Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice
On a regular \((d+1)\)-tree, where \(d+1\) edges meet at each vertex, the author studies the problem of existence and uniqueness of Markov random fields for a given specification, which is supposed to be given by a symmetric translation invariant positive function, defined on pairs of nearest neighbor sites. A boundedness condition yields the existence. The uniqueness problem is discussed for attractive and repulsive specifications. For \(d=1\), the author is able to reduce some more general cases to \textit{H. Kesten}'s uniqueness result [Ann. Probab. 4, 557-569 (1976; Zbl 0367.60080)].
existence and uniqueness of Markov random fields
scientific article; zbMATH DE number 3938171