Random Walkers with Shrinking Steps in d-Dimensions and Their Long Term Memory

TL;DR

This paper investigates d-dimensional random walkers with exponentially shrinking steps, developing a 1/d expansion, and shows that a modified diffusion equation effectively describes their long-term behavior.

Contribution

It introduces a 1/d expansion technique for analyzing shrinking-step random walks and derives a continuum equation that captures their qualitative dynamics.

Findings

Continuum equation with decaying diffusion constant models long-term behavior.

1/d expansion provides insights into correlations and moments.

Continuum approximation is effective for qualitative analysis.

Abstract

We study, in d-dimensions, the random walker with geometrically shrinking step sizes at each hop. We emphasize the integrated quantities such as expectation values, cumulants and moments rather than a direct study of the probability distribution. We develop a 1/d expansion technique and study various correlations of the first step to the position as ti me goes to infinity. We also show and substantiate with a study of the cumulants that to order 1/d the system admits a continuum counterpart equation which can be obtained with a generalization of the ordinary technique to obtain the continuum limit. We also advocate that this continuum counterpart equation, which is nothing but the ordinary diffusion equation with a diffusion constant decaying exponentially in continuous time, captures all the qualitative aspects of t he discrete system and is often a good starting point for quantitative approximations.