CONDITIONAL HAAR MEASURES ON CLASSICAL COMPACT GROUPS

• 成果类型: Article
• 署名作者: Bourgade, P.
• 署名单位: Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP443
• 发表日期: 2009
• 页码: 1566-1586
• 关键词: random matrices

摘要

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension it. 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, Z(U)((p))/p! =(law) Pi(n-p)(l=1)(1-X(l)), 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.