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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 valuedfunctions that are equivariant under the adjoint action of the group, thetransform has a meaningful limit $\mathscr{G}_{s,t}$ as $N\to\infty$, which canbe identified as an operator on the space of complex Laurent polynomials. Weintroduce the space of {\em trace polynomials}, and use it to give effectivecomputational methods to determine the action of the heat operator, and thusthe Segal--Bargmann transform. We prove several concentration of measure andlimit theorems, giving a direct connection from the finite-dimensionaltransform $\mathbf{B}_{s,t}^N$ to its limit $\mathscr{G}_{s,t}$. Wecharacterize the operator $\mathscr{G}_{s,t}$ through its inverse action on thestandard polynomial basis. Finally, we show that, in the case $s=t$, the limittransform $\mathscr{G}_{t,t}$ is the ``free Hall transform'' $\mathscr{G}^t$introduced by Biane.

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