{
  "id": "2301.05516",
  "version": "v1",
  "published": "2023-01-13T12:35:45.000Z",
  "updated": "2023-01-13T12:35:45.000Z",
  "title": "CLT for real \\b{eta}-Ensembles at High Temperature",
  "authors": [
    "Charlie Dworaczek Guera",
    "Ronan Memin"
  ],
  "comment": "34 pages, comments are welcome",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We establish a central limit theorem for the fluctuations of the empirical measure in the $\\beta$- ensemble of dimension $N$ at a temperature proportional to $N$ and with convex, smooth potential. The space of test functions for which the CLT holds includes $\\mathcal{C}^1$, vanishing functions at infinity. It is obtained by the inversion of an operator which is a pertubation of a Sturm-Liouville operator. The method that we use is based on a change of variables introduced by Bekerman, Guionnet and Figalli in arXiv:1311.2315 and by Shcherbina in arXiv:1310.7835.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-13T12:35:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "high temperature",
      "central limit theorem",
      "smooth potential",
      "sturm-liouville operator",
      "test functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}