Phase transition in the exit boundary problem for random walks on groups
- Authors
- Type
- Published Article
- Journal
- Functional Analysis and Its Applications
- Publisher
- Springer US
- Publication Date
- Apr 01, 2015
- Volume
- 49
- Issue
- 2
- Pages
- 86–96
- Identifiers
- DOI: 10.1007/s10688-015-0090-3
- Source
- Springer Nature
- Keywords
- License
- Yellow
Abstract
We describe the full exit boundary of random walks on homogeneous trees, in particular, on 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, and probability and was fully stated in a series of recent papers by the first author [1]–[3]. 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.