Number variance of random zeros on complex manifolds, II: Smooth statistics

Bernard Shiffman*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

We consider the zero sets ZN of systems of m random polynomials of degree N in m complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over ZN. Our asymptotic formulas show that the variances for these smooth statistics have the growth Nm-2. We also prove analogues for the integrals of smooth test forms over the subvarieties dened by k < m random polynomials. Such linear statistics of random zero sets are smooth analogues of the random variables given by counting the number of zeros in an open set, which we proved elsewhere to have variances of order Nm-1/2. We use the variance a symptotics and off-diagonal estimates of Szego{double acute} kernels to extend the central limit theorem of Sodin-Tsirelson to the case of smooth linear statistics for zero sets of codimension one in any dimension m.

Original languageEnglish (US)
Pages (from-to)1145-1167
Number of pages23
JournalPure and Applied Mathematics Quarterly
Volume6
Issue number4
DOIs
StatePublished - Oct 2010

Keywords

  • Käahler manifold
  • Positive line bundle
  • Random holomorphic sections
  • Szego{double acute} kernel
  • Zeros of random polynomials

ASJC Scopus subject areas

  • Mathematics(all)

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