{
  "id": "1806.07651",
  "version": "v1",
  "published": "2018-06-20T10:27:37.000Z",
  "updated": "2018-06-20T10:27:37.000Z",
  "title": "Large deviations principle for the largest eigenvalue of the Gaussian beta-ensemble at high temperature",
  "authors": [
    "Cambyse Pakzad"
  ],
  "comment": "16 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the Gaussian beta-ensemble when $\\beta$ scales with $n$ the number of particles such that $\\displaystyle{{n}^{-1}\\ll \\beta\\ll 1}$. Under a certain regime for $\\beta$, we show that the largest particle satisfies a large deviations principle in $\\mathbb{R}$ with speed $n\\beta$ and explicit rate function. As a consequence, the largest particle converges in probability to $2$, the rightmost point of the semicircle law.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-20T10:27:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "60F10"
    ],
    "keywords": [
      "large deviations principle",
      "largest eigenvalue",
      "gaussian beta-ensemble",
      "high temperature",
      "largest particle converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}