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# Spectral distribution of the free unitary Brownian motion: another approach

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### 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 M\"obius 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.

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###### Journal
24 March 2011
1103.4693