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

Abstract

We consider a d-dimensional correlated percolation problem of sites 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 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 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 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 β from the L dependence of M_{1} and M_{2}, 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).