{
  "id": "2005.09809",
  "version": "v1",
  "published": "2020-05-20T01:06:57.000Z",
  "updated": "2020-05-20T01:06:57.000Z",
  "title": "A Semicircle Law for Derivatives of Random Polynomials",
  "authors": [
    "Jeremy G. Hoskins",
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.PR",
    "math.CA"
  ],
  "abstract": "Let $x_1, \\dots, x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \\dots, x_n$. We prove that for $\\ell \\in \\mathbb{N}$ fixed as $n \\rightarrow \\infty$, the $(n-\\ell)-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval,$${n^{\\ell/2}} \\frac{\\ell!}{n!} \\cdot p_n^{(n-\\ell)}\\left( \\frac{x}{\\sqrt{n}}\\right) \\rightarrow He_{\\ell}(x + \\gamma_n),$$ where $He_{\\ell}$ is the $\\ell-$th probabilists' Hermite polynomial and $\\gamma_n$ is a random variable converging to the standard $\\mathcal{N}(0,1)$ Gaussian as $n \\rightarrow \\infty$. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-20T01:06:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random polynomial",
      "semicircle law",
      "hermite polynomial",
      "wigner semicircle distribution",
      "derivative"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}