## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 1 (1994), 219-243.

### Markov Chains Indexed by Trees

Itai Benjamini and Yuval Peres

#### Abstract

We study a variant of branching Markov chains in which the branching is governed by a fixed deterministic tree $T$ rather than a Galton-Watson process. Sample path properties of these chains are determined by an interplay of the tree structure and the transition probabilities. For instance, there exists an infinite path in $T$ with a bounded trajectory iff the Hausdorff dimension of $T$ is greater than $\log(1/\rho)$ where $\rho$ is the spectral radius of the transition matrix.

#### Article information

**Source**

Ann. Probab. Volume 22, Number 1 (1994), 219-243.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176988857

**Digital Object Identifier**

doi:10.1214/aop/1176988857

**Mathematical Reviews number (MathSciNet)**

MR1258875

**Zentralblatt MATH identifier**

0793.60080

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J15

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Trees Markov chains branching random walks recurrence Hausdorff dimension packing dimension

#### Citation

Benjamini, Itai; Peres, Yuval. Markov Chains Indexed by Trees. Ann. Probab. 22 (1994), no. 1, 219--243. doi:10.1214/aop/1176988857. http://projecteuclid.org/euclid.aop/1176988857.