{
  "id": "2402.11233",
  "version": "v1",
  "published": "2024-02-17T09:38:16.000Z",
  "updated": "2024-02-17T09:38:16.000Z",
  "title": "Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation",
  "authors": [
    "Yu Chen",
    "Shuai-Xia Xu",
    "Yu-Qiu Zhao"
  ],
  "comment": "41 pages, 14 figures",
  "doi": "10.1142/S2010326324500035",
  "categories": [
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "In the paper, we consider the extended Gross-Witten-Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the $(i,j)$-entry being the modified Bessel functions of order $i-j-\\nu$, $\\nu\\in\\mathbb{C}$. When the degree $n$ is finite, we show that the Toeplitz determinant is described by the isomonodromy $\\tau$-function of the Painlev\\'{e} III equation. As a double scaling limit, %In the double scaling limit as the degree $n\\to\\infty$, we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings-McLeod solution of the inhomogeneous Painlev\\'{e} II equation with parameter $\\nu+\\frac{1}{2}$. The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift-Zhou nonlinear steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point $z=-1$, where the $\\psi$-function of the Jimbo-Miwa Lax pair for the inhomogeneous Painlev\\'{e} II equation is involved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-17T09:38:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33E17",
      "34M55",
      "41A60"
    ],
    "keywords": [
      "modified bessel functions",
      "asymptotic",
      "toeplitz determinant",
      "deift-zhou nonlinear steepest descent method",
      "orthogonal polynomials"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "World Scientific"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}