Abstract. 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(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 M(t). We show that, to leading order as t gets large, these fluctuations are all proportional to a single universal random process, η(t), normalized to η 2 (t) = 1. For t → ∞ the probability law of η(t) tends to that of Varadhan's renormalized local time of self-intersections. An implication is that in the long time limit all M(t) are proportional to S(t).
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