The asymptotics of interlacing sequences and the growth of continual Young diagrams

We obtain a series of concrete results establishing a somewhat unexpected connection between the asymptotic representation theory of symmetric groups and the classical results for one-dimensional problems of mathematical physics and function theory. In particular:
1) The universal character of the division of roots for a wide class of orthogonal polynomials is shown.
2) A connection between the Plancherel measure of the infinite symmetric group and Markov's moment problem is established.
3) Asymptotics of the Plancherel measure turns out to be connected with the soliton-like solution of the simplest quasilinear equation,
$$ R'_t+RR'_x=0. $$
Bibliography: 14 titles.