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Lyapunov exponents of random walks in small random potential: the lower bound

We consider the simple random walk on Z^d, d > 2, evolving in a potential of the form βV, where (V(x), x \in Z^d) are i.i.d. random variables taking values in [0,+\infty), and β > 0. When the potential is integrable, the asymptotic behaviours as β 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 β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -Δ + βV.