Regularity of biased 1D random walks in random environment

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}.

34 pages, 3 figure2. Corrected and extended version