{
  "id": "1912.07798",
  "version": "v1",
  "published": "2019-12-17T03:03:40.000Z",
  "updated": "2019-12-17T03:03:40.000Z",
  "title": "Glauber dynamics for Ising models on random regular graphs: cut-off and metastability",
  "authors": [
    "Van Hao Can",
    "Remco van der Hofstad",
    "Takashi Kumagai"
  ],
  "comment": "51 pages",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Consider random $d$-regular graphs, i.e., random graphs such that there are exactly $d$ edges from each vertex for some $d\\ge 3$. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a $d$-regular graph chosen uniformly at random from the collection of all $d$-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random $d$-regular graph, both in the quenched as well as the annealed settings. Let $\\beta$ be the inverse temperature, $\\beta_c$ be the critical temperature and $B$ be the external magnetic field. Concerning the annealed measure, we show that for $\\beta > \\beta_c$ there exists $\\hat{B}_c(\\beta)\\in (0,\\infty)$ such that the model is metastable (i.e., the mixing time is exponential in the graph size $n$) when $\\beta> \\beta_c$ and $0 \\leq B < \\hat{B}_c(\\beta)$, whereas it exhibits the cut-off phenomenon at $c_\\star n \\log n$ with a window of order $n$ when $\\beta < \\beta_c$ or $\\beta > \\beta_c$ and $B>\\hat{B}_c(\\beta)$. Interestingly, $\\hat{B}_c(\\beta)$ coincides with the critical external field of the Ising model on the $d$-ary tree (namely, above which the model has a unique Gibbs measure). Concerning the quenched measure, we show that there exists $B_c(\\beta)$ with $B_c(\\beta) \\leq \\hat{B}_c(\\beta)$ such that for $\\beta> \\beta_c$, the mixing time is at least exponential along some subsequence $(n_k)_{k\\geq 1}$ when $0 \\leq B < B_c(\\beta)$, whereas it is less than or equal to $Cn\\log n$ when $B>\\hat{B}_c(\\beta)$. The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-17T03:03:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random regular graphs",
      "ising model",
      "glauber dynamics",
      "external magnetic field",
      "mixing time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}