The Annals of Probability

Countable State Space Markov Random Fields and Markov Chains on Trees

Stan Zachary

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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)

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


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.

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