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 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.