{
  "id": "1912.07788",
  "version": "v1",
  "published": "2019-12-17T02:34:05.000Z",
  "updated": "2019-12-17T02:34:05.000Z",
  "title": "Dimension Results for the Spectral Measure of the Circular Beta Ensembles",
  "authors": [
    "Tom Alberts",
    "Raoul Normand"
  ],
  "comment": "47 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the dimension properties of the spectral measure of the Circular $\\beta$-Ensembles. For $\\beta \\geq 2$ it it was previously shown by Simon that the spectral measure is almost surely singular continuous with respect to Lebesgue measure on $\\partial \\mathbb{D}$ and the dimension of its support is $1 - 2/\\beta$. We reprove this result with a combination of probabilistic techniques and the so-called Jitomirskaya-Last inequalities. Our method is simpler in nature and mostly self-contained, with an emphasis on the probabilistic aspects rather than the analytic. We also extend the method to prove a large deviations principle for norms involved in the Jitomirskaya-Last analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-17T02:34:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52",
      "60F10"
    ],
    "keywords": [
      "spectral measure",
      "circular beta ensembles",
      "dimension results",
      "large deviations principle",
      "probabilistic techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}