We investigate the free energy of nearest-neighbor random walks on ℤd, 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.
"Coincidence of Lyapunov exponents for random walks in weak random potentials." Ann. Probab. 36 (4) 1528 - 1583, July 2008. https://doi.org/10.1214/00-AOP368