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.