Topology of the Support of the Two-Dimensional Lattice Random Walk

Abstract

The average number $〈I\left(t\right)〉$ of islands (connected sets of unvisited sites) enclosed by the support of a 2D simple random walk is studied for a large number $t$ of steps. On an infinite square lattice the result $〈I\left(t\right)〉\simeq \frac{1}{2}\pi (\pi -2\mathrm{})t/\ln {}^{2}8t$ is the difference between a creation and a destruction term that are both $\propto t/\ln 8t$. On a finite lattice of $N$ sites, for $N$ large, $〈I\left(t\right)〉$ attains a maximum value $\left[\right(\pi -2\mathrm{})/2e]N/\ln N$ for $t\simeq {\pi }^{-1\mathrm{}}N\ln N$, and goes down to $O\left(1\right)$ for $t\simeq {\pi }^{-1\mathrm{}}N{\ln }^{2}N$. The scaling function describing this behavior is given and compared with simulations by Coutinho et al.