UNIVERSAL FLUCTUATIONS IN THE SUPPORT OF THE RANDOM-WALK

A random walk starts from the origin of a d-dimensional lattice. The o ccupation number n(x, t) equals unity if after t steps site x has been visited by the walk, and zero otherwise. We study translationally inv ariant sums M(t) of observables defined locally on the field of occupa tion numbers. Examples are the number S(t) of visited sites, the area E(t) of the (appropriately defined) surface of the set of visited site s, and, in dimension d = 3, the Euler index of this surface. In d grea ter than or equal to 3, the averages (M) over bar(t) all increase line arly 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) over bar(t) are, up to a normalization, all identical to a single ''universal'' random variable. This result resembles an ea rlier one in dimension d = 2; we show that this universality breaks do wn for d > 3.