# Pearson Walk with Shrinking Steps in Two Dimensions

Journal Article

**Published:**Thursday, January 14, 2010

**Citation:**Journal of Statistical Mechanics: Theory and Experiment, Pages P01006

**AIP Permalink**: http://www.iop.org/EJ/abstract/1742-5468/2010/01/P01006

**Authors (2 total):** C. A. Serino,
S. Redner

**Abstract:**

We study the shrinking Pearson random walk in two dimensions and greater, in which the direction of the Nth step is random and its length equals lambda^{N-1}, with lambda<1. As lambda increases past a critical value lambda_c, 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 lambda_c. 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.