## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 3 (1983), 457-490.

### Random Walks on Discrete Groups: Boundary and Entropy

V. A. Kaimanovich and A. M. Vershik

#### Abstract

The paper is devoted to a study of the exit boundary of random walks on discrete groups and related topics. We give an entropic criterion for triviality of the boundary and prove an analogue of Shannon's theorem for entropy, obtain a boundary triviality criterion in terms of the limit behavior of convolutions and prove a conjecture of Furstenberg about existence of a nondegenerate measure with trivial boundary on any amenable group. We directly connect Kesten's and Folner's amenability criteria by consideration of the spectral measure of the Markov transition operator. Finally we give various examples, some of which disprove some old conjectures.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 3 (1983), 457-490.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993497

**Digital Object Identifier**

doi:10.1214/aop/1176993497

**Mathematical Reviews number (MathSciNet)**

MR704539

**Zentralblatt MATH identifier**

0641.60009

**JSTOR**

links.jstor.org

**Subjects**

Primary: 22D40: Ergodic theory on groups [See also 28Dxx]

Secondary: 28D20: Entropy and other invariants 43A07: Means on groups, semigroups, etc.; amenable groups 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 20F99: None of the above, but in this section 60J15 60J50: Boundary theory

**Keywords**

Random walk on group exit boundary entropy of random walk $n$-fold convolution invariant mean amenability

#### Citation

Kaimanovich, V. A.; Vershik, A. M. Random Walks on Discrete Groups: Boundary and Entropy. Ann. Probab. 11 (1983), no. 3, 457--490. doi:10.1214/aop/1176993497. https://projecteuclid.org/euclid.aop/1176993497