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 $\mathcal{N}=u{L}^d$. 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 $M_1$ and $M_2$, 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 $\approx 1.9$ in $d=4$ and $\approx 1.3$ in $d=5$ towards $\gamma =1$ expected in $d=6$, which is close to $\gamma =4/(d-2)$.