Critical exponents of correlated percolation of sites not visited by a random walk

We consider a d-dimensional correlated percolation problem of sites { not} visited by a random walk on a hypercubic lattice Lᵈ for $d=3$, 4 and 5. The length of the random walk is N=uLᵈ. Close to the critical value u=uc, many geometrical properties of the problem can be described as powers (critical exponents) of uc-u, such as β, which controls the strength of the spanning cluster, and γ, which characterizes the behavior of the mean finite cluster size S. We show that at uc the ratio between the mean mass of the largest cluster M₁ and the mass of the second largest cluster M₂ is independent of L and can be used to find uc. We calculate β from the L-dependence of M₂ and γ from the finite size scaling of S. The resulting exponent β remains close to 1 in all dimensions. The exponent γ decreases from ≈ 3.9 in $d=3$ to ≈1.9 in $d=4$ and ≈ 1.3 in $d=5$ towards γ=1 expected in $d=6$, which is close to γ=4/(d-2).