Star-cumulants of free unitary Brownian motion

TL;DR

This paper investigates joint free cumulants of free unitary Brownian motion and its adjoint, providing explicit calculations for special cases and analyzing asymptotic behavior as the process approaches a Haar unitary.

Contribution

It explicitly determines certain joint free cumulants of free unitary Brownian motion and introduces an 'infinitesimal determining sequence' related to R-diagonal elements.

Findings

Explicit formulas for special joint free cumulants.

Asymptotic analysis of cumulants as t approaches infinity.

Identification of an 'infinitesimal determining sequence' for R-diagonal elements.

Abstract

We study joint free cumulants of u_t and u_t^{*}, where u_t is a free unitary Brownian motion at time t. We determine explicitly some special families of such cumulants. On the other hand, for a general joint cumulant of u_t and u_t^{*}, we "calculate the derivative" for t going to infinity, when u_t approaches a Haar unitary. In connection to the latter calculation we put into evidence an "infinitesimal determining sequence" which naturally accompanies an arbitrary R-diagonal element in a tracial *-probability space.