Environment viewed from the particle and slowdown for ballistic RWRE in low dimensions

TL;DR

This paper studies the environment viewed from a particle in ballistic random walks in low dimensions, disproving a previous conjecture, and provides tail estimates for invariant measures and regeneration times.

Contribution

It establishes the existence of an invariant measure for the environment viewed from the particle in 2D and 3D, and derives tail bounds for Radon-Nikodym derivatives and regeneration times.

Findings

Invariant measure $Q$ exists for $d=2,3$, disproving prior conjecture.

Tail estimates for $dQ/dP$ are obtained.

Nearly sharp tail bounds for regeneration times in $d=3$.

Abstract

We consider a random walk in a random environment on $\mathbb{Z}^d$ under ballisticity condition $(T)$. We show the existence of the invariant measure $Q$ with respect to the environment viewed from the particle for $d=2$ and $d=3$, which disproves a conjecture made in arXiv:1405.6819 regarding the two-dimensional case. We also prove tail estimates for the Radon-Nikodym derivative $dQ/dP$, where $P$ is the original distribution on the environment. Lastly, we provide nearly sharp tail bounds for regeneration times for $d=3$.