A short note on strong convergence of $q$-Gaussians

TL;DR

This paper proves the strong convergence of $q$-Gaussians as the parameter $q$ varies, showing the spectrum of polynomials in these variables deforms continuously, with implications for quantum group asymptotics.

Contribution

It establishes strong convergence of $q$-Gaussians and links spectrum deformation to the asymptotic behavior of free orthogonal quantum groups.

Findings

Spectrum of polynomials in $q$-Gaussians deforms continuously with $q$

Strong convergence of $q$-Gaussians as $q$ varies

Asymptotic strong convergence of free orthogonal quantum groups

Abstract

In this note, we prove strong convergence of $q$-Gaussians with respect to a parameter $q$, which implies the spectrum of any self-adjoint non-commutative polynomial in $q$-Gaussians is continuously deformed with respect to $q$. With Bo\.{z}ejko's Haagerup-type inequality, we follow Brannan's approach to proving the asymptotic strong convergence of the free orthogonal quantum group.