Generalized Meixner-type free gamma distributions:convolution formulas and potential correspondence

TL;DR

This paper introduces a new class of generalized Meixner-type free gamma distributions, explores their properties, mixture structures, and potential correspondence, extending classical free probability concepts and identifying algebraic relations among free gamma variables.

Contribution

It defines and studies a new family of free gamma distributions, analyzing their properties, maximizing free entropy, and extending free probability correspondences beyond classical limits.

Findings

The distributions maximize Voiculescu's free entropy with specific potentials.

Identification of algebraic relations among free gamma distributed variables.

Extension of free probability correspondences beyond Bercovici-Pata bijection.

Abstract

We introduce and study a class of generalized Meixner-type free gamma distributions $\mu_{t,\theta,\lambda}$ ($t,\theta>0$ and $\lambda\ge 1$), which includes both the free gamma distributions introduced by Anshelevich and certain scaled free beta prime distributions introduced by Yoshida. We investigate fundamental properties and mixture structures of these distributions. In particular, we consider the Gibbs distribution $\frac{1}{\mathcal{Z}_{t,\theta,\lambda}} \exp\{-V_{t,\theta,\lambda}(x)\}$ associated with a family of potentials $V_{t,\theta,\lambda}$, and show that $\mu_{t,\theta,\lambda}$ maximizes Voiculescu's free entropy with potential $V_{t,\theta,\lambda}$ for parameters $t,\theta>0$ and $1\le \lambda<1+t/\theta$. This result substantially extends the range of classcal-free correspondences obtained the potential function, differing from those arising from the Bercovici-Pata bijection. Moreover, we identify algebraic relations involving noncommutative random variables distributed as free gamma distributions.