{
  "id": "2106.00593",
  "version": "v1",
  "published": "2021-06-01T16:00:44.000Z",
  "updated": "2021-06-01T16:00:44.000Z",
  "title": "Sparse matrices: convergence of the characteristic polynomial seen from infinity",
  "authors": [
    "Simon Coste"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove that the reverse characteristic polynomial $\\det(I_n - zA_n)$ of a random $n \\times n$ matrix $A_n$ with iid $\\mathrm{Bernoulli}(d/n)$ entries converges in distribution towards the random infinite product $\\prod_{\\ell = 1}^\\infty(1-z^\\ell)^{Y_\\ell}$ where $Y_\\ell$ are independent $\\mathrm{Poisson}(d^\\ell/\\ell)$ random variables. We show that this random function is a Poisson analog of the Gaussian holomorphic chaos and give some of its properties. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erd\\H{o}s-R\\'enyi digraphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-01T16:00:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "15B52",
      "30B20"
    ],
    "keywords": [
      "sparse matrices",
      "convergence",
      "gaussian holomorphic chaos",
      "reverse characteristic polynomial",
      "random infinite product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}