Coincidence of Lyapunov exponents for random walks in weak random potentials

Abstract

We investigate the free energy of nearest-neighbor random walks on ${\mathbb{Z}}^d$, endowed with a drift along the first axis and evolving in a nonnegative random potential given by i.i.d. random variables. Our main result concerns the ballistic regime in dimensions $d\ge 4$, at which we show that quenched and annealed Lyapunov exponents are equal as soon as the strength of the potential is small enough.

Published in at http://dx.doi.org/10.1214/00-AOP368 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)