{
  "id": "math-ph/0409082",
  "version": "v1",
  "published": "2004-09-30T19:46:50.000Z",
  "updated": "2004-09-30T19:46:50.000Z",
  "title": "Asymptotics of the partition function of a random matrix model",
  "authors": [
    "Pavel Bleher",
    "Alexander Its"
  ],
  "comment": "43 pages, 3 figures",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential $V(z)$. Our approach is based on a deformation $\\tau_tV(z)$ of $V(z)$ to $z^2$, $0\\le t<\\infty$ and on the use of the underlying integrable structures of the matrix model. The main results include (1) the existence of a full asymptotic expansion in powers of $N^{-2}$ of the recurrence coefficients of the related orthogonal polynomials, for a one-cut regular $V$; (2) the existence of a full asymptotic expansion in powers of $N^{-2}$ of the free energy, for a $V$, which admits a one-cut regular deformation $\\tau_tV$; (3) the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of $V$; (4) the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular $V$; (5) the double scaling asymptotics of the free energy for a singular quartic polynomial $V$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-09-30T19:46:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B23"
    ],
    "keywords": [
      "random matrix model",
      "free energy",
      "partition function",
      "full asymptotic expansion",
      "recurrence coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.ph...9082B"
    }
  }
}