Transient random walks on a strip in a random environment

Abstract

We consider transient random walks on a strip in a random environment. The model was introduced by Bolthausen and Goldsheid [Comm. Math. Phys. 214 (2000) 429--447]. We derive a strong law of large numbers for the random walks in a general ergodic setup and obtain an annealed central limit theorem in the case of uniformly mixing environments. In addition, we prove that the law of the "environment viewed from the position of the walker" converges to a limiting distribution if the environment is an i.i.d. sequence.Comment: Published in at http://dx.doi.org/10.1214/08-AOP393 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip

“…Sufficient conditions for linear growth of a RWRE on a strip were recently obtained by Roitershtein [21]. He also proves, under certain mixing conditions imposed on the environment, the annealed CLT for this model.…”

“…The case considered in the just mentioned papers is m = 1 and hence the matrices ζ n are trivial: ζ n = 1. In the case m > 1 this function has been recently used in [21]. It turns out that it is important to know whether the lim sup in (2.1) can be replaced by lim and, if it exists, to be able to control the speed of convergence to this limit.…”

“…The annealed CLT can be derived from the quenched CLT if the environment satisfies certain mixing conditions (e.g. such as those used in [21]). There is a reason to believe that in general just the ergodicity of the environment is insufficient for having the annealed CLT.…”

Random Walk in Quasi-Periodic Random Environment

“…In the transient situations and still for an independent environment, a CLT with random centering was shown by Goldsheid in [14]. Media satisfying a mixing condition were investigated by Roitershtein in [24], who proved an annealed CLT for transient walks on a strip. As mentionned in a remark in [14], some more correlated environments could also be treated.…”

Branching Structure for the Transient (1, R)-Random Walk in Random Environment and Its Applications

“…✷ Another application of the branching structure is to specify the density of the absolutely continuous invariant measure for the "environments viewed from the particle". Let us review the process discussed in Section 4 of [10]. Let ω n = θ ξn w, for n ≥ 0, and consider the process Z n := (ω n , Y n ), defined in (Ω × D, F ⊗ H ), where H as the set of all subsets of D, and the initial distribution P µ…”