# Segal-Bargmann transform: the $q$-deformation

## TL;DR

This paper explores the $q$-deformed Segal-Bargmann transform, establishing its connections to classical, free, and matrix models, and providing new identifications and definitions in the context of mixed $q$-Gaussian variables.

## Contribution

It introduces the $q$-deformed Segal-Bargmann transform, links it to classical and free transforms, and extends its definition to mixed $q$-Gaussian variables.

## Key findings

- Classical Segal-Bargmann converges to $q$-deformed version in large $N$ limit.
- $q$-deformed transform can be obtained as a mixture of classical and free transforms.
- Identifications of the $q$-deformed transform with matrix models are established.

## Abstract

We give identifications of the $q$-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed $q$-Gaussian variables. We prove that, when defined on the random matrix model of \'Sniady for the $q$-Gaussian variable, the classical Segal-Bargmann transform converges to the $q$-deformed Segal-Bargmann transform in the large $N$ limit. We also show that the $q$-deformed Segal-Bargmann transform can be recovered as a limit of a mixture of classical and free Segal-Bargmann transform.