Random walks in random medium on [formula omitted] and Lyapunov spectrum

Abstract

We consider a one-dimensional random walk with bounded steps in a stationary and ergodic random medium. We show that the algebraic structure of the random walk is given by geometrical invariants related to the description of a space of harmonic functions. We then prove a recurrence criterion similar to Key's Theorem [E.S. Key, Ann. Probab. 12 (2) (1984) 529] in terms of the sign of an intermediate Lyapunov exponent of a random matrix. We show that this exponent is simple and we relate it to the dominant exponents of two non-negative matrices associated to the random walks of left and right records. We also give an algorithm to compute that exponent. In a last part, we deduce from [J. Brémont, Ann. Probab. 30 (3) (2002) 1266] that the Law of Large Numbers is always valid.