The Annals of Probability

Conditional Haar measures on classical compact groups

P. Bourgade

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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 ZU(p) for the first nonzero derivative of the characteristic polynomial at 1,

\[\frac{Z_{U}^{(p)}}{p!}\stackrel{\mathrm{law}}{=}\prod_{\ell =1}^{n-p}(1-X_{\ell})\],

the X’s being explicit independent random variables. This implies a central limit theorem for log ZU(p) and asymptotics for the density of ZU(p) near 0. Similar limit theorems are given for the orthogonal and symplectic groups, relying on results of Killip and Nenciu.

Article information

Ann. Probab., Volume 37, Number 4 (2009), 1566-1586.

First available in Project Euclid: 21 July 2009

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Zentralblatt MATH identifier

Primary: 15A52 60F05: Central limit and other weak theorems 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture)

Random matrices characteristic polynomial the Weyl integration formula zeta and L-functions central limit theorem


Bourgade, P. Conditional Haar measures on classical compact groups. Ann. Probab. 37 (2009), no. 4, 1566--1586. doi:10.1214/08-AOP443.

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