  We study the support (i.e. the set of visited sites) of a t step random walk
on a two-dimensional square lattice in the large t limit. A broad class of
global properties M(t) of the support is considered, including, e.g., the
number S(t) of its sites; the length of its boundary; the number of islands of
unvisited sites that it encloses; the number of such islands of given shape,
size, and orientation; and the number of occurrences in space of specific local
patterns of visited and unvisited sites. On a finite lattice we determine the
scaling functions that describe the averages <M(t)> on appropriate lattice size
dependent time scales. On an infinite lattice we first observe that the <M(t)>
all increase with t as t/\log^k t, where k is an M dependent positive integer.
We then consider the class of random processes constituted by the fluctuations
around average Delta M(t). We show that to leading order as t gets large these
fluctuations are all proportional to a single universal random process eta(t),
normalized to <eta^2(t)>=1$. For t--> infinity the probability law of eta(t)
tends to that of Varadhan's renormalized local time of self-intersections. An
implication is that in the long time limit all Delta M(t) are proportional to
Delta S(t).
