Directional decay of the green's function for a random nonnegative potential on ℤd

Abstract

We derive a shape theorem type result for the almost sure exponential decay of the Green's function of -σ + V, where the potentials V(x), x ∈ ℤd, are i.i.d. nonnegative random variables. This result implies a large deviation principle governing the position of a d-dimensional random walk moving in the same potential.