Topology of the support of the two-dimensional random walk

TL;DR

This paper investigates the large-scale geometric and topological properties of the support of a two-dimensional random walk, revealing universal fluctuation behavior and scaling laws for various global features of the visited sites.

Contribution

It introduces a comprehensive analysis of the support's topology, including scaling functions and universal fluctuation processes, extending understanding of random walk support structures.

Findings

Average support size scales as t / log^k t

Fluctuations are proportional to a universal process eta(t)

Long-term fluctuations relate to Varadhan's local time

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, 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/\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 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(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).