  This paper describes results of the author with B. K. Driver and T. Kemp
concerning the large-N limit of the Segal--Bargmann transform for the unitary
group U(N). We consider the transform on matrix-valued functions that are
polynomials in a single variable in U(N). We show that in the large-N limit,
the transform maps functions of this type to single-variable polynomial
functions on the complex group GL(N;C). This result was conjectured by Ph.
Biane and was also proved independently by G. C\'ebron. The first main
ingredient in our proof of this result is an "asymptotic product rule" for the
Laplacian on U(N), which allows us to compute explicitly the leading-order
large-N behavior of the heat operator on U(N). The second main ingredient in
the proof is the phenomenon of "concentration of traces," in which the relevant
heat kernel measures are concentrating onto sets where the trace of any power
of the variable is constant.
