{
  "id": "math-ph/0002039",
  "version": "v2",
  "published": "2000-02-16T23:27:27.000Z",
  "updated": "2000-10-17T21:08:04.000Z",
  "title": "Universality and scaling of zeros on symplectic manifolds",
  "authors": [
    "Pavel Bleher",
    "Bernard Shiffman",
    "Steve Zelditch"
  ],
  "comment": "Added results on the decay of connected correlations; corrected typographical errors . To appear in the Proceedings of the 1999 MSRI Workshop on Random Matrices and Their Applications",
  "journal": "Random Matrix Models and Their Applications, MSRI Publications 40, Cambridge Univ. Press, 2001, pp. 31-69.",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR",
    "math.SG"
  ],
  "abstract": "This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L \\to M$ over any complex manifold. Our main interest is in the statistics of zeros of $k$ independent sections (generalized polynomials) of degree $N$ as $N\\to\\infty$. We fix a point $P$ and focus on the ball of radius $1/\\sqrt{N}$ about $P$. Under a microscope magnifying the ball by the factor $\\sqrt{N}$, the statistics of the configurations of simultaneous zeros of random $k$-tuples of sections tends to a universal limit independent of $P,M,L$. We review this result and generalize it further to the case of pre-quantum line bundles over almost-complex symplectic manifolds $(M,J,\\omega)$. Following [SZ2], we replace $H^0(M,L^N)$ in the complex case with the `asymptotically holomorphic' sections defined by Boutet de Monvel-Guillemin and (from another point of view) by Donaldson and Auroux. Using a generalization to an $m$-dimensional setting of the Kac-Rice formula for zero correlations together with the results of [SZ2], we prove that the scaling limits of the correlation functions for zeros of random $k$-tuples of asymptotically holomorphic sections belong to the same universality class as in the complex case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2000-10-17T21:08:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "universality",
      "complex case",
      "random holomorphic sections",
      "random holomorphic polynomials",
      "universal limit independent"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.ph...2039B"
    }
  }
}