## Abstract

Let ? be a homogenous Markov specification associated with a countable state space S and countably infinite parameter space A possessing a neighbor relation ~ such that (A,~) is the regular tree with d +1 edges meeting at each vertex. Let g(p)be the simplex of corresponding Markov random fields. We show that if ? satisfies a 'boundedness' condition then g(p).We further study the structure of g(p) when ? is either attractive or repulsive with respect to a linear ordering on S. When d = 1, so that (A, ~) is the one-dimensional lattice, we relax the requirement of homogeneity to that of stationarity; here we give sufficient conditions for g(p) and for g(p)to have precisely one member. © 1985.

Original language | English |
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Pages (from-to) | 247-256 |

Number of pages | 10 |

Journal | Stochastic Processes and their Applications |

Volume | 20 |

Issue number | 2 |

Publication status | Published - Sep 1985 |

## Keywords

- attractive specifications
- Markov chains on infinite trees
- Markov random fields
- phase transition
- repulsive specifications