Gibbsian Distribution of Random Fields Whose Conditional Probabilities May Vanish

The article considers Markov random fields with an arbitrary set of values on a finite or infinite graph. It is shown that such random fields admit a Gibbsian description when their conditional probabilities can vanish with certain constraints. The concept of a random field with vacuum state is introduced to describe these constraints. Moreover, for potentials of fairly general form the article investigates a class of transformations that do not alter the Gibbsian conditional distribution.