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Discrete fractal dimensions of the ranges of random walks in \({{\mathbb Z}^d}\) associate with random conductances
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  • Published: 17 March 2012

Discrete fractal dimensions of the ranges of random walks in \({{\mathbb Z}^d}\) associate with random conductances

  • Yimin Xiao1 &
  • Xinghua Zheng2 

Probability Theory and Related Fields volume 156, pages 1–26 (2013)Cite this article

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Abstract

Let \({X= \{X_t, t \ge 0\}}\) be a continuous time random walk in an environment of i.i.d. random conductances \({\{\mu_e \in [1,\infty), e \in E_d\}}\) , where E d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice \({\mathbb{Z}^d}\) and d ≥ 3. Let \({{\rm R} = \{x \in \mathbb{Z}^d: X_t = x {\rm \,for\, some}\,t \ge 0\}}\) be the range of X. It is proved that, for almost every realization of the environment, dimH R = dimP R = 2 almost surely, where dimH and dimP denote, respectively, the discrete Hausdorff and packing dimension. Furthermore, given any set \({A \subseteq \mathbb{Z}^d}\) , a criterion for A to be hit by X t for arbitrarily large t > 0 is given in terms of dimH A. Similar results for Bouchoud’s trap model in \({\mathbb{Z}^d}\) (d ≥ 3) are also proven.

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Author information

Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, A-413 Wells Hall, East Lansing, MI, 48824, USA

    Yimin Xiao

  2. Department of ISOM, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Xinghua Zheng

Authors
  1. Yimin Xiao
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  2. Xinghua Zheng
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Corresponding author

Correspondence to Xinghua Zheng.

Additional information

Y. Xiao was partially supported by NSF grant DMS-1006903; X. Zheng was partially supported by GRF 606010 of the HKSAR.

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Cite this article

Xiao, Y., Zheng, X. Discrete fractal dimensions of the ranges of random walks in \({{\mathbb Z}^d}\) associate with random conductances . Probab. Theory Relat. Fields 156, 1–26 (2013). https://doi.org/10.1007/s00440-012-0418-3

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  • Received: 13 October 2011

  • Revised: 25 February 2012

  • Accepted: 27 February 2012

  • Published: 17 March 2012

  • Issue Date: June 2013

  • DOI: https://doi.org/10.1007/s00440-012-0418-3

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Keywords

  • Random conductance model
  • Bouchoud’s trap model
  • Range
  • Discrete Hausdorff dimension
  • Discrete packing dimension
  • Transience

Mathematics Subject Classification

  • 60K37
  • 60F17
  • 82C41
  • 31C20
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