Spectral distribution of the free unitary Brownian motion: Another approach

Nizar Demni, Taoufik Hmidi

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any τ ∈ (0,4) a Jordan curve γt around the origin, not intersecting the semi-axis [1, ∞] and whose image under some meromorphic function ht 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.

Original languageEnglish (US)
Title of host publicationSeminaire de Probabilites XLIV
PublisherSpringer Verlag
Number of pages16
ISBN (Print)9783642274602
StatePublished - 2012

Publication series

NameLecture Notes in Mathematics
ISSN (Print)0075-8434

ASJC Scopus subject areas

  • Algebra and Number Theory


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