Schur–Weyl duality and the heat kernel measure on the unitary group

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.