Statistical properties of the set of sites visited by the two-dimensional random walk

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 for example 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) over bar(t), on appropriate lattice size-dependent time scales. On an infinite lattice we first observe that the (M) over bar(t) all increase with t as similar to t/log(k), 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))over bar>(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).