Free Bessel laws

TL;DR

This paper introduces free Bessel laws, a new family of probability measures related to free Poisson laws, exploring their properties, analytic and combinatorial aspects, and connections to random matrices and quantum groups.

Contribution

It defines and analyzes free Bessel laws, establishing their fundamental properties and relationships with free Poisson laws, and discusses their relevance to random matrices and quantum groups.

Findings

Defined free Bessel laws and their basic properties

Analyzed supports, atoms, and densities of these measures

Explored connections with random matrices and quantum groups

Abstract

We introduce and study a remarkable family of real probability measures $\pi_{st}$, that we call free Bessel laws. These are related to the free Poisson law $\pi$ via the formulae $\pi_{s1}=\pi^{\boxtimes s}$ and $\pi_{1t}=\pi^{\boxplus t}$. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.