I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated Lie algebra.
On the probabilistic side, this relates to the fact that uniformly random tilings of a hexagon (e.g., see http://www.math.brown.edu/~rkenyon/gallery/bppsim.gif) in a scaling limit show the behavior of GUE random matrices (more precisely, the GUE minors, or GUE corners, process).
In more detail, tilings of the hexagon are in 1-1 correspondence with vectors of the Gelfand-Tsetlin basis of a suitable representation of $U(N)$. The scaling limit means taking this representation to be very large. The asymptotics are therefore described by probability measures on the space of Hermitian matrices invariant under conjugation by unitary matrices.
If the height of the hexagon is constant, the resulting measure on Hermitian matrices is supported by a single $U(N)$-orbit; if the height also goes to infinity, it is possible to observe a Gaussian limit - a Gaussian mixture of such measures, which exactly corresponds to the GUE random matrices.
Update:
One of the papers I found which most closely reflects this:
G.J. Heckmann. Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups. Invent. Math., 67(2):333–356, 1982.
