The Large-$N$ Limit of the Segal--Bargmann Transform on $\mathbb{U}_N$

TL;DR

This paper investigates the large-$N$ limit of the Segal--Bargmann transform on the unitary group, establishing a connection to free probability and providing computational tools for analyzing the transform's asymptotic behavior.

Contribution

It introduces the limit operator $ extscr{G}_{s,t}$ for the Segal--Bargmann transform on $ extscr{U}_N$, develops trace polynomial techniques, and links finite-dimensional transforms to free probability theory.

Findings

Established the limit $ extscr{G}_{s,t}$ as $N o fty$

Developed computational methods using trace polynomials

Connected the limit transform to the free Hall transform

Abstract

We study the (two-parameter) Segal--Bargmann transform $\mathbf{B}_{s,t}^N$ on the unitary group $\mathbb{U}_N$, for large $N$. Acting on matrix valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit $\mathscr{G}_{s,t}$ as $N\to\infty$, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of {\em trace polynomials}, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal--Bargmann transform. We prove several concentration of measure and limit theorems, giving a direct connection from the finite-dimensional transform $\mathbf{B}_{s,t}^N$ to its limit $\mathscr{G}_{s,t}$. We characterize the operator $\mathscr{G}_{s,t}$ through its inverse action on the standard polynomial basis. Finally, we show that, in the case $s=t$, the limit transform $\mathscr{G}_{t,t}$ is the ``free Hall transform'' $\mathscr{G}^t$ introduced by Biane.