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Spectral Distribution of the Free Unitary Brownian Motion: Another Approach

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

Abstract

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.

Keywords

Spectral Distribution Implicit Function Theorem Jordan Curve Local Diffeomorphism Unique Critical Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgements

This work was partially supported by Agence Nationale de la recherche grant ANR-09-BLAN-0084-01.

References

  1. 1.
    P. Biane, Free Brownian Motion, Free Stochastic Calculus and Random Matrices. Fields Institute Communications, 12, (American Mathematical Society Providence, RI, 1997), pp. 1–19Google Scholar
  2. 2.
    P. Biane, Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems. J. Funct. Anal. 144(1), 232–286 (1997)Google Scholar
  3. 3.
    T. Lévy, Schur-Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218(2), 537–575 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.IRMARUniversité de Rennes 1Rennes cedexFrance

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