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

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, , on appropriate lattice size-dependent time scales. On an infinite lattice we first observe that the all increase with t as , where k is an M-dependent positive integer. We then consider the class of random processes constituted by the fluctuations around average . We show that, to leading order as t gets large, these fluctuations are all proportional to a single universal random process, , normalized to . For the probability law of tends to that of Varadhan's renormalized local time of self-intersections. An implication is that in the long time limit all are proportional to .