Limit shapes for growing extreme characters of U(.)

We prove the existence of a limit shape and give its explicit description for certain probability distribution on signatures (or highest weights for unitary groups). The distributions have representation theoretic origin.they encode decomposition on irreducible characters of the restrictions of certain extreme characters of the infinite-dimensional unitary group U(.) to growing finite-dimensional unitary subgroups U(N). The characters of U(.) are allowed to depend on N. In a special case, this describes the hydrodynamic behavior for a family of random growth models in (2+1)-dimensions with varied initial conditions.