{
  "id": "2402.01553",
  "version": "v3",
  "published": "2024-02-02T16:46:06.000Z",
  "updated": "2024-07-09T04:58:08.000Z",
  "title": "Q-factor: A measure of competition between the topper and the average in percolation and in SOC",
  "authors": [
    "Asim Ghosh",
    "S. S. Manna",
    "Bikas K. Chakrabarti"
  ],
  "comment": "9 pages, 13 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We define the $Q$-factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability $p$ is increased, the $Q$-factor for the system size $L$ grows systematically to its maximum value $Q_{max}(L)$ at a specific value $p_{max}(L)$ and then gradually decays. Our numerical study of site percolation problems on the square, triangular and the simple cubic lattices exhibits that the asymptotic values of $p_{max}$ though close, are distinctly different from the corresponding percolation thresholds of these lattices. We have also shown using the scaling analysis that at $p_{max}$ the value of $Q_{max}(L)$ diverges as $L^d$ ($d$ denoting the dimension of the lattice) as the system size approaches to their asymptotic limit. We have further extended this idea to the non-equilibrium systems such as the sandpile model of self-organized criticality. Here, the $Q(\\rho,L)$-factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches; $\\rho$ being the drop density of the driving mechanism. This study has been prompted by some observations in Sociophysics.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-07-09T04:58:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "competition",
      "site percolation problems",
      "simple cubic lattices",
      "occupation probability",
      "largest avalanche"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}