{
  "id": "2102.07135",
  "version": "v1",
  "published": "2021-02-14T12:12:15.000Z",
  "updated": "2021-02-14T12:12:15.000Z",
  "title": "Nested Closed Paths in Two-Dimensional Percolation",
  "authors": [
    "Yu-Feng Song",
    "Xiao-Jun Tan",
    "Xin-Hang Zhang",
    "Jesper Lykke Jacobsen",
    "Bernard Nienhuis",
    "Youjin Deng"
  ],
  "comment": "5 pages, 5 figures plus supplemental material, 1 page 2 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "For two-dimensional percolation on a domain with the topology of a disc, we introduce a nested-path operator (NP) and thus a continuous family of one-point functions $W_k \\equiv \\langle \\mathcal{R} \\cdot k^\\ell \\rangle $, where $\\ell$ is the number of independent nested closed paths surrounding the center, $k$ is a path fugacity, and $\\mathcal{R}$ projects on configurations having a cluster connecting the center to the boundary. At criticality, we observe a power-law scaling $W_k \\sim L^{X_{\\rm NP}}$, with $L$ the linear system size, and we determine the exponent $X_{\\rm NP}$ as a function of $k$. On the basis of our numerical results, we conjecture an analytical formula, $X_{\\rm NP} (k) = \\frac{3}{4}\\phi^2 -\\frac{5}{48}\\phi^2/ (\\phi^2-\\frac{2}{3})$ where $k = 2 \\cos(\\pi \\phi)$, which reproduces the exact results for $k=0,1$ and agrees with the high-precision estimate of $X_{\\rm NP}$ for other $k$ values. In addition, we observe that $W_2(L)=1$ for site percolation on the triangular lattice with any size $L$, and we prove this identity for all self-matching lattices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-14T12:12:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B43",
      "60K35"
    ],
    "keywords": [
      "two-dimensional percolation",
      "nested closed paths surrounding",
      "nested-path operator",
      "one-point functions",
      "path fugacity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}