{
  "id": "2305.13184",
  "version": "v1",
  "published": "2023-05-22T16:12:05.000Z",
  "updated": "2023-05-22T16:12:05.000Z",
  "title": "A matrix model of a non-Hermitian $β$-ensemble",
  "authors": [
    "Francesco Mezzadri",
    "Henry Taylor"
  ],
  "comment": "23 pages, 2 figures",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We introduce the first random matrix model of a complex $\\beta$-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite $\\beta$-ensembles discovered by Dumitriu and Edelman (J. Math. Phys., Vol. 43, 5830 (2002)). The main feature of the model is that the exponent $\\beta$ of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in $\\mathbb{R}_+$. However, when $\\beta=2$, the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-22T16:12:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first random matrix model",
      "joint probability density function",
      "vandermonde determinant",
      "multidimensional integral",
      "non-hermitian analogue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}