Random walks on supercritical percolation clusters. (English) Zbl 1067.60101

Gaussian upper and lower bounds are obtained on the transition density of the continuous time simple random walk on a supercritical percolation cluster in the Euclidean space. The paper begins from a resume of random walks on graphs. Next, percolation estimates are established and Poincaré inequalities are derived. For a general graph that satisfies approporiate volume growth and Poincaré inequalities, the two-sided bound is derived for the transition density. These bounds are analogous to Aronsen’s bounds for uniformly elliptic divergence from diffusions and depend on the percolation probability. The irregular nature of the medium means that the bound for the transition density holds only for times exceeding certain percolation configuration-dependent value.


60K37 Processes in random environments
58J35 Heat and other parabolic equation methods for PDEs on manifolds
82B43 Percolation
05C80 Random graphs (graph-theoretic aspects)
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[1] Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036–1048. · Zbl 0871.60089
[2] Barlow, M. T. and Bass, R. F. (1989). The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincaré Probab. Statist. 25 225–257. · Zbl 0691.60070
[3] Barlow, M. T. and Bass, R. F. (1992). Transition densities for Brownian motion on the Sierpinski carpet. Probab. Theory Related Fields 91 307–330. · Zbl 0739.60071
[4] Barlow, M. T. and Bass, R. F. (2004). Stability of parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356 1501–1533. · Zbl 1034.60070
[5] Bass, R. F. (2002). On Aronsen’s upper bounds for heat kernels. Bull. London Math. Soc. 34 415–419. · Zbl 1032.35008
[6] Benjamini, I. and Mossel, E. (2003). On the mixing time of simple random walk on the super critical percolation cluster. Probab. Theory Related Fields 125 408–420. · Zbl 1020.60037
[7] Benjamini, I., Lyons, R. and Schramm, O. (1999). Percolation perturbations in potential theory and random walks. In Random Walks and Discrete Potential Theory 56–84 (M. Dicardello and W. Woess, eds.). Cambridge Univ. Press. · Zbl 0958.05121
[8] Carlen, E. A., Kusuoka, S. and Stroock, D. W. (1987). Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23 245–287. · Zbl 0634.60066
[9] Coulhon, T. and Grigor’yan, A. (2003). Pointwise estimates for transition probabilities of random walks on infinite graphs. In Fractals (P. Grabner and W. Woess, eds.) 119–134. Birkhäuser, Boston. · Zbl 1048.60081
[10] Couronné, O. and Messikh, R. J. (2003). Surface order large deviations for 2D FK-percolation and Potts models. · Zbl 1080.60020
[11] Davies, E. B. (1993). Large deviations for heat kernels on graphs. J. London Math. Soc. (2) 47 65–72. · Zbl 0799.58086
[12] De Gennes, P. G. (1976). La percolation: Un concept unificateur. La Recherche 7 919–927.
[13] Delmotte, T. (1999). Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 181–232. · Zbl 0922.60060
[14] De Masi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 787–855. · Zbl 0713.60041
[15] Deuschel, J.-D. and Pisztora, A. (1996). Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 467–482. · Zbl 0842.60023
[16] Fabes, E. B. and Stroock, D. W. (1986). A new proof of Moser’s parabolic Harnack inequality via the old ideas of Nash. Arch. Rational. Mech. Anal. 96 327–338. · Zbl 0652.35052
[17] Grigor’yan, A. A. (1992). Heat equation on a noncompact Riemannian manifold. Math. USSR-Sb. 72 47–77. · Zbl 0776.58035
[18] Grimmett, G. R. (1999). Percolation , 2nd ed. Springer, Berlin. · Zbl 0926.60004
[19] Grimmett, G. R., Kesten, H. and Zhang, Y. (1993). Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 33–44. · Zbl 0791.60095
[20] Heicklen, D. and Hoffman, C. (2000). Return probabilities of a simple random walk on percolation clusters. · Zbl 1070.60067
[21] Jerison, D. (1986). The weighted Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53 503–523. · Zbl 0614.35066
[22] Kaimanovitch, V. A. (1990). Boundary theory and entropy of random walks in random environment. In Probability Theory and Mathematical Statistics 573–579. Mokslas, Vilnius.
[23] Kesten, H. (1986a). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369–394. · Zbl 0584.60098
[24] Kesten, H. (1986b). Subdiffusive behavior of random walks on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 425–487. · Zbl 0632.60106
[25] Kusuoka, S. and Zhou, X. Y. (1992). Dirichlet form on fractals: Poincaré constant and resistance. Probab. Theory Related Fields 93 169–196. · Zbl 0767.60076
[26] Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Probab. 25 71–95. · Zbl 0882.60046
[27] Mathieu, P. and Remy, E. (2004). Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 100–128. · Zbl 1078.60085
[28] Nash, J. (1958). Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 931–954. · Zbl 0096.06902
[29] Penrose, M. D. and Pisztora, A. (1996). Large deviations for discrete and continuous percolation. Adv. in Appl. Probab. 28 29–52. · Zbl 0853.60085
[30] Pisztora, A. (1996). Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104 427–466. · Zbl 0842.60022
[31] Saloff-Coste, L. (1992). A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices 2 27–38. · Zbl 0769.58054
[32] Saloff-Coste, L. (1997). Lectures on finite Markov chains. Lectures on Probability Theory and Statistics . Ecole d’Éte de Probabilités de Saint-Flour XXVI . Lecture Notes in Math. 1665 301–408. Springer, Berlin. · Zbl 0885.60061
[33] Saloff-Coste, L. and Stroock, D. W. (1991). Opérateurs uniformément sous-elliptiques sur les groupes de Lie. J. Funct. Anal. 98 97–121. · Zbl 0734.58041
[34] Sidoravicius, V. and Sznitman, A.-S. (2003). Quenched invariance principles for walks on clusters of percolation or amoung random conductances. · Zbl 1070.60090
[35] Stroock, D. W. and Zheng, W. (1997). Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 33 619–649. · Zbl 0885.60065
[36] Thomassen, C. (1992). Isoperimetric inequalities and transient random walks on graphs. Ann. Probab. 20 1592–1600. JSTOR: · Zbl 0756.60065
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