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 $X=\prod\limits_{i\in V}X_i$, 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
$$ \{P_i(x_i/X_{V\setminus i})\},\quad i\in V,\quad x_i\in X_i,\quad x_{V\setminus i}\in\prod_{j\in V\setminus i}X_j. $$
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.