Pearson Walk with Shrinking Steps in Two Dimensions
Abstract
We study the shrinking Pearson random walk in two dimensions and greater, in which the direction of the Nth is random and its length equals lambdaN-1, with lambda<1. As lambda increases past a critical value lambdac, the endpoint distribution in two dimensions, P(r), changes from having a global maximum away from the origin to being peaked at the origin. The probability distribution for a single coordinate, P(x), undergoes a similar transition, but exhibits multiple maxima on a fine length scale for lambda close to lambdac. We numerically determine P(r) and P(x) by applying a known algorithm that accurately inverts the exact Bessel function product form of the Fourier transform for the probability distributions.
0