  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 $U(\infty)$ to growing
finite-dimensional unitary subgroups $U(N)$. The characters of $U(\infty)$ 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.
