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.