On the limiting velocity of random walks in mixing random environment

TL;DR

This paper studies the behavior of random walks in complex, strongly-mixing environments, providing new proofs for existing laws and establishing the uniqueness of nonzero limiting velocities in high dimensions.

Contribution

It offers an alternative proof of the conditional law of large numbers for mixing environments and proves the uniqueness of nonzero limiting velocities in dimensions five and higher.

Findings

Alternative proof of the conditional law of large numbers for mixing environments

Uniqueness of nonzero limiting velocity in dimensions ≥ 5

Application of coupling techniques to analyze limiting velocities

Abstract

We consider random walks in strong-mixing random Gibbsian environments in $\mathbb{Z}^d, d\ge 2$. Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha's conditional law of large numbers (CLLN) for mixing environment Rassoul-Agha (2005). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ($d\ge 5$).