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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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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

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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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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Keywords

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

Mathematics Subject Classification

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