Pearson Walk with Shrinking Steps in Two Dimensions

TL;DR

This paper investigates the behavior of a two-dimensional shrinking Pearson random walk, analyzing how the endpoint distribution transitions as the shrinking factor crosses a critical value, using numerical methods to accurately determine the distributions.

Contribution

It introduces a detailed numerical analysis of the endpoint distributions in a shrinking Pearson walk, revealing phase-like transitions in the distribution shape as lambda varies.

Findings

Distribution P(r) shifts from off-center to centered at the origin at lambda_c.

P(x) exhibits multiple maxima near lambda_c.

Numerical methods accurately invert Bessel function products for distribution calculation.

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.