Coincidence of Lyapunov exponents for random walks in weak random potentials

Abstract

We investigate the free energy of nearest-neighbor random walks on 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≥4, at which we show that quenched and annealed Lyapunov exponents are equal as soon as the strength of the potential is small enough.