On the recurrence of some random walks in random environment

TL;DR

This paper investigates the recurrence properties of certain two-dimensional random walks in random environments, introducing new estimates for return probabilities and demonstrating infinite meetings of multiple walkers.

Contribution

It provides novel estimates for quenched return probabilities in non-reversible 2D RWRE and analyzes recurrence in models involving products of 1D recurrent RWREs.

Findings

Multiple independent walkers meet infinitely often at the origin.

Models with 1D recurrent RWRE are more recurrent than symmetric walks.

New valley construction method for potential in recurrence proofs.

Abstract

This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They imply that k independent walkers in the same one-dimensional (recurrent) environment will meet in the origin infinitely often, for any k. We also consider direct products of one-dimensional recurrent RWRE with another RWRE or with a RW. We point out the that models involving one-dimensional recurrent RWRE are more recurrent than the corresponding models involving simple symmetric walk.