{
  "id": "0905.3282",
  "version": "v2",
  "published": "2009-05-20T11:30:17.000Z",
  "updated": "2011-09-09T07:45:20.000Z",
  "title": "Central limit theorem for the heat kernel measure on the unitary group",
  "authors": [
    "Thierry Lévy",
    "Mylène Maïda"
  ],
  "comment": "44 pages",
  "journal": "Journal of Functional Analysis 259, 12 (2010) 3163-3204",
  "doi": "10.1016/j.jfa.2010.08.005",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove that for a finite collection of real-valued functions $f_{1},...,f_{n}$ on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of $(\\tr f_{1},...,\\tr f_{n})$ under the properly scaled heat kernel measure at a given time on the unitary group $\\U(N)$ has Gaussian fluctuations as $N$ tends to infinity, with a covariance for which we give a formula and which is of order $N^{-1}$. In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-09-09T07:45:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central limit theorem",
      "unitary group",
      "properly scaled heat kernel measure",
      "covariance",
      "finite collection"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.3282L"
    }
  }
}