{
  "id": "2405.14950",
  "version": "v1",
  "published": "2024-05-23T18:00:59.000Z",
  "updated": "2024-05-23T18:00:59.000Z",
  "title": "Critical exponents of correlated percolation of sites not visited by a random walk",
  "authors": [
    "Raz Halifa Levi",
    "Yacov Kantor"
  ],
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We consider a $d$-dimensional correlated percolation problem of sites {\\em not} visited by a random walk on a hypercubic lattice $L^d$ for $d=3$, 4 and 5. The length of the random walk is ${\\cal N}=uL^d$. Close to the critical value $u=u_c$, many geometrical properties of the problem can be described as powers (critical exponents) of $u_c-u$, such as $\\beta$, which controls the strength of the spanning cluster, and $\\gamma$, which characterizes the behavior of the mean finite cluster size $S$. We show that at $u_c$ the ratio between the mean mass of the largest cluster $M_1$ and the mass of the second largest cluster $M_2$ is independent of $L$ and can be used to find $u_c$. We calculate $\\beta$ from the $L$-dependence of $M_2$ and $\\gamma$ from the finite size scaling of $S$. The resulting exponent $\\beta$ remains close to 1 in all dimensions. The exponent $\\gamma$ decreases from $\\approx 3.9$ in $d=3$ to $\\approx1.9$ in $d=4$ and $\\approx 1.3$ in $d=5$ towards $\\gamma=1$ expected in $d=6$, which is close to $\\gamma=4/(d-2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-23T18:00:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walk",
      "critical exponents",
      "dimensional correlated percolation problem",
      "second largest cluster",
      "mean finite cluster size"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}