The large-N limit of the Segal-Bargmann transform on UN

We study the (two-parameter) Segal-Bargmann transform B-s,t(N) on the unitary group 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 g(s,t) as N -> infinity, 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 B-s,t(N) to its limit g(s,t). We characterize the operator g(s,t) through its inverse action on the standard polynomial basis. Finally, we show that, in the case s = t, the limit transform g(s,t) is the "free Hall transform" g(t) introduced by Biane. (C) 2013 Elsevier Inc. All rights reserved.