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

We consider the simple random walk on ${mathbb{Z}^d}$ Z d , d > 3, evolving in a potential of the form β V, where ${(V(x))_{x in mathbb{Z}^d}}$ ( V ( x ) ) x ∈ Z d are i.i.d. random variables taking values in [0, + ∞), 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 ${-triangle + beta V}$ - ? + β V .