Lyapunov Exponents of Random Walks in Small Random Potential: The Lower Bound

We consider the simple random walk on , d > 3, evolving in a potential of the form beta V, where are i.i.d. random variables taking values in [0, + a), and beta > 0. When the potential is integrable, the asymptotic behaviours as beta tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small beta. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian .