Percolation of sites not removed by a random walker in d dimensions

Abstract

How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of $N=u{L}^d$ steps on a $d$-dimensional hypercubic lattice of size $L^d$ (with periodic boundaries). We systematically explore dependence of the probability ${\mathrm{\Pi }}_d(L,u)$ of percolation (existence of a spanning cluster) of sites not removed by the RW on $L$ and $u$. The concentration of unvisited sites decays exponentially with increasing $u$, while the visited sites are highly correlated—their correlations decaying with the distance $r$ as $1/r^{d-2}$ (in $d>2$). On increasing $L$, the percolation probability ${\mathrm{\Pi }}_d(L,u)$ approaches a step function, jumping from 1 to 0 when $u$ crosses a percolation threshold $u_c$ that is close to 3 for all $3\le d\le 6$. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with $\nu =2/(d-2)$. There is no percolation threshold at the lower critical dimension of $d=2$, with the percolation probability approaching a smooth function ${\mathrm{\Pi }}_2(\infty ,u)>0$.