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

TL;DR

This study investigates the critical behavior of sites not visited by a random walk in correlated percolation across dimensions 3 to 5, revealing dimension-dependent critical exponents and a relation for the exponent gamma.

Contribution

It introduces a method to determine the critical point and critical exponents for correlated percolation of unvisited sites by a random walk in multiple dimensions.

Findings

The ratio of the largest to second largest cluster mass is dimension-independent at criticality.

The exponent beta remains close to 1 across dimensions.

The exponent gamma decreases with dimension, approaching a theoretical value in six dimensions.

Abstract

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 $\gamma=4/(d-2)$.