STATISTICAL PROPERTIES OF THE SET OF SITES VISITED BY THE 2-DIMENSIONAL RANDOM-WALK

We study the support (i.e. the set of visited sites) of a t-step rando m walk on a two-dimensional square lattice in the large t limit. A bro ad class of global properties, M(t), of the support is considered, inc luding for example the number, S(t), of its sites; the length of its b oundary; the number of islands of unvisited sites that it encloses; th e number of such islands of given shape, size, and orientation; and th e number of occurrences in space of specific local patterns of visited and unvisited sites. On a finite lattice we determine the scaling fun ctions that describe the averages, (M) over bar(t), on appropriate lat tice size-dependent time scales. On an infinite lattice we first obser ve 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 cla ss of random processes constituted by the fluctuations around average Delta M(t). We show that, to leading order as t gets large,these fluct uations are all proportional to a single universal random process, eta (t), normalized to <(eta(2))over bar>(t) = 1. For t --> infinity the p robability law of eta(t) tends to that of Varadhan's renormalized loca l time of self-intersections. An implication is that in the long time limit all Delta M(t) are proportional to Delta S(t).