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Random walk on the infinite cluster of the percolation model
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  • Published: March 1993

Random walk on the infinite cluster of the percolation model

  • G. R. Grimmett1,
  • H. Kesten2 &
  • Y. Zhang3 

Probability Theory and Related Fields volume 96, pages 33–44 (1993)Cite this article

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Summary

We consider random walk on the infinite cluster of bond percolation on ℤd. We show that, in the supercritical regime whend≧3, this random walk is a.s. transient. This conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance. It is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite.

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References

  • DeMasi, A., Ferrari, P.A., Goldstein, S., Wick, W.D.: Invariance principle for reversible Markov processes with application to diffusion in the percolation regime. In: Durrett, R. (ed.) Particle systems, random media and large deviations. (Contemp. Math., vol. 41, pp. 71–85) Providence, RI: Am. Math. Soc. 1985

    Google Scholar 

  • DeMasi, A., Ferrari, P.A., Goldstein, S., Wick, W.D.: An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys.55, 787–855 (1989)

    Google Scholar 

  • Doyle, P.G., Snell, E.J.: Random walk on electric networks. (Carus Math. Monogr. no. 22) Washington, DC: AMA 1984

    Google Scholar 

  • Grimmett, G.R.: Percolation. Berlin Heidelberg New York: Springer 1989

    Google Scholar 

  • Grimmett, G.R., Marstrand, J.M.: The supercritical phase of percolation is well behaved. Proc. R. Soc. London Ser. A430, 439–457 (1990)

    Google Scholar 

  • Hammersley, J.M.: Mesoadditive processes and the specific conductivity of lattices. J. Appl. Probab.25A, 347–358 (1988)

    Google Scholar 

  • Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982

    Google Scholar 

  • Kesten, H., Zhang, Y.: The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab.18, 537–555 (1990)

    Google Scholar 

  • Lyons, T.: A simple criterion for transience of a reversible Markov chain. Ann. Probab.11, 393–402 (1983)

    Google Scholar 

  • Nash-Williams, C.St.J.A.: Random walk and electric currents in networks. Proc. Camb. Philos. Soc.55, 181–194 (1959)

    Google Scholar 

  • Zhikov, V.V.: Effective conductivity of random homogeneous sets. Mat. Zametki45, 34–45, 288–296 in translation (1989)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Statistical Laboratory, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, UK

    G. R. Grimmett

  2. Mathematics Department, Cornell University, 14853, Ithaca, NY, USA

    H. Kesten

  3. Mathematics Department, University of Colorado, 80933, Colorado Springs, CO, USA

    Y. Zhang

Authors
  1. G. R. Grimmett
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  2. H. Kesten
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  3. Y. Zhang
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Additional information

G.R.G. acknowledges support from Cornell University, and also partial support by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University

H.K. was supported in part by the N.S.F. through a grant to Cornell University

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Grimmett, G.R., Kesten, H. & Zhang, Y. Random walk on the infinite cluster of the percolation model. Probab. Th. Rel. Fields 96, 33–44 (1993). https://doi.org/10.1007/BF01195881

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  • Received: 05 February 1992

  • Revised: 12 November 1992

  • Issue Date: March 1993

  • DOI: https://doi.org/10.1007/BF01195881

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Mathematics Subject Classification (1991)

  • 60J15
  • 60K35
  • 82B43
  • 82D30
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