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 . It arises in representation theory as the space of indecomposable characters of the infinite-dimensional unitary group U(∞). Alternatively, 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.