 181–232 (1999)
Ethier, S.M., Kurtz, T.G.: Markov processes. John Wiley & Sons, New York, 1986
Grigoryan, A., Telcs, A.: Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324, 521–556 (2002)
Grimmett, G.: Percolation. Second edition, Springer, Berlin, 1999
Grimmett, G., Kesten, H., Zhang, Y.: Random walk on the infinite cluster of the percolation model. Probab. Theory Relat. Fields 96, 33–44 (1993)
Grimmett, G., Marstrand, J.: The supercritical phase of percolation is well behaved. Proc. Royal Society (London), Ser. A. 4306, 429–457 (1990)
Heicklen, D., Hoffmann, C.: Return probabilities of a simple random walk on percolation clusters. Preprint
Helland, I.S.: Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9, 79–94 (1982)
Kipnis, C., Varadhan, S.R.S.: A central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104, 1–19 (1986)
Kozlov, S.M.: The method of averaging and walks in inhomogeneous environments. Russian Math. Surveys 40 (2), 73–145 (1985)
Krengel, U.: Ergodic theorems. Walter de Gruyter, Berlin, 1985
Kunnemann, R.: The diffusion limit for reversible jump processes in ℤd with ergodic bond conductivities. Commun. Math. Phys. 90, 27–68 (1983)
Kurtz, T.G.: Approximation of population processes. CBMS-NSF regional conferences series in applied mathematics 36, SIAM, Philadelphia, 1981
Landis, E.M.: Second order equations of elliptic and parabolic type. Translations of Mathematical Monographs, AMS, Providence, 1998
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 (3-4), 787–855 (1989)
Mathieu, P., Remy, E.: Isoperimetry and heat decay in percolation clusters. To appear in Ann. Probab.
Osada, H.: Homogenization of diffusion processes with random stationary coefficients. In: Probability Theory and Mathematical Statistics, Tbilissi, 1982. Lecture Notes in Math. 1021, Springer, Berlin, 1983, pp. 507–517
Stroock, D.W.: Probability theory. An analytic view. Cambridge University Press, 1993
Author information
Authors and Affiliations
Corresponding author
Additional information
V. Sidoravicius would like to thank the FIM for financial support and hospitality during his multiple visits to ETH. His research was also partially supported by FAPERJ and CNPq.
Rights and permissions
About this article
Cite this article
Sidoravicius, V., Sznitman, AS. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Relat. Fields 129, 219–244 (2004). https://doi.org/10.1007/s00440-004-0336-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0336-0