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 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.

8 pages, 6 figures. Version 2: various corrections in response for referees. This is the final version for publication in JSTAT

References in corpus (2)

• On the Inelastic Collapse of a Ball Bouncing on a Randomly Vibrating Platform
• Random Walkers with Shrinking Steps in d-Dimensions and Their Long Term Memory

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• Statistics of Projected Motion in one dimension of a d-dimensional Random Walker