Discrete fractal dimensions of the ranges of random walks in Z(d) associate with random conductances

Let be a continuous time random walk in an environment of i.i.d. random conductances , where E (d) is the set of nonoriented nearest neighbor bonds on the Euclidean lattice and d a parts per thousand yen 3. Let 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 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 (d a parts per thousand yen 3) are also proven