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Random Walk on the High-Dimensional IIC

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Abstract

We study the asymptotic behavior of the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by Kumagai and Misumi (J Theor Probab 21:910–935, 2008). We do this by getting bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander–Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation.

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Authors and Affiliations

  1. Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA, Leiden, The Netherlands

    Markus Heydenreich

  2. Centrum voor Wiskunde and Informatica, P.O. Box 94079, 1090 GB, Amsterdam, The Netherlands

    Markus Heydenreich

  3. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Remco van der Hofstad

  4. Department of Mathematics, The University of British Columbia, Vancouver, Canada

    Tim Hulshof

  5. Pacific Institute for Mathematical Sciences, The University of British Columbia, Vancouver, Canada

    Tim Hulshof

Authors
  1. Markus Heydenreich
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  2. Remco van der Hofstad
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  3. Tim Hulshof
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Correspondence to Tim Hulshof.

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Communicated by H. Spohn

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Heydenreich, M., van der Hofstad, R. & Hulshof, T. Random Walk on the High-Dimensional IIC. Commun. Math. Phys. 329, 57–115 (2014). https://doi.org/10.1007/s00220-014-1931-2

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  • DOI: https://doi.org/10.1007/s00220-014-1931-2

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