{
  "id": "1811.08340",
  "version": "v2",
  "published": "2018-11-20T16:11:42.000Z",
  "updated": "2018-12-06T19:32:52.000Z",
  "title": "On the eigenvalues of truncations of random unitary matrices",
  "authors": [
    "Elizabeth Meckes",
    "Kathryn Stewart"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the empirical eigenvalue distribution of an $m\\times m$ principle submatrix of an $n\\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and R\\'effy identified the limiting spectral measure if $\\frac{m}{n}\\to\\alpha$, as $n\\to\\infty$; under suitable scaling, the family $\\{\\mu_\\alpha\\}_{\\alpha\\in(0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $\\alpha$) and uniform measure on the unit circle (as $\\alpha\\to1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $\\mu_\\alpha$ is typically of order $\\sqrt{\\frac{\\log(m)}{m}}$ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new \"Coulomb transport inequality\" due to Chafa\\\"i, Hardy, and Ma\\\"ida.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-12-06T19:32:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random unitary matrix",
      "uniform measure",
      "truncations",
      "coulomb transport inequality",
      "two-dimensional coulomb gases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}