# 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.

## Key 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.