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July 2009 Conditional Haar measures on classical compact groups
P. Bourgade
Ann. Probab. 37(4): 1566-1586 (July 2009). DOI: 10.1214/08-AOP443


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.


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P. Bourgade. "Conditional Haar measures on classical compact groups." Ann. Probab. 37 (4) 1566 - 1586, July 2009.


Published: July 2009
First available in Project Euclid: 21 July 2009

zbMATH: 1172.43005
MathSciNet: MR2546755
Digital Object Identifier: 10.1214/08-AOP443

Primary: 14G10 , 15A52 , 60F05

Keywords: central limit theorem , characteristic polynomial , random matrices , the Weyl integration formula , zeta and L-functions

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 4 • July 2009
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