{
  "id": "2112.10162",
  "version": "v2",
  "published": "2021-12-19T14:54:03.000Z",
  "updated": "2022-04-16T13:05:46.000Z",
  "title": "Backbone and shortest-path exponents of the two-dimensional $Q$-state Potts model",
  "authors": [
    "Sheng Fang",
    "Da Ke",
    "Wei Zhong",
    "Youjin Deng"
  ],
  "comment": "14 pages, 9 figures",
  "journal": "Phys. Rev. E 105, 044122(2022)",
  "doi": "10.1103/PhysRevE.105.044122",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional $Q$-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents $d_{\\rm B} = 1.732 \\, 0(3)$ and $1.794(2)$ for $Q=2,3$ respectively. However, for large $Q$, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O$(n)$ loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with $Q=n^2$. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops $d_{\\rm B} =1.643\\,39(5), 1.732\\,27(8), 1.793\\,8(3), 1.838\\,4(5), 1.875\\,3(6)$ for $Q = 1, 2, 3, 2 \\! + \\! \\sqrt{3}, 4$, respectively, and $d_{\\rm min} = 1.094\\,5(2), 1.067\\,5(3), 1.047\\,5(3), 1.032\\,2(4)$ for $Q=2,3, 2+\\sqrt{3}, 4$ respectively. Our estimates significantly improve over the existing results for both $d_{\\rm B}$ and $d_{\\rm min}$. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of $n$ for the leading correction exponent.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-16T13:05:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "state potts model",
      "shortest-path exponents",
      "two-dimensional",
      "critical potts model",
      "highly efficient cluster algorithm"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. E"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}