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Simple random walk on long range percolation clusters I: heat kernel bounds
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  • Published: 04 August 2011

Simple random walk on long range percolation clusters I: heat kernel bounds

  • Nicholas Crawford1 &
  • Allan Sly2 

Probability Theory and Related Fields volume 154, pages 753–786 (2012)Cite this article

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  • 16 Citations

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Abstract

In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any d ≥ 1 and for any exponent \({s \in (d, (d+2) \wedge 2d)}\) giving the rate of decay of the percolation process, we show that the return probability decays like \({t^{-{d}/_{s-d}}}\) up to logarithmic corrections, where t denotes the time the walk is run. Our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. The bounds and accompanying understanding of the geometry of the cluster play a crucial role in the companion paper (Crawford and Sly in Simple randomwalk on long range percolation clusters II: scaling limit, 2010) where we establish the scaling limit of the random walk to be α-stable Lévy motion.

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References

  1. Aizenman M., Newman C.: Discontinuity of the percolation density in one dimensional \({\frac{1}{(x-y)^2}}\) percolation models. Commun. Math. Phys. 107, 611 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barlow M.T.: Random walks on supercritical percolation clusters. Ann. Probab. 32, 3024–3084 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benjamini I., Berger N.: The diameter of long-range percolation clusters on finite cycles. Random Struct. Algorithms 19(2), 102–111 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benjamini I., Berger N., Yadin A.: Long-range percolation mixing time. Combin. Probab. Comput. 17(4), 487–494 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benjamini, I., Berger, N., Yadin, A.: Long-range percolation mixing time. http://arxiv.org/abs/math/0703872 (2009)

  6. Benjamini, I., Kozma, G., Wormald, N.: The mixing time of the giant component of a random graph. Preprint. arXiv:math/0610459 (2006)

  7. Berger N.: Transience, recurrence and critical behavior for long-range percolation. Commun. Math. Phys. 226(3), 531–558 (2002)

    Article  MATH  Google Scholar 

  8. Berger N., Biskup M.: Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Relat. Fields 137(1–2), 83–120 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Biskup, M.: Graph diameter in long-range percolation. ArXiv Mathematics e-prints, June 2004

  10. Biskup M.: On the scaling of the chemical distance in long-range percolation models. Ann. Probab. 32(4), 2938–2977 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Burton R.M., Keane M.: Density and uniqueness in percolation. Commun. Math. Phys. 121, 501–505 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Coppersmith, D., Gamarnik, D., Sviridenko, M.: The diameter of a long range percolation graph. In: SODA ’02: Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 329–337, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics (2002)

  13. Coulhon T., Grigor’yan A.: Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8, 656–701 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Coulhon, T., Grigor’yan, A.: Pointwise estimates for transition probabilities of random walks on infinite graphs. In: Trends Math.: Fractals in Graz, pp. 119–134. Birkäuser, Basel (2003)

  15. Crawford, N., Sly, A.: Simple random walk on long range percolation clusters II: scaling limits. http://arxiv.org/abs/0911.5668 (2010)

  16. Delmotte T.: Parabolic Harnack inequality and estimates of markov chains on graphs. Revista Matematica Iberoamericana 15, 181–232 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fountoulakis N., Reed B.A.: The evolution of the mixing rate of a simple random walk on the giant component of a random graph. Random Struct. Algorithms 33, 68–86 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Friedman, J.: A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Am. Math. Soc. 195(910), viii+100 (2008)

  19. Hoffman C., Heicklen D.: Return times of a simple random walk on percolation clusters. Electron. J. Probab. 10(8), 250–302 (2005)

    MathSciNet  Google Scholar 

  20. Kesten H., Aizenman M., Newman C.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Commun. Math. Phys. 111, 505–532 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Krivelevich M., Benjamini I., Haber S., Lubetzky E.: The isoperimetric constant of the random graph process. Random Struct. Algorithms 32, 101–114 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kumagai, T., Misumi, J.: Heat kernel estimates for strongly recurrent random walk on random media. ArXiv e-prints, June 2008

  23. Lubetzky E., Sly A.: Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153, 475–510 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mathieu P., Remy E.: Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32(1A), 100–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Milgram S.: The small world problem. Psychol. Today 2, 60–67 (1967)

    Google Scholar 

  26. Newman C.M., Schulman L.S.: One dimensional \({1/\|j-i\|^s}\) percolation models: the existence of a transition for s = 2. Commun. Math. Phys. 180, 483–504 (1986)

    MathSciNet  Google Scholar 

  27. Pu F.C., Zhang Z.Q., Li B.Z.: Long-range percolation in one dimension. J. Phys. A: Math. Gen. 16, L85–L89 (1983)

    Article  MathSciNet  Google Scholar 

  28. Schulman L.S.: Long-range percolation in one dimension. J. Phys. A 16, L639–L641 (1983)

    Article  MathSciNet  Google Scholar 

  29. Sinclair A.: Improved bounds for mixing rates of markov-chains and multicommodity flow. Lecture Notes in Computer Science 583, 474–487 (1992)

    Article  Google Scholar 

  30. Trapman, P.: The growth of the infinite long-range percolation cluster. ArXiv e-prints, January 2009

  31. Watts D., Strogatz S.: Collective dynamics of small-world networks. Nature 363, 202–204 (1998)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Industrial Engineering, The Technion, Haifa, Israel

    Nicholas Crawford

  2. Theory Group, Microsoft Research, Redmond, WA, USA

    Allan Sly

Authors
  1. Nicholas Crawford
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  2. Allan Sly
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Corresponding author

Correspondence to Nicholas Crawford.

Additional information

N. Crawford supported in part at the Technion by an Marilyn and Michael Winer Fellowship. Work completed while the authors were at UC Berkeley Department of Statistics.

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Crawford, N., Sly, A. Simple random walk on long range percolation clusters I: heat kernel bounds. Probab. Theory Relat. Fields 154, 753–786 (2012). https://doi.org/10.1007/s00440-011-0383-2

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  • Received: 09 February 2010

  • Revised: 08 July 2011

  • Published: 04 August 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0383-2

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Mathematics Subject Classification (2000)

  • 60FXX
  • 82B43
  • 05C81
  • 74QXX
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