Numerical Simulations of Random Walk in Random Environment

Abstract

This paper is concerned with the numerical simulation of a random walk in a random environment in dimension d = 2. Consider a nearest neighbor random walk on the 2-dimensional integer lattice. The transition probabilities at each site are assumed to be themselves random variables, but fixed for all time. This is the random environment. Consider a parallel strip of radius R centered on an axis through the origin. Let X R be the probability that the walk that started at the origin exits the strip through one of the boundary lines. Then X R is a random variable, depending on the environment. In dimension d = 1, the variable X R converges in distribution to the Bernoulli variable, X ∞ = 0, 1 with equal probability, as R → ∞. Here the 2-dimensional problem is studied using Gauss-Seidel and multigrid algorithms.