Spectral Distribution of the Free Unitary Brownian Motion: Another Approach

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.