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

We consider the percolation problem of sites on an L x 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 kth 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 approximate to 1.83 and q approximate to 1.20. We suggest that these are effective exponents that drift towards their asymptotic values with increasing L as slowly as 1/ln L approaches zero.