The large-N limit of the Segal–Bargmann transform on UNUN

We study the (two-parameter) Segal–Bargmann transform Bs,tN on the unitary group UNUN, for large N  . Acting on matrix-valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit Gs,tGs,t as N→∞N→∞, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of 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 Bs,tN to its limit Gs,tGs,t. We characterize the operator Gs,tGs,t through its inverse action on the standard polynomial basis. Finally, we show that, in the case s=ts=t, the limit transform Gt,tGt,t is the “free Hall transform” GtGt introduced by Biane.