On the Strong Recurrence of Recurrent RWRE

TL;DR

This paper studies the return probabilities of recurrent one-dimensional random walks in random environments, revealing decay rates and providing examples that highlight strong recurrence phenomena compared to classical symmetric random walks.

Contribution

It offers new insights into the decay of return probabilities in recurrent RWRE and presents examples illustrating strong recurrence in multidimensional cases.

Findings

Decay rates of quenched return probabilities established

Examples demonstrating strong recurrence behavior

Comparison with symmetric random walk on 

Abstract

We consider a recurrent RWRE $(X_n)_{n \in \mathbb{N}_0}$ on $\mathbb{Z}$ and investigate the quenched return probabilities of the RWRE to the origin for which we state results on their decay in terms of summability. Additionally, we give some examples for recurrent RWRE with a multidimensional state space which give reason for the part "strong recurrence" in the title of this paper when we compare the behaviour of RWRE with the behaviour of the symmetric random walk on $\mathbb{Z}^d$.