Conditional Haar measures on classical compact groups

Bourgade, P.

Ann. Probab., Tome 37 (2009) no. 1, p. 1566-1586 / Harvested from Project Euclid

Résumé

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

Détail

Publié le : 2009-07-15
Classification:  Random matrices,  characteristic polynomial,  the Weyl integration formula,  zeta and L-functions,  central limit theorem,  15A52,  60F05,  14G10