{
  "id": "1509.07159",
  "version": "v1",
  "published": "2015-09-23T21:06:24.000Z",
  "updated": "2015-09-23T21:06:24.000Z",
  "title": "From gap probabilities in random matrix theory to eigenvalue expansions",
  "authors": [
    "Thomas Bothner"
  ],
  "comment": "50 pages, 10 figures",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR",
    "math.SP",
    "nlin.SI"
  ],
  "abstract": "We present a method to derive asymptotics of eigenvalues for trace-class integral operators $K:L^2(J;d\\lambda)\\circlearrowleft$, acting on a single interval $J\\subset\\mathbb{R}$, which belong to the ring of integrable operators \\cite{IIKS}. Our emphasis lies on the behavior of the spectrum $\\{\\lambda_i(J)\\}_{i=0}^{\\infty}$ of $K$ as $|J|\\rightarrow\\infty$ and $i$ is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant $\\det(I-\\gamma K)|_{L^2(J)}$ as $|J|\\rightarrow\\infty$ and $\\gamma\\uparrow 1$ in a Stokes type scaling regime. Concrete asymptotic formul\\ae\\, are obtained for the eigenvalues of Airy and Bessel kernels in random matrix theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-23T21:06:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "45C05",
      "45M05",
      "82B26",
      "33C10",
      "33C45"
    ],
    "keywords": [
      "random matrix theory",
      "eigenvalue expansions",
      "gap probabilities",
      "trace-class integral operators",
      "stokes type scaling regime"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/49/7/075204",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2016,
      "month": "Feb",
      "volume": 49,
      "number": 7,
      "pages": "075204"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016JPhA...49g5204B"
    }
  }
}