Directional decay of the Green's function for a random nonnegative potential on ${\bf Z}\sp d$

Abstract

We derive a shape theorem type result for the almost sure exponential decay of the Green's function of $-\Delta + V$, where the potentials $V(x), x \epsilon \mathbb{Z}^d$ 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.