  In this paper we study the asymptotic of multiplicities of irreducible
representations in large tensor products of finite dimensional representations
of simple Lie algebras and their statistics with respect to Plancherel and
character probability measures. We derive the asymptotic distribution of
irreducible components for the Plancherel measure, generalizing results of
Biane and Tate and Zelditch. We also derive the asymptotic of the character
measure for generic parameters and an intermediate scaling in the vicinity of
the Plancherel measure. It is interesting that the asymptotic measure is
universal and after suitable renormalization does not depend on which
representations were multiplied but depends significantly on the degeneracy of
the parameter in the character distribution.
