Geodesic
A central limit theorem for Plancherel representations of the infinite-dimensional unitary group
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 19-34

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We study the asymptotics of traces of (noncommutative) monomials formed by the images of certain elements of the universal enveloping algebra of the infinite-dimensional unitary group in its Plancherel representations. We prove that they converge to (commutative) moments of a Gaussian process which can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian free fields. The limiting process has previously arisen via the global scaling limit of spectra for submatrices of Wigner Hermitian random matrices. This note is an announcement, proofs will appear elsewhere. 
@article{ZNSL_2012_403_a1,
     author = {A. M. Borodin and A. I. Bufetov},
     title = {A central limit theorem for {Plancherel} representations of the infinite-dimensional unitary group},
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     publisher = {mathdoc},
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     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a1/}
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A. M. Borodin; A. I. Bufetov. A central limit theorem for Plancherel representations of the infinite-dimensional unitary group. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 19-34. http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a1/