Two-dimensional random walk in a bounded domain

TL;DR

This paper investigates the behavior of two-dimensional random walks in bounded domains, revealing that the distribution of returning walkers forms a fractal Devil's staircase pattern, linked to non-linear stochastic maps.

Contribution

It demonstrates that the cumulative probability distribution of returning walkers is a Devil's staircase, connecting it to non-linear stochastic maps and fractal properties.

Findings

Distribution is a Devil's staircase.

Fractal dimension of the PDF boundary is approximately 1.75.

Universal feature of the constrained random walk.

Abstract

In a recent Letter Ciftci and Cakmak [EPL 87, 60003 (2009)] showed that the two dimensional random walk in a bounded domain, where walkers which cross the boundary return to a base curve near origin with deterministic rules, can produce regular patterns. Our numerical calculations suggest that the cumulative probability distribution function of the returning walkers along the base curve is a Devil's staircase, which can be explained from the mapping of these walks to a non-linear stochastic map. The non-trivial probability distribution function(PDF) is a universal feature of CCRW characterized by the fractal dimension d=1.75(0) of the PDF bounding curve.