The 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 theNth step is random andits length equals λN−1, with λ<1. Asλ increases past acritical value λc, the endpoint distribution in two dimensions,P(r), changes from having a global maximum away from the origin to beingpeaked 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 forλ close toλc. We numericallydetermine P(r) and P(x) by applying a known algorithm that accurately inverts the exact Bessel function productform of the Fourier transform for the probability distributions.