{
  "id": "1808.08902",
  "version": "v1",
  "published": "2018-08-27T16:10:02.000Z",
  "updated": "2018-08-27T16:10:02.000Z",
  "title": "Phase transition for the interchange and quantum Heisenberg models on the Hamming graph",
  "authors": [
    "Radosław Adamczak",
    "Michał Kotowski",
    "Piotr Miłoś"
  ],
  "comment": "54 pages, 3 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study a family of random permutation models on the $2$-dimensional Hamming graph $H(2,n)$, containing the interchange process and the cycle-weighted interchange process with parameter $\\theta > 0$. This family contains the random representation of the quantum Heisenberg ferromagnet. We show that in these models the cycle structure of permutations undergoes a \\textit{phase transition} -- when the number of transpositions defining the permutation is $\\leq c n^2$, for small enough $c > 0$, all cycles are microscopic, while for more than $\\geq C n^2$ transpositions, for large enough $C > 0$, macroscopic cycles emerge with high probability. We provide bounds on values $C,c$ depending on the parameter $\\theta$ of the model, in particular for the interchange process we pinpoint exactly the critical time of the phase transition. Our results imply also the existence of a phase transition in the quantum Heisenberg ferromagnet on $H(2,n)$, namely for low enough temperatures spontaneous magnetization occurs, while it is not the case for high temperatures. At the core of our approach is a novel application of the cyclic random walk, which might be of independent interest. By analyzing explorations of the cyclic random walk, we show that sufficiently long cycles of a random permutation are uniformly spread on the graph, which makes it possible to compare our models to the mean-field case, i.e., the interchange process on the complete graph, extending the approach used earlier by Schramm.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-27T16:10:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J27"
    ],
    "keywords": [
      "quantum heisenberg models",
      "phase transition",
      "hamming graph",
      "interchange process",
      "cyclic random walk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}