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 XR be the probability that the walk that started at the origin exits the strip through one of the boundary lines. Then XR is a random variable, depending on the environment. In dimension d = 1, the variable XR 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.