# Random walks in a strongly sparse random environment

## TL;DR

This paper studies a random walk on a line with strongly sparse marked sites, revealing new limit theorems that describe its diverse diffusive and subdiffusive behaviors in such environments.

## Contribution

It establishes novel distributional limit theorems for random walks in strongly sparse environments, extending previous results to this specific case.

## Key findings

- Exhibits diffusive and subdiffusive scaling behaviors.
- Limit distributions are non-stable.
- Complements prior work on moderate and weak sparsity.

## Abstract

The integer points (sites) of the real line are marked by the positions of a standard random walk. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the standard random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity we consider a nearest neighbor random walk on the set of integers having jumps $\pm 1$ 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. (2018+) 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.