The Annals of Probability
- Ann. Probab.
- Volume 37, Number 4 (2009), 1566-1586.
Conditional Haar measures on classical compact groups
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,
the Xℓ’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.
Ann. Probab., Volume 37, Number 4 (2009), 1566-1586.
First available in Project Euclid: 21 July 2009
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Bourgade, P. Conditional Haar measures on classical compact groups. Ann. Probab. 37 (2009), no. 4, 1566--1586. doi:10.1214/08-AOP443. https://projecteuclid.org/euclid.aop/1248182148