{
  "id": "1502.01427",
  "version": "v1",
  "published": "2015-02-05T03:43:55.000Z",
  "updated": "2015-02-05T03:43:55.000Z",
  "title": "Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials",
  "authors": [
    "Pavel M. Bleher",
    "Yushi Homma",
    "Roland K. W. Roeder"
  ],
  "comment": "24 pages, 1 figure",
  "categories": [
    "math-ph",
    "math.AG",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study the two-point correlation functions for the zeroes of systems of $SO(n+1)$-invariant Gaussian random polynomials on $\\mathbb{RP}^n$ and systems of ${\\rm isom}(\\mathbb{R}^n)$-invariant Gaussian analytic functions. Our result reflects the same \"repelling,\" \"neutral,\" and \"attracting\" short-distance asymptotic behavior, depending on the dimension, as was discovered in the complex case by Bleher, Shiffman, and Zelditch. For systems of the ${\\rm isom}(\\mathbb{R}^n)$-invariant Gaussian analytic functions we also obtain a fast decay of correlations at long distances. We then prove that the correlation function for the ${\\rm isom}(\\mathbb{R}^n)$-invariant Gaussian analytic functions is \"universal,\" describing the scaling limit of the correlation function for the restriction of systems of the $SO(k+1)$-invariant Gaussian random polynomials to any $n$-dimensional $C^2$ submanifold $M \\subset \\mathbb{RP}^k$. This provides a real counterpart to the universality results that were proved in the complex case by Bleher, Shiffman, and Zelditch. (Our techniques also apply to the complex case, proving a special case of the universality results of Bleher, Shiffman, and Zelditch.)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-05T03:43:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "two-point correlation functions",
      "invariant gaussian analytic functions",
      "invariant gaussian random polynomials",
      "complex case",
      "universality results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150201427B"
    }
  }
}