Percolation Perspective on Sites Not Visited by a Random Walk in Two Dimensions

We consider the percolation problem of sites on an $L\times L$ square lattice with periodic boundary conditions which were unvisited by a random walk of $N=uL^2$ steps, i.e. are vacant. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with $u$. The clusters of vacant sites are not fractal but have fractal boundaries of dimension 4/3. The lattice size $L$ is the only large length scale in this problem. The typical mass (number of sites $s$) in the largest cluster is proportional to $L^2$, and the mean mass of the remaining (smaller) clusters is also proportional to $L^2$. The normalized (per site) density $n_s$ of clusters of size (mass) $s$ is proportional to $s^{-\tau}$, while the volume fraction $P_k$ occupied by the $k$th largest cluster scales as $k^{-q}$. We put forward a heuristic argument that $\tau=2$ and $q=1$. However, the numerically measured values are $\tau\approx1.83$ and $q\approx1.20$. We suggest that these are effective exponents that drift towards their asymptotic values with increasing $L$ as slowly as $1/\ln L$ approaches zero.