Universal fluctuations in the support of the random walk

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.

17 pages, LaTeX, 2 figures included

Cited by in corpus (6)

• Exact distributions of the number of distinct and common sites visited by N independent random walkers
• Number of distinct sites visited by a resetting random walker
• Critical scaling in standard biased random walks
• Lattice gases with a point source
• Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice
• Information Loss in Coarse Graining of Polymer Configurations via Contact Matrices