Coincidence of Lyapunov exponents for random walks in weak random   potentials
  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 $d\geq4$, at which we show that quenched and annealed Lyapunov exponents are equal as soon as the strength of the potential is small enough. 
Markus Flury