Random Walks in a Sparse Random Environment

TL;DR

This paper introduces a new model of random walks in sparse environments on integers, analyzing their asymptotic behaviors, including recurrence, speed, and limit theorems, with a focus on the critical recurrent regime.

Contribution

It presents a novel model combining features of existing random media models, providing new insights into asymptotic properties and generalizing Sinai's scaling in the recurrent regime.

Findings

Model exhibits recurrence and transience properties.

In the recurrent regime, the walk's location scales as $( ext{log } n)^eta$.

The parameter $eta$ depends on impurity distribution.

Abstract

We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a "locally strong" perturbation of a simple random walk by a random potential induced by "rare impurities," which are randomly distributed over the integer lattice. Interestingly, in the critical (recurrent) regime, our model generalizes Sinai's scaling of $(\log n)^2$ for the location of the random walk after $n$ steps to $(\log n)^\alpha,$ where $\alpha>0$ is a parameter determined by the distribution of the distance between two successive impurities. Similar scaling factors have appeared in the literature in different contexts and have been discussed in [28] and [30].