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

Xiao, Yimin; Zheng, Xinghua
June 2013
Probability Theory & Related Fields;Jun2013, Vol. 156 Issue 1/2, p1
Academic Journal
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 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, dim R = dim R = 2 almost surely, where dim and dim 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 for arbitrarily large t > 0 is given in terms of dim A. Similar results for Bouchoud's trap model in $${\mathbb{Z}^d}$$ ( d ≥ 3) are also proven.


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