Zero-one law for directional transience of one-dimensional random walks in dynamic random environments

TL;DR

This paper establishes a comprehensive zero-one law for the directional behavior of one-dimensional random walks in dynamic random environments, covering various models under broad assumptions.

Contribution

It proves a trichotomy law for transience and recurrence in dynamic environments, extending previous results to more general and non-uniformly elliptic models.

Findings

Transience to the right or left occurs with probability zero or one.

Recurrence is guaranteed for symmetric models.

The results apply to both uniformly and non-uniformly elliptic cases.

Abstract

We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.