  We describe the full exit boundary of random walks on homogeneous trees, in
particular, on the free groups. This model exhibits a phase transition, namely,
the family of Markov measures under study loses ergodicity as a parameter of
the random walk changes.
  The problem under consideration is a special case of the problem of
describing the invariant (central) measures on branching graphs, which covers a
number of problems in combinatorics, representation theory, probability and was
fully stated in a series of recent papers by the first author \cite{V1,V2,V3}.
On the other hand, in the context of the theory of Markov processes, close
problems were discussed as early as 1960s by E.~B.~Dynkin.
