Universal fluctuations in the support of the random walk

Details Universal fluctuations in the support of the random walk Universal fluctuations in the support of the random walk

1996-11-26

arxiv.org/abs/cond-mat/9611202v1

Physics

Condensed Matter

Statistical Mechanics

17 pages, LaTeX, 2 figures included

Scientific paper

10.1007/BF02770757

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 (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)-(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.

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