The spectral edge of unitary Brownian motion

• 成果类型: Article
• 署名作者: Collins, Benoit; Dahlqvist, Antoine; Kemp, Todd
• 署名单位: Kyoto University; Centre National de la Recherche Scientifique (CNRS); University of Cambridge; University of California System; University of California San Diego
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0753-x
• 发表日期: 2018
• 页码: 49-93
• 关键词: heat kernel measure; RANDOM MATRICES; free entropy; asymptotic freeness; LARGEST EIGENVALUE; jacobi ensembles; INFORMATION; liberation; calculus

摘要

The Brownian motion on the unitary group converges, as a process, to the free unitary Brownian motion as . In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time , we prove that the unitary Brownian motion has a spectral edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.