{
  "id": "1207.4734",
  "version": "v2",
  "published": "2012-07-19T17:08:20.000Z",
  "updated": "2012-08-29T11:14:45.000Z",
  "title": "Correlations and Pairing Between Zeros and Critical Points of Gaussian Random Polynomials",
  "authors": [
    "Boris Hanin"
  ],
  "comment": "35 pages, 3 figures. Some typos corrected and Introduction revised",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "We study the asymptotics of correlations and nearest neighbor spacings between zeros and holomorphic critical points of $p_N$, a degree N Hermitian Gaussian random polynomial in the sense of Shiffman and Zeldtich, as N goes to infinity. By holomorphic critical point we mean a solution to the equation $\\frac{d}{dz}p_N(z)=0.$ Our principal result is an explicit asymptotic formula for the local scaling limit of $\\E{Z_{p_N}\\wedge C_{p_N}},$ the expected joint intensity of zeros and critical points, around any point on the Riemann sphere. Here $Z_{p_N}$ and $C_{p_N}$ are the currents of integration (i.e. counting measures) over the zeros and critical points of $p_N$, respectively. We prove that correlations between zeros and critical points are short range, decaying like $e^{-N\\abs{z-w}^2}.$ With $\\abs{z-w}$ on the order of $N^{-1/2},$ however, $\\E{Z_{p_N}\\wedge C_{p_N}}(z,w)$ is sharply peaked near $z=w,$ causing zeros and critical points to appear in rigid pairs. We compute tight bounds on the expected distance and angular dependence between a critical point and its paired zero.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-08-29T11:14:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "correlations",
      "holomorphic critical point",
      "hermitian gaussian random polynomial",
      "explicit asymptotic formula",
      "nearest neighbor spacings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.4734H"
    }
  }
}