Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice

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.