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=uL^{d} steps on a d-dimensional hypercubic lattice of size L^{d} (with periodic boundaries). We systematically explore dependence of the probability Π_{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 Π_{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≤d≤6. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with ν=2/(d-2). There is no percolation threshold at the lower critical dimension of d=2, with the percolation probability approaching a smooth function Π_{2}(∞,u)>0.