{
  "id": "2003.09590",
  "version": "v1",
  "published": "2020-03-21T06:37:44.000Z",
  "updated": "2020-03-21T06:37:44.000Z",
  "title": "Percolation between $k$ separated points in two dimensions",
  "authors": [
    "S. S. Manna",
    "Robert M. Ziff"
  ],
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech"
  ],
  "abstract": "We consider a percolation process in which $k$ widely separated points simultaneously connect together ($k>1$), or a single point at the center of a system connects to the boundary ($k=1$), through adjacent connected points of a single cluster. These processes yield new thresholds $\\overline p_{ck}$ defined as the average value of $p$ at which the desired connections first occur. These thresholds are not sharp as the distribution of values of $p_{ck}$ remains broad in the limit of $L \\to \\infty$. We study $\\overline p_{ck}$ for bond percolation on the square lattice, and find that $\\overline p_{ck}$ are above the normal percolation threshold $p_c = 1/2$ and represent specific supercritical states. The $\\overline p_{ck}$ can be related to integrals over powers of the function $P_\\infty(p) =$ the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of $P_\\infty(p)$ on $L\\times L$ systems that for $L \\to \\infty$, $\\overline p_{c1} = 0.51761(3)$, $\\overline p_{c2} = 0.53220(3)$, $\\overline p_{c3} = 0.54458(3)$, and $\\overline p_{c4} = 0.55530(3)$. The percolation thresholds $\\overline p_{ck}$ remain the same, even when the $k$ points are randomly selected within the lattice. We show that the finite-size corrections scale as $L^{-1/\\nu_k}$ where $\\nu_k = \\nu/(k \\beta +1)$, with $\\beta=5/36$ and $\\nu=4/3$ being the ordinary percolation critical exponents, so that $\\nu_1= 48/41$, $\\nu_2 = 24/23$, $\\nu_3 = 16/17$, $\\nu_4 = 6/7$, etc. We also study three-point correlations in the system, and show how for $p>p_c$, the correlation ratio goes to 1 (no net correlation) as $L \\to \\infty$, while at $p_c$ it reaches the known value of 1.021$\\ldots.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-21T06:37:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dimensions",
      "normal percolation threshold",
      "desired connections first occur",
      "represent specific supercritical states",
      "ordinary percolation critical exponents"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}