Dobrushin-Shlosman phase uniqueness criterion and applications to hard squares

The rigorous Dobrushin-Shlosman phase uniqueness criterion is reviewed, then applied to the hard square model to prove that only a single phase exists at activity z = 1.185. The criterion is violated (for a five-site by five-site lattice cell) at z = 1.35557, but this does not imply phase nonuniqueness. This work complements that of Dobrushin, Kolafa, and Shlosman, who proved phase uniqueness for all z ≤ 1. Certain experimentally discovered regularities are presented as conjectures: one for a more general problem and two for the application to hard squares. Even with these regularities, however, substantial further improvements in the algorithmic implementation of the criterion will be required before it can become a practical tool for locating phase transitions