It is proved here that at sufficiently low temperatures, a phase transition occurs in the model of a lattice gas with pairwise interaction of the particles, if a constraint, meaning roughly that the negative part of the potential in some sense outweighs its positive part, is imposed on the interaction potential; or if the potential is nonzero, nonpositive, and decreases sufficiently rapidly at infinity. The proof is based on a further development of the method introduced independently by the author in [1], [2] for the proof of the existence of a phase transition in the Ising model of a lattice gas, and by Griffiths [3] for the solution of a similar problem. Using the same method, Berezin and Sinai [4] proved the existence of a phase transition in models of a lattice gas with a nonpositive finite potential, which is negative in the segment [0, R].All the constructions presented below are carried out analogously for lattices of any dimensionality greater than one (as is known, there are no phase transitions in one-dimensional lattices). For greater clarity, we carry out the reasoning for two-dimensional lattices (the generalization to higher dimensions is described in detail in [2]).
