Overcrowding and hole probabilities for random zeros on complex manifolds

Bernard Shiffman*, Steve Zelditch, Scott Zrebiec

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a random holomorphic section of the N-th power of a positive line bundle on a compact Kahler manifold. In particular, we show that for all δ > 0, the probability that this volume differs by more than δ N from its average value is less than exp(-Cδ,UNm+1), for some constant Cδ,U > 0. As a consequence, the "hole probability" that a random section does not vanish in U has an upper bound of the form exp(-CUNm+1).

Original languageEnglish (US)
Pages (from-to)1977-1997
Number of pages21
JournalIndiana University Mathematics Journal
Volume57
Issue number5
DOIs
StatePublished - Dec 16 2008

Keywords

  • Bergman kernel
  • Large deviations
  • Positive line bundle
  • Random zeros
  • Szego kernel

ASJC Scopus subject areas

  • Mathematics(all)

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