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 λ ∈ R. For ergodic shift-invariant environments, we show that the limiting velocity v(λ) is always increasing and that it is everywhere analytic except at most in two points λ− and λ+. When λ− and λ+ are distinct, v(λ) might fail to be continuous. We refine the assumptions in [?] for having a recentered CLT with diffusivity σ 2 (λ) and give explicit conditions for σ 2 (λ) to be analytic. For the random conductance model we show that, in contrast with the deterministic case, σ 2 (λ) is not monotone on the positive (resp. negative) half-line and that it is not differentiable at λ = 0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of Lam and Depaw (2016).