### Markov Chains Indexed by Trees

Itai Benjamini and Yuval Peres
Source: Ann. Probab. Volume 22, Number 1 (1994), 219-243.

#### 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.

First Page:
Primary Subjects: 60J15
Secondary Subjects: 60J10, 60J80
Full-text: Open access