Electronic Journal of Probability

Return Probabilities of a Simple Random Walk on Percolation Clusters

Deborah Heicklen and Christopher Hoffman

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Abstract

We bound the probability that a continuous time simple random walk on the infinite percolation cluster on $Z^d$ returns to the origin at time $t$. We use this result to show that in dimensions 5 and higher the uniform spanning forest on infinite percolation clusters supported on graphs with infinitely many connected components a.s.

Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 8, 250-302.

Dates
Accepted: 4 March 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816809

Digital Object Identifier
doi:10.1214/EJP.v10-240

Mathematical Reviews number (MathSciNet)
MR2120245

Zentralblatt MATH identifier
1070.60067

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Heicklen, Deborah; Hoffman, Christopher. Return Probabilities of a Simple Random Walk on Percolation Clusters. Electron. J. Probab. 10 (2005), paper no. 8, 250--302. doi:10.1214/EJP.v10-240. https://projecteuclid.org/euclid.ejp/1464816809


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References

  • Aizenman, M.; Kesten, H.; Newman, C. M. Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111 (1987), no. 4, 505–531.
  • Antal, Peter; Pisztora, Agoston. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996), no. 2, 1036–1048.
  • Barlow, Martin T. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004), no. 4, 3024–3084.
  • Barlow, Martin T.; Perkins, Edwin A. Symmetric Markov chains in $bold Zsp d$: how fast can they move? Probab. Theory Related Fields 82 (1989), no. 1, 95–108.
  • Barsky, David J.; Grimmett, Geoffrey R.; Newman, Charles M. Dynamic renormalization and continuity of the percolation transition in orthants. Spatial stochastic processes, 37–55, Progr. Probab., 19, Birkhäuser Boston, Boston, MA, 1991.
  • Benjamini, Itai; Lyons, Russell; Peres, Yuval; Schramm, Oded. Uniform spanning forests. Ann. Probab. 29 (2001), no. 1, 1–65.
  • Benjamini, Itai; Lyons, Russell; Schramm, Oded. Percolation perturbations in potential theory and random walks. Random walks and discrete potential theory (Cortona, 1997), 56–84, Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge, 1999.
  • Benjamini, Itai; Pemantle, Robin; Peres, Yuval. Unpredictable paths and percolation. Ann. Probab. 26 (1998), no. 3, 1198–1211.
  • Burton, R. M.; Keane, M. Density and uniqueness in percolation. Comm. Math. Phys. 121 (1989), no. 3, 501–505.
  • Carne, Thomas Keith. A transmutation formula for Markov chains. Bull. Sci. Math. (2) 109 (1985), no. 4, 399–405.
  • Chayes, J. T.; Chayes, L.; Newman, C. M. Bernoulli percolation above threshold: an invasion percolation analysis. Ann. Probab. 15 (1987), no. 4, 1272–1287.
  • Doyle, Peter G.; Snell, J. Laurie. Random walks and electric networks. Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, 1984. xiv+159 pp. ISBN: 0-88385-024-9.
  • Erdös, P.; Taylor, S. J. Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar 11 1960 137–162. (unbound insert).
  • Feige, Uriel. A tight upper bound on the cover time for random walks on graphs. Random Structures Algorithms 6 (1995), no. 1, 51–54.
  • Grimmett, Geoffrey. Percolation. Springer-Verlag, New York, 1989. xii+296 pp. ISBN: 0-387-96843-1.
  • Grimmett, G. R.; Kesten, H.; Zhang, Y. Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 (1993), no. 1, 33–44.
  • 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.
  • Häggström, Olle. Random-cluster measures and uniform spanning trees. Stochastic Process. Appl. 59 (1995), no. 2, 267–275.
  • Häggström, Olle; Mossel, Elchanan. Nearest-neighbor walks with low predictability profile and percolation in $2+epsilon$ dimensions. Ann. Probab. 26 (1998), no. 3, 1212–1231.
  • Hoffman, Christopher. Energy of flows on $Zsp 2$ percolation clusters. Random Structures Algorithms 16 (2000), no. 2, 143–155.
  • Kaui manovich, V. A. Boundary and entropy of random walks in random environment. Probability theory and mathematical statistics, Vol. I (Vilnius, 1989), 573–579, "Mokslas", Vilnius, 1990.
  • Kesten, Harry. The critical probability of bond percolation on the square lattice equals ${1over 2}$. Comm. Math. Phys. 74 (1980), no. 1, 41–59.
  • Lawler, Gregory F. Intersections of random walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1991. 219 pp. ISBN: 0-8176-3557-2.
  • Mathieu, Pierre; Remy, Elisabeth. Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004), no. 1A, 100–128.
  • Pemantle, Robin. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 (1991), no. 4, 1559–1574.
  • Russo, Lucio. A note on percolation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43 (1978), no. 1, 39–48.
  • Seymour, P. D.; Welsh, D. J. A. Percolation probabilities on the square lattice. Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). Ann. Discrete Math. 3 (1978), 227–245.
  • Varopoulos, Nicholas Th. Long range estimates for Markov chains. Bull. Sci. Math. (2) 109 (1985), no. 3, 225–252.
  • Shorack, Galen R.; Wellner, Jon A. Empirical processes with applications to statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. xxxviii+938 pp. ISBN: 0-471-86725-X.