Spectral Distribution of the Free Unitary Brownian Motion: Another Approach

  • Nizar Demni
  • Taoufik Hmidi
Part of the Lecture Notes in Mathematics book series (LNM, volume 2046)


We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any \(t\,\in \,(0,4)\) a Jordan curve \({\gamma }_{t}\) around the origin, not intersecting the semi-axis \([1,\infty [\) and whose image under some meromorphic function h t lies in the circle. Our construction is naturally suggested by a residue-type integral representation of the moments and h t is up to a Möbius transformation the main ingredient used in the original proof. Once we did, the spectral measure is described as the push-forward of a complex measure under a local diffeomorphism yielding its absolute-continuity and its support. Our approach has the merit to be an easy yet technical exercise from real analysis.


Spectral Distribution Implicit Function Theorem Jordan Curve Local Diffeomorphism Unique Critical Point 
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This work was partially supported by Agence Nationale de la recherche grant ANR-09-BLAN-0084-01.


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    T. Lévy, Schur-Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218(2), 537–575 (2008)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.IRMARUniversité de Rennes 1Rennes cedexFrance

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