{
  "id": "2403.19157",
  "version": "v1",
  "published": "2024-03-28T05:42:07.000Z",
  "updated": "2024-03-28T05:42:07.000Z",
  "title": "Correlation functions between singular values and eigenvalues",
  "authors": [
    "Matthias Allard",
    "Mario Kieburg"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure on $j$ eigenvalues and $k$ singular values that we coin $j,k$-point correlation measure. We fully derive all $j,k$-point correlation measures in the simplest cases for one- and two-dimensional matrices. For $n>2$, we find a general formula for the $1,1$-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a P\\'{o}lya ensemble and extend known results between their eigenvalue and singular value statistics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-28T05:42:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52",
      "43A90",
      "42B10",
      "42C05"
    ],
    "keywords": [
      "point correlation measure",
      "correlation functions",
      "eigenvalue",
      "singular value statistics",
      "invariant complex random matrix ensembles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}