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

, Mehran Kardar
Published: 20 August 2019

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=uLd steps on a d-dimensional hypercubic lattice of size Ld (with periodic boundaries). We systematically explore dependence of the probability Πd(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/rd2 (in d>2). On increasing L, the percolation probability Πd(L,u) approaches a step function, jumping from 1 to 0 when u crosses a percolation threshold uc that is close to 3 for all 3d6. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with ν=2/(d2). There is no percolation threshold at the lower critical dimension of d=2, with the percolation probability approaching a smooth function Π2(,u)>0.
Keywords: function / percolation / threshold / walk / sites not removed / dimension / step
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