Pearson Walk with Shrinking Steps in Two Dimensions

Details Pearson Walk with Shrinking Steps in Two Dimensions Pearson Walk with Shrinking Steps in Two Dimensions

2009-10-05

arxiv.org/abs/0910.0852v3

J. Stat. Mech. P01006 (2010)

Physics

Data Analysis, Statistics and Probability

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

Scientific paper

10.1088/1742-5468/2010/01/P01006

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.

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