{
  "id": "1907.00018",
  "version": "v1",
  "published": "2019-06-28T18:08:33.000Z",
  "updated": "2019-06-28T18:08:33.000Z",
  "title": "Percolation of sites not removed by a random walker in $d$ dimensions",
  "authors": [
    "Yacov Kantor",
    "Mehran Kardar"
  ],
  "comment": "RevTex, 12 pages, 13 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "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 $\\Pi_d(L,u)$ 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 $d>2$). Upon increasing $L$, the percolation probability $\\Pi_d(L,u)$ approaches a step function, jumping from 1 to 0 when $u$ crosses a percolation threshold $u_c$ that is close to 3 for all $3\\le d\\le6$. 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 $\\Pi_2(\\infty,u)>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-28T18:08:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walker",
      "percolation threshold",
      "percolation probability",
      "dimensional hypercubic lattice",
      "correlated site percolation"
    ],
    "note": {
      "typesetting": "RevTeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}