{
  "id": "2101.06655",
  "version": "v1",
  "published": "2021-01-17T12:09:44.000Z",
  "updated": "2021-01-17T12:09:44.000Z",
  "title": "Percolation Perspective on Sites Not Visited by a Random Walk in Two Dimensions",
  "authors": [
    "Amit Federbush",
    "Yacov Kantor"
  ],
  "comment": "14 pages, 13 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We consider the percolation problem of sites on an $L\\times L$ square lattice with periodic boundary conditions which were {\\em unvisited} by a random walk of $N=uL^2$ steps, i.e. are {\\em vacant}. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with $u$. The clusters of vacant sites are {\\em not} fractal but have fractal boundaries of dimension 4/3. The lattice size $L$ is the only large length scale in this problem. The typical mass (number of sites $s$) in the largest cluster is proportional to $L^2$, and the mean mass of the remaining (smaller) clusters is also proportional to $L^2$. The normalized (per site) density $n_s$ of clusters of size (mass) $s$ is proportional to $s^{-\\tau}$, while the volume fraction $P_k$ occupied by the $k$th largest cluster scales as $k^{-q}$. We put forward a heuristic argument that $\\tau=2$ and $q=1$. However, the numerically measured values are $\\tau\\approx1.83$ and $q\\approx1.20$. We suggest that these are effective exponents that drift towards their asymptotic values with increasing $L$ as slowly as $1/\\ln L$ approaches zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-17T12:09:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walk",
      "percolation perspective",
      "th largest cluster scales",
      "periodic boundary conditions",
      "proportional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}