  The classification results for the extreme characters of two basic "big"
groups, the infinite symmetric group S(infinity) and the infinite-dimensional
unitary group U(infinity), are remarkably similar. It does not seem to be
possible to explain this phenomenon using a suitable extension of the
Schur-Weyl duality to infinite dimension. We suggest an explanation of a
different nature that does not have analogs in the classical representation
theory.
  We start from the combinatorial/probabilistic approach to characters of "big"
groups initiated by Vershik and Kerov. In this approach, the space of extreme
characters is viewed as a boundary of a certain infinite graph. In the cases of
S(infinity) and U(infinity), those are the Young graph and the Gelfand-Tsetlin
graph, respectively. We introduce a new related object that we call the Young
bouquet. It is a poset with continuous grading whose boundary we define and
compute. We show that this boundary is a cone over the boundary of the Young
graph, and at the same time it is also a degeneration of the boundary of the
Gelfand-Tsetlin graph.
  The Young bouquet has an application to constructing infinite-dimensional
Markov processes with determinantal correlation functions.
