{
  "id": "2103.04229",
  "version": "v1",
  "published": "2021-03-07T01:48:06.000Z",
  "updated": "2021-03-07T01:48:06.000Z",
  "title": "Painlevé IV, $σ-$Form and the Deformed Hermite Unitary Ensembles",
  "authors": [
    "Mengkun Zhu",
    "Dan Wang",
    "Yang Chen"
  ],
  "journal": "Journal of Mathematical Physics,2021",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the Hankel determinant generated by a deformed Hermite weight with one jump $w(z,t,\\gamma)=e^{-z^2+tz}|z-t|^{\\gamma}(A+B\\theta(z-t))$, where $A\\geq 0$, $A+B\\geq 0$, $t\\in\\textbf{R}$, $\\gamma>-1$ and $z\\in\\textbf{R}$. By using the ladder operators for the corresponding monic orthogonal polynomials, and their relative compatibility conditions, we obtain a series of difference and differential equations to describe the relations among $\\alpha_n$, $\\beta_n$, $R_n(t)$ and $r_n(t)$. Especially, we find that the auxiliary quantities $R_n(t)$ and $r_n(t)$ satisfy the coupled Riccati equations, and $R_n(t)$ satisfies a particular Painlev\\'{e} IV equation. Based on above results, we show that $\\sigma_n(t)$ and $\\hat{\\sigma}_n(t)$, two quantities related to the Hankel determinant and $R_n(t)$, satisfy the continuous and discrete $\\sigma-$form equations, respectively. In the end, we also discuss the large $n$ asymptotic behavior of $R_n(t)$, which produce the expansion of the logarithmic of the Hankel determinant and the asymptotic of the second order differential equation of the monic orthogonal polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-07T01:48:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "42C05",
      "33E17"
    ],
    "keywords": [
      "deformed hermite unitary ensembles",
      "hankel determinant",
      "second order differential equation",
      "corresponding monic orthogonal polynomials",
      "ladder operators"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "J. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}