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