Abstract
We investigate in this work the asymptotic behavior of an anisotropic random walk on the supercritical cluster for bond percolation on ℤd, d≥2. In particular we show that for small anisotropy the walk behaves in a ballistic fashion, whereas for strong anisotropy the walk is sub-diffusive. For arbitrary anisotropy, we also prove the directional transience of the walk and construct a renewal structure.
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Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs. http://www.stat.Berkeley.EDV/users/aldous/book.html
Antal, P., Pisztora, A.: On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24(2), 1036–1048 (1996)
Benjamini, I., Mossel, E.: On the mixing time of simple random walk on the supercritical percolation cluster. Probab. Theory Relat. Fields 125, 408–420 (2003)
Berger, N., Gantert, N., Peres, Y.: The speed of biased random walk on percolation clusters. Probab. Theory Relat. Fields 126, 221–242 (2003)
Bouchaud, J.P.: Weak ergodicity breaking and aging in disordered systems. J. Phys, I. France 2, 1705–1713 (1992)
Bramson, M.: Random walk in random environment: A counterexample without potential. J. Statist. Phys. 62(314), 863–875 (1991)
Bramson, M., Durrett, R.: Random walk in random environment: A counterexample. Commun. Math. Phys. 119, 199–211 (1988)
Carne, T.K.: A transmutation formula for Markov chains. Bull. Sc. Math. 109, 399–405 (1985)
Chung, K.L.: Markov chains with stationary transition probabilities. Berlin: Springer, 1960
Dembo, A., Gandoldi, A., Kesten, H.: Greedy lattice animals: Negative values and unconstrained maxima. Ann. Prob. 29(1), 205–241 (2001)
Dhar, D., Stauffer D.: Drift and trapping in biased diffusion on disordered lattices. Int. J. Mod. Phys. C 9(2), 349–355 (1998)
Durrett, R.: Probability: Theory and Examples. Pacific Grove, CA: Wadsworth and Brooks/Cole, 1991
Ethier, S.M., Kurtz, T.G.: Markov processes. New York: John Wiley & Sons, 1986
Grimmett, G.: Percolation. Second edition, Berlin: Springer, 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)
Havlin, S., Bunde, A.: Percolation I, II. In: Fractals and disordered systems, A. Bunde, S. Havlin, (ed.), Berlin: Springer, 1991, pp. 51–95, 96–149
Kesten, H., Kozlov, M.V., Spitzer, F.: A limit law for random walk in a random environment. Compositio Mathematica 30(2), 145–168 (1975)
Lyons, R., Pemantle, R., Peres, Y.: Unsolved problems concerning random walks on trees. In: Classical and modern branching processes, K.B. Athreya, P. Jagers, (eds.), IMA volume 84, Berlin-Heidelberg-New York: Springer, 1997, pp. 223–237
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 kernel decay on percolation clusters. Preprint
Mathieu, P., Remy, E.: Décroissance du noyau de la chaleur et isopérimétrie sur un amas de percolation. C.R. Acad. Sci. Paris, t. 332, Serie 1, 927–931 (2001)
Saloff-Coste, L.: Lectures on finite Markov chains. Volume 1665. Ecole d'Eté de Probabilités de Saint Flour, P. Bernard, (ed.), Lectures Notes in Mathematics, Berlin: Springer, 1997
Shen, L.: Asymptotic properties of certain anisotropic walks in random media. Ann. Appl. Probab. 12(2), 477–510 (2002)
Sinai, Ya.G.: The limiting behavior of a one-dimensional random walk in a random environment. Theory Prob. Appl. 27(2), 247–258 (1982)
Solomon, F.: Random walk in a random environment. Ann. Probab. 3, 1–31 (1975)
Sznitman, A.S.: Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. 2, 93–143 (2000)
Sznitman, A.S.: Topics in random walk in random environment. To appear in ICTP Lecture Notes Series. School and Conference on Probability Theory, Trieste, 2002 www.math.ethz.ch/∼sznitman/preprint.shtml
Sznitman, A.S., Zerner, M.P.W.: A law of large numbers for random walks in random environment. Ann. Probab. 27(4), 1851–1869 (1999)
Zeitouni, O.: Notes of Saint Flour lectures 2001. Preprint, www-ee.technion.ac.il/∼zeitouni/ps/notes1.ps
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Communicated by J.L. Lebowitz
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Sznitman, AS. On the Anisotropic Walk on the Supercritical Percolation Cluster. Commun. Math. Phys. 240, 123–148 (2003). https://doi.org/10.1007/s00220-003-0896-3
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DOI: https://doi.org/10.1007/s00220-003-0896-3