  We 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. These expectations turn out to be the generating series of certain
paths in the Cayley graph of the symmetric group. We then compute the
asymptotic distribution of a random unitary matrix under the heat kernel
measure on the unitary group U(N) as N tends to infinity, and prove a result of
asymptotic freeness result for independent large unitary matrices, thus
recovering results obtained previously by Xu and Biane. We give an
interpretation of our main expansion in terms of random ramified coverings of a
disk. Our approach is based on the Schur-Weyl duality and we extend some of our
results to the orthogonal and symplectic cases.
