## 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_{ℓ}'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 language | English (US) |
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Pages (from-to) | 1566-1586 |

Number of pages | 21 |

Journal | Annals of Probability |

Volume | 37 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2009 |

## Keywords

- 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