Universal fluctuations in the support of the random walk

TL;DR

This paper investigates the fluctuations of the support of a random walk in various dimensions, revealing a universal behavior in three dimensions that does not extend to higher dimensions.

Contribution

It demonstrates a universal fluctuation behavior in three-dimensional random walks and clarifies its breakdown in higher dimensions.

Findings

Universal fluctuations in 3D random walks

Breakdown of universality in dimensions greater than 3

Linear growth of average support in dimensions > 3

Abstract

A random walk starts from the origin of a d-dimensional lattice. The occupation number n(x,t) equals unity if after t steps site x has been visited by the walk, and zero otherwise. We study translationally invariant sums M(t) of observables defined locally on the field of occupation numbers. Examples are the number S(t) of visited sites; the area E(t) of the (appropriately defined) surface of the set of visited sites; and, in dimension d=3, the Euler index of this surface. In d > 3, the averages <M>(t) all increase linearly with t as t-->infinity. We show that in d=3, to leading order in an asymptotic expansion in t, the deviations from average Delta M(t)= M(t)-<M>(t) are, up to a normalization, all identical to a single "universal" random variable. This result resembles an earlier one in dimension d=2; we show that this universality breaks down for d>3.