On some random walks on Z in random medium

Abstract

We consider random walks on $\Z$ in a stationary random medium, defined by an ergodic dynamical system, in case when possible jumps are $\{-L,\ldots,-1,0,+1\}$ for some fixed integer L. We provide a recurrence criterion expressed in terms of sign of maximal Liapounov exponent of a certain random matrix and give an algorithm of calculation of that exponent. Next, we characterize existence of absolutely continuous invariant measure for Markov chain of the environments viewed from particle and also characterize, in transient cases, existence of a nonzero drift. To study validity of central limit theorem, we consider notion of harmonic coordinates introduced by Kozlov. We characterize existence of both invariant measure and harmonic coordinates and show in recurrent case that existence of those two objects is equivalent to validity of an invariance principle. We give sufficient conditions for validity of central limit theorem in transient cases. Finally, we consider previous results in context of a random medium defined by an irrational rotation on circle and study their realization in terms of regularity and Diophantine approximation.