On favourite sites of a random walk in moderately sparse random environment

TL;DR

This paper investigates the behavior of a symmetric random walk in a sparse random environment, focusing on its favorite sites and how the environment's sparsity influences the walk's long-term distribution and limiting behavior.

Contribution

It extends known results for i.i.d. environments to sparse environments and uncovers new behaviors caused by environment sparsity.

Findings

Limit theorems for favorite sites in different environmental regimes

Generalization of known results to sparse environments

Discovery of new behavior due to environment sparsity

Abstract

We study the favourite sites of a random walk evolving in a sparse random environment on the set of integers. The walker moves symmetrically apart from some randomly chosen sites where we impose random drift. We prove annealed limit theorems for the time the walk spends in its favourite sites in two cases. The first one, in which it is the distribution of the drift that determines the limiting behaviour of the walk, is a generalization of known results for a random walk in i.i.d. random environment. In the second case a new behaviour appears, caused by the sparsity of the environment.