The Annals of Probability

Random walks on supercritical percolation clusters

Martin T. Barlow

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We obtain Gaussian upper and lower bounds on the transition density qt(x,y) of the continuous time simple random walk on a supercritical percolation cluster ${\mathcal{C}}_{\infty}$ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for qt(x,⋅) holds only for tSx(ω), where the constant Sx(ω) depends on the percolation configuration ω.

Article information

Ann. Probab., Volume 32, Number 4 (2004), 3024-3084.

First available in Project Euclid: 8 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 58J35: Heat and other parabolic equation methods

Percolation random walk heat kernel


Barlow, Martin T. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004), no. 4, 3024--3084. doi:10.1214/009117904000000748.

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  • Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036--1048.
  • 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.
  • Barlow, M. T. and Bass, R. F. (1992). Transition densities for Brownian motion on the Sierpinski carpet. Probab. Theory Related Fields 91 307--330.
  • Barlow, M. T. and Bass, R. F. (2004). Stability of parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356 1501--1533.
  • Bass, R. F. (2002). On Aronsen's upper bounds for heat kernels. Bull. London Math. Soc. 34 415--419.
  • 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.
  • 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.
  • 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.
  • 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.
  • Couronné, O. and Messikh, R. J. (2003). Surface order large deviations for 2D FK-percolation and Potts models. Preprint.
  • Davies, E. B. (1993). Large deviations for heat kernels on graphs. J. London Math. Soc. (2) 47 65--72.
  • De Gennes, P. G. (1976). La percolation: Un concept unificateur. La Recherche 7 919--927.
  • Delmotte, T. (1999). Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 181--232.
  • 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.
  • Deuschel, J.-D. and Pisztora, A. (1996). Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 467--482.
  • 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.
  • Grigor'yan, A. A. (1992). Heat equation on a noncompact Riemannian manifold. Math. USSR-Sb. 72 47--77.
  • Grimmett, G. R. (1999). Percolation, 2nd ed. Springer, Berlin.
  • 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.
  • Heicklen, D. and Hoffman, C. (2000). Return probabilities of a simple random walk on percolation clusters. Preprint.
  • Jerison, D. (1986). The weighted Poincaré inequality for vector fields satisfying Hörmander's condition. Duke Math. J. 53 503--523.
  • Kaimanovitch, V. A. (1990). Boundary theory and entropy of random walks in random environment. In Probability Theory and Mathematical Statistics 573--579. Mokslas, Vilnius.
  • Kesten, H. (1986a). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369--394.
  • Kesten, H. (1986b). Subdiffusive behavior of random walks on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 425--487.
  • Kusuoka, S. and Zhou, X. Y. (1992). Dirichlet form on fractals: Poincaré constant and resistance. Probab. Theory Related Fields 93 169--196.
  • Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Probab. 25 71--95.
  • Mathieu, P. and Remy, E. (2004). Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 100--128.
  • Nash, J. (1958). Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 931--954.
  • Penrose, M. D. and Pisztora, A. (1996). Large deviations for discrete and continuous percolation. Adv. in Appl. Probab. 28 29--52.
  • Pisztora, A. (1996). Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104 427--466.
  • Saloff-Coste, L. (1992). A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices 2 27--38.
  • 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.
  • Saloff-Coste, L. and Stroock, D. W. (1991). Opérateurs uniformément sous-elliptiques sur les groupes de Lie. J. Funct. Anal. 98 97--121.
  • Sidoravicius, V. and Sznitman, A.-S. (2003). Quenched invariance principles for walks on clusters of percolation or amoung random conductances. Preprint.
  • Stroock, D. W. and Zheng, W. (1997). Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 33 619--649.
  • Thomassen, C. (1992). Isoperimetric inequalities and transient random walks on graphs. Ann. Probab. 20 1592--1600.