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

Abstract

We consider the simple random walk on Z^d, d > 2, evolving in a potential of the form \beta V, where (V(x), x \in Z^d) are i.i.d. random variables taking values in [0,+\infty), 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 -\Delta\ + \beta V.