  We consider a $d$-dimensional correlated percolation problem of sites {\em
not} visited by a random walk on a hypercubic lattice $L^d$ for $d=3$, 4 and 5.
The length of the random walk is ${\cal N}=uL^d$. Close to the critical value
$u=u_c$, many geometrical properties of the problem can be described as powers
(critical exponents) of $u_c-u$, such as $\beta$, which controls the strength
of the spanning cluster, and $\gamma$, which characterizes the behavior of the
mean finite cluster size $S$. We show that at $u_c$ the ratio between the mean
mass of the largest cluster $M_1$ and the mass of the second largest cluster
$M_2$ is independent of $L$ and can be used to find $u_c$. We calculate $\beta$
from the $L$-dependence of $M_2$ and $\gamma$ from the finite size scaling of
$S$. The resulting exponent $\beta$ remains close to 1 in all dimensions. The
exponent $\gamma$ decreases from $\approx 3.9$ in $d=3$ to $\approx1.9$ in
$d=4$ and $\approx 1.3$ in $d=5$ towards $\gamma=1$ expected in $d=6$, which is
close to $\gamma=4/(d-2)$.
