# 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 School of Mathematical and Physical Sciences > Mathematics Probability and Statistics Research Group Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics Antoine Dahlqvist 25 Mar 2019 15:34 16 Feb 2021 15:21 http://sro.sussex.ac.uk/id/eprint/82484