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Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions

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Abstract

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on \({\mathbb{Z}}^{d} \times {\mathbb{Z}}_+\). In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is \(\frac {4}{3}\), and thereby prove a version of the Alexander–Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d > 6, by extending results about critical oriented percolation obtained previously via the lace expansion.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Martin T. Barlow & Gordon Slade

  2. Carleton University, School of Mathematics and Statistics, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Antal A. Járai

  3. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan

    Takashi Kumagai

Authors
  1. Martin T. Barlow
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  2. Antal A. Járai
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  3. Takashi Kumagai
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  4. Gordon Slade
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Correspondence to Antal A. Járai.

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Communicated by M. Aizenman

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Barlow, M.T., Járai, A.A., Kumagai, T. et al. Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions. Commun. Math. Phys. 278, 385–431 (2008). https://doi.org/10.1007/s00220-007-0410-4

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  • DOI: https://doi.org/10.1007/s00220-007-0410-4

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