Semi-classical limit for random walks

Let (G, mu) be a discrete symmetric random walk on a compact Lie group G wi th step distribution mu and let T-mu be the associated transition operator on L-2 (G). The irreducibles V-rho of the left regular representation of G on L-2(G) are finite dimensional invariant subspaces for T-mu and the spect rum of T-mu is the union of the sub-spectra sigma(T-mu\V-rho) on the irredu cibles, which consist of real eigenvalues {lambda(rho 1),...,lambda(rho) di m V-rho}. Our main result is an asymptotic expansion for the spectral measu res [GRAPHICS] along rays of representations in a positive Weyl chamber t(+)*, i.e. for se quences of representations k(rho), k is an element of N with k --> infinity . As a corollary we obtain some estimates on the spectral radius of the ran dom walk. We also analyse the fine structure of the spectrum for certain ra ndom walks on U(n) (for which T-mu is essentially a direct sum of Harper op erators).