{
  "id": "1607.00243",
  "version": "v1",
  "published": "2016-07-01T13:42:16.000Z",
  "updated": "2016-07-01T13:42:16.000Z",
  "title": "On the maximum of the C$β$E field",
  "authors": [
    "Reda Chhaibi",
    "Joseph Najnudel",
    "Thomas Madaule"
  ],
  "comment": "73 pages ; v1: Preliminary version. All comments are welcome",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CA",
    "math.CV",
    "math.MP"
  ],
  "abstract": "In this paper, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (C$\\beta$E). More precisely, if $X_n$ is this characteristic polynomial and $\\mathbb{U}$ the unit circle, we prove that: $$\\sup_{z \\in \\mathbb{U} } \\Re \\log X_n(z) = \\sqrt{\\frac{2}{\\beta}} \\left(\\log n - \\frac{3}{4} \\log \\log n + \\mathcal{O}(1) \\right)\\ ,$$ as well as an analogous statement for the imaginary part. The notation $\\mathcal{O}(1)$ means that the corresponding family of random variables, indexed by $n$, is tight. This answers a conjecture of Fyodorov, Hiary and Keating, originally formulated for the case where $\\beta$ equals to $2$, which corresponds to the CUE field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-01T13:42:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "characteristic polynomial",
      "random unitary matrix",
      "extremal values",
      "unit circle",
      "cue field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 73,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}