Conditional haar measures on classical compact groups

Research output: Contribution to journalArticle

Abstract

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension n. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension p. The developed method leads to the following result: for this conditional measure, writing Z(p)U for the first nonzero derivative of the characteristic polynomial at 1, the X's being explicit independent random variables. This implies a central limit theorem for log Z(p)U and asymptotics for the density of Z(p)U near 0. Similar limit theorems are given for the orthogonal and symplectic groups, relying on results of Killip and Nenciu.

Original languageEnglish (US)
Pages (from-to)1566-1586
Number of pages21
JournalAnnals of Probability
Volume37
Issue number4
DOIs
StatePublished - Jul 2009

Fingerprint

Haar Measure
Classical Groups
Compact Group
Symplectic Group
Orthogonal Group
Unitary group
Characteristic polynomial
Independent Random Variables
Limit Theorems
Central limit theorem
Subspace
Imply
Derivative
Limit theorems
Polynomials
Derivatives
Random variables

Keywords

  • Central limit theorem
  • Characteristic polynomial
  • Random matrices
  • The Weyl integration formula
  • Zeta and L-functions

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Statistics and Probability

Cite this

Conditional haar measures on classical compact groups. / Bourgade, Paul.

In: Annals of Probability, Vol. 37, No. 4, 07.2009, p. 1566-1586.

Research output: Contribution to journalArticle

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