1996/09/02 by Frédéric van Wijland, F. van Wijland, van Wijland, F. +4
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat
paper · pdf · doi:10.48550/arxiv.cond-mat/9609018
34 pages, LaTeX, uses amstex and righttag, figure available upon request at [email protected]
arxiv created 1996/09/02 · arxiv updated 2009/11/30
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/logk 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 <eta2(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).