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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 $N=u{L}^{d}$ steps on a $d$-dimensional hypercubic lattice of size ${L}^{d}$ (with periodic boundaries). We systematically explore dependence of the probability ${\mathrm{\Pi }}_{d}\left(L,u\right)$ 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 ${\mathrm{\Pi }}_{d}\left(L,u\right)$ 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 $\nu =2/\left(d-2\right)$. There is no percolation threshold at the lower critical dimension of $d=2$, with the percolation probability approaching a smooth function ${\mathrm{\Pi }}_{2}\left(\infty ,u\right)>0$.
Keywords: function / percolation / threshold / walk / sites not removed / dimension / step