The spectral edge of unitary Brownian motion

Collins, Benoît, Dahlqvist, Antoine and Kemp, Todd (2018) The spectral edge of unitary Brownian motion. Probability Theory and Related Fields, 170 (1-2). pp. 49-93. ISSN 0178-8051

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Abstract

The Brownian motion $$(UN_t)_{t\backslashge 0}$$(UtN)t≥0on the unitary group converges, as a process, to the free unitary Brownian motion $$(u_t)_{t\backslashge 0}$$(ut)t≥0as $$N\backslashrightarrow \backslashinfty $$N→∞. 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 $$t>0$$t>0, 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Probability and Statistics Research Group
Subjects: Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
Depositing User: Antoine Dahlqvist
Date Deposited: 25 Mar 2019 15:34
Last Modified: 19 Jun 2019 14:46
URI: http://sro.sussex.ac.uk/id/eprint/82484
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