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=uL2 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 L2, and the mean mass of the remaining (smaller) clusters is also proportional to L2. The normalized (per site) density ns of clusters of size (mass) s is proportional to s-τ, while the volume fraction Pk 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.