Inventiones mathematicae

, Volume 142, Issue 2, pp 351–395

Universality and scaling of correlations between zeros on complex manifolds

Authors

  • Pavel Bleher
    • Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202, USA¶(e-mail: bleher@math.iupui.edu)
  • Bernard Shiffman
    • Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA¶(e-mail: shiffman@math.jhu.edu, zel@math.jhu.edu)
  • Steve Zelditch
    • Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA¶(e-mail: shiffman@math.jhu.edu, zel@math.jhu.edu)

DOI: 10.1007/s002220000092

Cite this article as:
Bleher, P., Shiffman, B. & Zelditch, S. Invent. math. (2000) 142: 351. doi:10.1007/s002220000092

Abstract.

We study the limit as N→∞ of the correlations between simultaneous zeros of random sections of the powers LN of a positive holomorphic line bundle L over a compact complex manifold M, when distances are rescaled so that the average density of zeros is independent of N. We show that the limit correlation is independent of the line bundle and depends only on the dimension of M and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we prove that Hannay’s limit pair correlation function for SU(2) polynomials holds for all compact Riemann surfaces.

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© Springer-Verlag Berlin Heidelberg 2000