{
  "id": "2011.13723",
  "version": "v1",
  "published": "2020-11-27T13:13:54.000Z",
  "updated": "2020-11-27T13:13:54.000Z",
  "title": "An edge CLT for the log determinant of Gaussian ensembles",
  "authors": [
    "Iain M. Johnstone",
    "Yegor Klochkov",
    "Alexei Onatski",
    "Damian Pavlyshyn"
  ],
  "comment": "39 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We derive a Central Limit Theorem (CLT) for $\\log \\left\\vert\\det \\left( M_{N}/\\sqrt{N}-2\\theta_{N}\\right)\\right\\vert,$ where $M_{N}$ is from the Gaussian Unitary or Gaussian Orthogonal Ensemble (GUE and GOE), and $2\\theta_{N}$ is local to the edge of the semicircle law. Precisely, $2\\theta_{N}=2+N^{-2/3}\\sigma_N$ with $\\sigma_N$ being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that $\\sigma_N \\ll \\log^{2} N$. For slowly growing $\\sigma_N$, our proofs hold for general Gaussian $\\beta$-ensembles. We also extend our CLT to cover spiked GUE and GOE.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-27T13:13:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60B20"
    ],
    "keywords": [
      "log determinant",
      "gaussian ensembles",
      "edge clt",
      "central limit theorem",
      "general gaussian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}