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

0. Introduction and notation. We consider the following model of a random walk evolving in a random environment on Zd (d > 1). Let (S"),r be a time discrete, symmetric, nearest-neighbor random walk on the hypercubic lattice Zd with start in x and denote by Px and Ex the probability measure and the expectation, respectively, of the underlying probability space. The environment is assumed to be independent of the random walk and is given by the potentials W(x), x E Ed, which are supposed to be independent and identically distributed nonnegative random variables. Then for a fixed realization w of the environment, the Green's function of x, y E Zd is defined as