{
  "id": "2309.09389",
  "version": "v1",
  "published": "2023-09-17T22:21:05.000Z",
  "updated": "2023-09-17T22:21:05.000Z",
  "title": "A limit law for the maximum of subcritical DG-model on a hierarchical lattice",
  "authors": [
    "Marek Biskup",
    "Haiyu Huang"
  ],
  "comment": "46 pages, preliminary version",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the extremal properties of the \"integer-valued Gaussian\" a.k.a. DG-model on the hierarchical lattice $\\Lambda_n:=\\{1,\\dots,b\\}^n$ (with $b\\ge2$) of depth $n$. This is a law on $\\varphi\\in\\mathbb Z^{\\Lambda_n}$ proportional to $e^{\\frac12\\beta(\\varphi,\\Delta_n\\varphi)}\\prod_{x\\in\\Lambda_n}\\#(d\\varphi_x)$, where $\\Delta_n$ is the hierarchical Laplacian, $\\beta$ is the inverse temperature and $\\#$ is the counting measure on $\\mathbb Z$. Denoting $\\beta_c:=2\\pi^2/\\log b$ and $m_n:=\\beta^{-1/2}[(2\\log b)^{1/2}n-\\frac32(2\\log b)^{-1/2}\\log n]$, for $0<\\beta<\\beta_c$ we prove that, along increasing sequences of $n$ such that the fractional part of $m_{n}$ converges to an $s\\in[0,1)$, the centered maximum $\\max_{x\\in\\Lambda_n}\\varphi_x-\\lfloor m_n\\rfloor$ tends (as $n\\to\\infty$) in law to a discrete variant of a randomly shifted Gumbel law with the shift depending non-trivially on $s$. The convergence extends to the extremal process whose law tends to a decorated Poisson point process with a random intensity measure. The proofs rely on renormalization-group analysis which enables a tight coupling of the DG-model to a Gaussian Free Field. The interval $(0,\\beta_c]$ marks the full range of values of $\\beta$ for which the renormalization-group iterations tend to a \"trivial\" fixed point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-17T22:21:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "60G15",
      "82B28"
    ],
    "keywords": [
      "hierarchical lattice",
      "limit law",
      "subcritical dg-model",
      "gaussian free field",
      "random intensity measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}