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

Abstract

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 [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.

Абсолют Конечно Порожденных Групп: II. Лапласова И Вырожденная Части

“…Then f ν does not depend on the choice of a path: if for some path Q g representing the same element g, the values ν(Q g ) · A −|Qg | ν and ν(P g ) · A −|Pg | ν differed, then, extending P g and Q g by paths with the same projections to the Cayley graph to paths representing the identity, we would obtain a contradiction with (3).…”

“…Let us present a complete description of the absolute (Theorem 2.1 in [3]): the absolute of the free group with respect to the natural generators is the direct product of the boundary of the free group and an interval:…”

“…The general problem of describing even the Laplacian part of the absolute of an arbitrary discrete finitely generated group is difficult, and it is not yet clear how wide is the class of groups for which it can be reduced to finding the PF boundary. We have demonstrated such a reduction for free groups [3]: in this case, the Laplacian part of the absolute is the product of the PF boundary and a semiopen interval. We cite this result in the section containing examples.…”

The theory of filtrations of subalgebras, standardness, and independence

“…While the Poisson-Furstenberg boundary reduces to a single point for many groups, the absolute does it very rarely; it has a structure of a generalized fiber bundle over the Poisson-Furstenberg boundary. For the free group, the absolute is calculated in [89]. In this case, a natural phase transition from ergodicity to nonergodicity is discovered.…”

Asymptotic theory of path spaces of graded graphs and its applications

“…Theorem 15 (Vershik-Malyutin [71]). For q ≥ 2, the set Erg(Γ(T q+1 , v 0 )) of all ergodic central measures on the space T (Γ(T q+1 , v 0 )) of infinite paths in the dynamic graph Γ(T q+1 , v 0 ) over the (q + 1)-homogeneous tree T q+1 (i.e., the absolute boundary) coincides with the following family of Markov measures: Λ q := {λ ω,r | ω ∈ ∂T q+1 , r ∈ [1/2, 1]} .…”