Conditional haar measures on classical compact groups

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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
Issue number4
StatePublished - Jul 2009


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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