Random walks in a strongly sparse random environment

Abstract

The integer points (sites) of the real line are marked by the positions of a standard random walk with positive integer jumps. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity and assuming additionally that the distribution tail of the jumps is regularly varying at infinity we consider a nearest neighbor random walk on the set of integers having jumps ±1 with probability 1∕2 at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2019) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.