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The spectral edge of unitary Brownian motion

journal contribution
posted on 2023-06-09, 17:13 authored by Benoît Collins, Antoine DahlqvistAntoine Dahlqvist, Todd Kemp
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?8. 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.

History

Publication status

  • Published

Journal

Probability Theory and Related Fields

ISSN

0178-8051

Publisher

Springer Verlag

Issue

1-2

Volume

170

Page range

49-93

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Probability and Statistics Research Group Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2019-03-25

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