  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.
