it is known that the distribution of a random unitary matrix, under the heat kernel measure on the unitary group u(n), converges as n tends to infinity. i will discuss the convergence of the distribution of a random unitary matrix arising from a levy process on the unitary group u(n). the approach is based on the schur-weyl duality, and we will see that the asymptotic distribution is closely related to the counting of certain paths in the symmetric group.