Probability Theory and Related Fields

, Volume 170, Issue 1–2, pp 49–93 | Cite as

The spectral edge of unitary Brownian motion



The Brownian motion \((U^N_t)_{t\ge 0}\) on the unitary group converges, as a process, to the free unitary Brownian motion \((u_t)_{t\ge 0}\) as \(N\rightarrow \infty \). 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 \(t>0\), 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.

Mathematics Subject Classification

15B52 46L54 60F15 60G55 60J65 



This paper was initiated during a research visit of BC to TK in late 2012; substantial subsequent progress was made during a one-month visit of AD to BC at AIMR, Sendai during the fall of 2013. We would like to express our great thanks to UC San Diego and AIMR for their warm hospitality, fruitful work environments, and financial support that aided in the completion of this work. We would like to thank Ioana Dumitriu for helping to design the simulations that produced the figures. Finally, we express our gratitude to the anonymous referees whose careful reading and detailed comments significantly improved the presentation of this manuscript (particularly in Sect. 3.2).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Benoît Collins
    • 1
    • 2
  • Antoine Dahlqvist
    • 3
  • Todd Kemp
    • 4
  1. 1.Department of MathematicsKyoto UniversityKyotoJapan
  2. 2.CNRSLyonFrance
  3. 3.Statistical LaboratoryCentre for Mathematical SciencesCambridgeUK
  4. 4.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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