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Abstract: We prove that for a finite collection of real-valued functions on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of under the properly scaled heat kernel measure at a given time on the unitary group has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order . In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S.N. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results. [ABSTRACT FROM AUTHOR]

Abstract: We investigate a relation between the Brownian motion on the unitary group and the most natural random walk on the symmetric group, based on Schur–Weyl duality. We use this relation to establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure. This expectation turns out to be the generating series of certain paths in the Cayley graph of the symmetric group. Using our expansion, we recover asymptotic results of Xu, Biane and Voiculescu. We give an interpretation of our main expansion in terms of random ramified coverings of a disk. [Copyright &y& Elsevier]