  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.
