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November, 1983 Countable State Space Markov Random Fields and Markov Chains on Trees
Stan Zachary
Ann. Probab. 11(4): 894-903 (November, 1983). DOI: 10.1214/aop/1176993439


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 $|\mathscr{M}(\Pi)| = \infty$.


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Stan Zachary. "Countable State Space Markov Random Fields and Markov Chains on Trees." Ann. Probab. 11 (4) 894 - 903, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0524.60056
MathSciNet: MR714953
Digital Object Identifier: 10.1214/aop/1176993439

Primary: 60G60
Secondary: 60J10 , 60K35 , 82A25

Keywords: entrance laws , Markov chains on infinite trees , Markov random fields , phase transition

Rights: Copyright © 1983 Institute of Mathematical Statistics


Vol.11 • No. 4 • November, 1983
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