{
  "id": "1803.10954",
  "version": "v1",
  "published": "2018-03-29T08:15:43.000Z",
  "updated": "2018-03-29T08:15:43.000Z",
  "title": "Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite $n$ to Double Scaling",
  "authors": [
    "Chao Min",
    "Yang Chen"
  ],
  "comment": "20 pages",
  "journal": "Studies in Applied Mathematics 140 (2018) 202-220",
  "doi": "10.1111/sapm.12198",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely the probability that the interval $(-a,a)\\:(0<a<1)$ is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by $H_{n}(a)$, $R_{n}(a)$ and $r_{n}(a)$. We find that each one satisfies a second order differential equation. We show that after a double scaling, the large second order differential equation in the variable $a$ with $n$ as parameter satisfied by $H_{n}(a)$, can be reduced to the Jimbo-Miwa-Okamoto $\\sigma$ form of the Painlev\\'{e} V equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-29T08:15:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "42C05",
      "33E17"
    ],
    "keywords": [
      "jacobi unitary ensemble",
      "gap probability distribution",
      "elementary treatment",
      "double scaling",
      "large second order differential equation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Wiley-Blackwell",
      "journal": "J. Finance"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}