A note on the antisymmetry in the speed of a random walk in reversible dynamic random environment

TL;DR

This paper demonstrates that in a reversible dynamic random environment, the speed of a one-dimensional random walk exhibits antisymmetry with respect to the bias parameter, using a coupling argument in the discrete setting.

Contribution

It establishes the antisymmetry property of the walk's speed in reversible environments, a result not previously formalized.

Findings

Proves that v(-ε) = -v(ε) in reversible environments.

Uses a coupling argument specific to the discrete setting.

Applicable under conditions where the weak LLN holds.

Abstract

In this short note, we prove that $v(-\epsilon)=-v(\epsilon)$. Here, $v(\epsilon)$ is the speed of a one-dimensional random walk in a dynamic \emph{reversible} random environment, that jumps to the right (resp. to the left) with probability $1/2+\epsilon$ (resp. $1/2-\epsilon$) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where $v(\epsilon), v(-\epsilon)$ are well-defined, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only in the discrete setting.