Discrete Fractal Dimensions of the Ranges of Random Walks in $\Z^d$ Associate with Random Conductances

TL;DR

This paper investigates the fractal dimensions of the range of a random walk in a random conductance environment on Z^d, establishing that the Hausdorff and packing dimensions are almost surely 2, with criteria for hitting sets and extensions to trap models.

Contribution

It proves that the Hausdorff and packing dimensions of the random walk's range are almost surely 2 in high-dimensional environments and provides criteria for hitting sets, extending results to trap models.

Findings

Hausdorff and packing dimensions of the range are almost surely 2

Criteria for sets to be hit by the walk based on Hausdorff dimension

Extension of results to Bouchaud's trap model in Z^d

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 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.