Semi-classical limit for random walks

Let (G,μ) be a discrete symmetric random walk on a compact Lie group G with step distribution μ and let Tμ be the associated transition operator on L²(G). The irreducibles V_p of the left regular representation of G on L²(G) are finite dimensional invariant subspaces for T_μ and the spectrum of T_μ is the union of the sub-spectra simg;a(T_μ 1v_ρ) on the irreducibles, which consist of real eigenvalues {λ_ρ1,..., λ_ρdim;v_ρ}. Our main result is an asymptotic expansion for the spectral measures Matrix Equation along rays of representations in a positive Weyl chamber t., i.e. for sequences of representations kp, k ∈ N with k Ρ ∞. As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on U(n) (for which T_μ, is essentially a direct sum of Harper operators).