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Phase transition in the exit boundary problem for random walks on groups

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.