  The goal of the paper is to describe new connections between representation
theory and algebraic combinatorics on one side, and probability theory on the
other side.
  The central result is a construction, by essentially algebraic tools, of a
family of Markov processes. The common state space of these processes is an
infinite dimensional (but locally compact) space Omega. It arises in
representation theory as the space of indecomposable characters of the
infinite-dimensional unitary group U(infinity).
  Alternatively, Omega can be defined in combinatorial terms as the boundary of
the Gelfand-Tsetlin graph --- an infinite graded graph that encodes the
classical branching rule for characters of the compact unitary groups U(N).
  We also discuss two other topics concerning the Gelfand-Tsetlin graph:
  (1) Computation of the number of trapezoidal Gelfand-Tsetlin schemes (one
could also say, the number of integral points in a truncated Gelfand-Tsetlin
polytope). The formula we obtain is well suited for asymptotic analysis.
  (2) A degeneration procedure relating the Gelfand-Tsetlin graph to the Young
graph by means of a new combinatorial object, the Young bouquet.
  At the end we discuss a few related works and further developments.
