Free Jacobi process associated with one projection: local inverse of the flow

TL;DR

This paper analyzes the spectral dynamics of the free Jacobi process linked to a projection, using Lagrange inversion to compute local inverse flow coefficients, revealing connections to free unitary Brownian motion.

Contribution

It introduces a method to compute the local inverse of the flow for the free Jacobi process using Lagrange inversion, extending understanding of spectral flow behavior.

Findings

Computed Taylor coefficients of the local inverse flow around zero.

Derived a contour integral representation for the first derivative of the Taylor series.

Connected the case of rank 1/2 to free unitary Brownian motion.

Abstract

We pursue the study started in \cite{Dem-Hmi} of the dynamics of the spectral distribution of the free Jacobi process associated with one orthogonal projection. More precisely, we use Lagrange inversion formula in order to compute the Taylor coefficients of the local inverse around $z=0$ of the flow determined in \cite{Dem-Hmi}. When the rank of the projection equals $1/2$, the obtained sequence reduces to the moment sequence of the free unitary Brownian motion. For general ranks in $(0,1)$, we derive a contour integral representation for the first derivative of the Taylor series which is a major step toward the analytic extension of the flow in the open unit disc.