## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 4 (1983), 894-903.

### Countable State Space Markov Random Fields and Markov Chains on Trees

#### Abstract

Let $S$ and $A$ be countable sets and let $\mathscr{G}(\Pi)$ be the set of Markov random fields on $S^A$ (with the $\sigma$-field generated by the finite cylinder sets) corresponding to a specification $\Pi$, Markov with respect to a tree-like neighbour relation in $A$. We define the class $\mathscr{M}(\Pi)$ of Markov chains in $\mathscr{G}(\Pi)$, and generalise results of Spitzer to show that every extreme point of $\mathscr{G}(\Pi)$ belongs to $\mathscr{M}(\Pi)$. We establish a one-to-one correspondence between $\mathscr{M}(\Pi)$ and a set of "entrance laws" associated with $\Pi$. These results are applied to homogeneous Markov specifications on regular infinite trees. In particular for the case $|S| = 2$ we obtain a quick derivation of Spitzer's necessary and sufficient condition for $|\mathscr{G}(\Pi)| = 1$, and further show that if $|\mathscr{M}(\Pi)| > 1$ then $|\mathscr{M}(\Pi)| = \infty$.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 4 (1983), 894-903.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993439

**Digital Object Identifier**

doi:10.1214/aop/1176993439

**Mathematical Reviews number (MathSciNet)**

MR714953

**Zentralblatt MATH identifier**

0524.60056

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G60: Random fields

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82A25 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Phase transition Markov random fields Markov chains on infinite trees entrance laws

#### Citation

Zachary, Stan. Countable State Space Markov Random Fields and Markov Chains on Trees. Ann. Probab. 11 (1983), no. 4, 894--903. doi:10.1214/aop/1176993439. https://projecteuclid.org/euclid.aop/1176993439