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