Percolation perspective on sites not visited by a random walk in two dimensions

We consider the percolation problem of sites on an L×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^{-τ}, while the volume fraction P_{k} occupied by the kth largest cluster scales as k^{-q}. We put forward a heuristic argument that τ=2 and q=1. However, the numerically measured values are τ≈1.83 and q≈1.20. We suggest that these are effective exponents that drift towards their asymptotic values with increasing L as slowly as 1/lnL approaches zero.