Constructive Criterion for the Uniqueness of Gibbs Field

In this report we consider v -dimensional lattice systems with a given translation-invariant potential U. Our main result is the construction of a set of conditions CV on the potential U, where V⊂Z v is any finite volume, such that if for some V the condition CV holds for the interaction U, then the Gibbs State with this interaction is unique. The complexity of the conditions CV increases, of course, with the volume V. The one-point criterion, c{t}, was introduced earlier by one of us [1]. It was intensively used later (see [2]-[7]).