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Discrete Fractal Dimensions of the Ranges of Random Walks in Associate with Random Conductances
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 Z^d and d\ge
3. Let R = {x \in Z^d: X_t = x for some t \ge 0} be the range of X. It is
proved that, for almost every realization of the environment, dim_H (R) = dim_P
(R) = 2 almost surely, where dim_H and dim_P denote respectively the discrete
Hausdorff and packing dimension. Furthermore, given any set A \subseteq Z^d, a
criterion for A to be hit by X_t for arbitrarily large t>0 is given in terms of
dim_H(A). Similar results for Bouchoud's trap model in Z^d (d \ge 3) are also
proven
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