## Abstract

This paper is a companion to a series of papers devoted to the study of the spectral distribution of the free Jacobi process associated with a single projection. Actually, we note that the flow derived in [N. Demni and T. Hmidi, Spectral distribution of the free Jacobi process associated with one projection, Colloq. Math. 137(2) (2014) 271-296] solves a radial Löwner equation and as such, the general theory of Löwner equations implies that it is univalent in some connected region in the open unit disc. We also prove that its inverse defines the Aleksandrov-Clark measure at z = 1 of some Herglotz function which is absolutely-continuous with an essentially bounded density. As a by-product, we deduce that z = 1 does not belong to the continuous singular spectrum of the unitary operator whose spectral dynamics are governed by the flow. Moreover, we use a previous result due to the first author in order to derive an explicit, yet complicated, expression of the moments of both the unitary and the free Jacobi processes. The paper is closed with some remarks on the boundary behavior of the flow's inverse.

Original language | English (US) |
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Article number | 1850001 |

Journal | Random Matrices: Theory and Application |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2018 |

## Keywords

- Aleksandrov-Clark measures
- Free Jacobi process
- free unitary Brownian motion
- Herglotz transform
- radial Löwner equation
- univalent map

## ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Discrete Mathematics and Combinatorics