Random zeros on complex manifolds: Conditional expectations

Bernard Shiffman*, Steve Zelditch, Qi Zhong

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We study the conditional distribution KκNof zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros but we show that it has quite a different small distance behaviour. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension 1. To prove this, we give universal scaling asymptotics for KκN around p. The key tool is the conditional Szeg kernel and its scaling asymptotics.

Original languageEnglish (US)
Pages (from-to)753-783
Number of pages31
JournalJournal of the Institute of Mathematics of Jussieu
Volume10
Issue number3
DOIs
StatePublished - Jul 1 2011

Keywords

  • Kähler manifold
  • Random holomorphic sections
  • Szego kernel
  • holomorphic line bundle
  • zeros of random polynomials

ASJC Scopus subject areas

  • Mathematics(all)

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