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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 anonnegative random potential given by i.i.d. random variables. Our main resultconcerns the ballistic regime in dimensions $d\geq4$, at which we show thatquenched and annealed Lyapunov exponents are equal as soon as the strength ofthe potential is small enough.

Coincidence of Lyapunov exponents for random walks in weak random potentials