Spectral distribution of the free unitary Brownian motion: another approach

TL;DR

This paper presents a new approach to describing the spectral measure of free unitary Brownian motion using complex analysis, Jordan curves, and meromorphic functions, providing an explicit geometric construction.

Contribution

It introduces a novel geometric method involving Jordan curves and meromorphic functions to analyze the spectral distribution of free unitary Brownian motion.

Findings

Constructed Jordan curves $oldsymbol{rac{t}{2}}$ around the origin for $t \u2208 (0,4)$.

Described the spectral measure as a push-forward of a complex measure.

Proved the spectral measure is absolutely continuous with a specific support.

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.