Sufficient conditions for the uniqueness of a probability field and estimates for correlations

In this article we will investigate probability fields (probability distributions) on spaces of the form \(X = \mathop \prod \limits_{i \in V} X_i\) , where X_i={0,1} and V is countable and deduce criteria for the uniqueness of a probability field having a given set of conditional probabilities $$\{ P_{i.} ^ - (x_i /x_{V\backslash i} )\} ,i \in V,x_i \in x_i ,x_{V\backslash i} \in \mathop \prod \limits_{j \in V\backslash i} X_j .$$ The results obtained here are convenient for the estimates of probability fields of a sufficiently general form (e.g., with an arbitrary conjugate potential). In the case of a Markov field an exponential estimate for the correlations is derived.