Completely analytic interactions with infinite values

Summary In this note we extend the notion of completely analytic interactions of Gibbs random fields that is known for finite interactions with finite range to interactions that can have infinite values, too. We formulate a set of ten conditions on such interactions in terms of analyticity properties of the partition functions, or correlation decay. The main theorem states that all these conditions are equivalent. Therefore, an interaction is called a completely analytic interaction, if it satisfies one of these conditions.