{
  "id": "1307.4357",
  "version": "v3",
  "published": "2013-07-16T17:48:19.000Z",
  "updated": "2014-04-30T01:52:29.000Z",
  "title": "Local universality of zeroes of random polynomials",
  "authors": [
    "Terence Tao",
    "Van Vu"
  ],
  "comment": "56 pages, no figures, to appear, IMRN. A gap in an argument invoking Gromov type theorems (in the treatment of the Kac model) has been fixed",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\\sum_{i=1}^n c_i \\xi_i z^i$ and $\\tilde f =\\sum_{i=1}^n c_i \\tilde \\xi_i z^i$, where the $\\xi_i$ and $\\tilde \\xi_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our results show, under some light conditions on the coefficients $c_i$ and the tails of $\\xi_i, \\tilde \\xi_i$, that the correlation functions of the zeroes of $f$ and $\\tilde f$ are approximately the same. As an application, we give some answers to the classical question `\"How many zeroes of a random polynomials are real?\" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions $f$ and $\\tilde f$ if their log magnitudes $\\log |f|, \\log|\\tilde f|$ are close in distribution, and if some non-concentration bounds are obeyed.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-30T01:52:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26C10",
      "30C15"
    ],
    "keywords": [
      "correlation functions",
      "random matrix theory",
      "random polynomial models",
      "general replacement principle",
      "iid random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.4357T"
    }
  }
}