{
  "id": "2301.06973",
  "version": "v1",
  "published": "2023-01-17T15:52:28.000Z",
  "updated": "2023-01-17T15:52:28.000Z",
  "title": "Almost sure behavior of the critical points of random polynomials",
  "authors": [
    "Jürgen Angst",
    "Dominique Malicet",
    "Guillaume Poly"
  ],
  "comment": "16 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(Z_k)_{k\\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $\\mu$ and let $P_n(X):=\\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\\mathbb C[X]$. In [Kab15], the author established the conjecture stated by Pemantle and Rivin in [PR13] that the empirical measure $\\nu_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $\\mu$. In this note, we establish that the convergence in fact holds in the almost sure sense. Our result positively answers a question raised by Z. Kabluchko and formalized as a conjecture in the recent paper [MV22].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-17T15:52:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15",
      "60G57",
      "60B10"
    ],
    "keywords": [
      "critical points",
      "sure behavior",
      "identically distributed complex random variables",
      "conjecture",
      "common distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}