Regularity of biased 1D random walks in random environment

TL;DR

This paper investigates the asymptotic behavior of 1D biased random walks in random environments, revealing properties of their velocity and diffusivity, including analyticity, discontinuities, and the Einstein relation.

Contribution

It provides new insights into the regularity of the velocity and diffusivity functions, including explicit conditions for analyticity and the first proof of the Einstein relation in the random conductance model.

Findings

Velocity $v()$ is increasing and mostly analytic, with possible discontinuities.

Diffusivity $\sigma^2()$ can be non-monotone and non-differentiable at zero.

The Einstein relation holds for the random conductance model in both discrete and continuous time.

Abstract

We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity $\lambda\in\mathbb{R}$. For ergodic shift-invariant environments, we show that the limiting velocity $v(\lambda)$ is always increasing and that it is everywhere analytic except at most in two points $\lambda_-$ and $\lambda_+$. When $\lambda_-$ and $\lambda_+$ are distinct, $v(\lambda)$ might fail to be continuous. We refine the assumptions in \cite{Z} for having a recentered CLT with diffusivity $\sigma^2(\lambda)$ and give explicit conditions for $\sigma^2(\lambda)$ to be analytic. For the random conductance model we show that, in contrast with the deterministic case, $\sigma^2(\lambda)$ is not monotone on the positive (resp.~negative) half-line and that it is not differentiable at $\lambda=0$. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of \cite{LD16}.