Gibbs measures and Markov random fields with association. I (Q1810166)

In [``Gibbs states on countable sets'' (1974; Zbl 0297.60054)] \textit{C. J. Preston} introduced the notions of a Gibbs state with potential of the nearest neighbor, of a Markov random field, and of the state of the nearest neighbor on a finite graph, and he proved that these three definitions are equivalent. In the present paper the notions of Gibbs measure with corresponding potential with association \(I\) (where \(I\) is a subset of the set \(\{1,2,\dots,k\}\)) of a Markov random field with memory \(I\) and measure with association \(I\) are introduced. It is proven that these three notions are equivalent.