### 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) |
---|---|

Pages (from-to) | 1566-1586 |

Number of pages | 21 |

Journal | Annals of Probability |

Volume | 37 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2009 |

### Fingerprint

### 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

*Annals of Probability*,

*37*(4), 1566-1586. https://doi.org/10.1214/08-AOP443

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

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 37, no. 4, pp. 1566-1586. https://doi.org/10.1214/08-AOP443

}

TY - JOUR

T1 - Conditional haar measures on classical compact groups

AU - Bourgade, Paul

PY - 2009/7

Y1 - 2009/7

N2 - 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 Xℓ'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.

AB - 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 Xℓ'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.

KW - Central limit theorem

KW - Characteristic polynomial

KW - Random matrices

KW - The Weyl integration formula

KW - Zeta and L-functions

UR - http://www.scopus.com/inward/record.url?scp=69249214042&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69249214042&partnerID=8YFLogxK

U2 - 10.1214/08-AOP443

DO - 10.1214/08-AOP443

M3 - Article

VL - 37

SP - 1566

EP - 1586

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -