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This paper describes results of the author with B. K. Driver and T. Kempconcerning the large-N limit of the Segal--Bargmann transform for the unitarygroup U(N). We consider the transform on matrix-valued functions that arepolynomials 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 polynomialfunctions 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 mainingredient in our proof of this result is an "asymptotic product rule" for theLaplacian on U(N), which allows us to compute explicitly the leading-orderlarge-N behavior of the heat operator on U(N). The second main ingredient inthe proof is the phenomenon of "concentration of traces," in which the relevantheat kernel measures are concentrating onto sets where the trace of any powerof the variable is constant.

The Segal-Bargmann transform for unitary groups in the large-N limit