The hard-core model attracted interest at an early stage of the progress in the rigorous Statistical Mechanics; its popularity increased after recent successes in studying dense-packing configurations of hard spheres in R^d, for d = 2,3,8,24 and in view of new applications, in particular in Computer Science and Biology. A lattice version of the same problem emerges when the sphere centers are positioned at sites of a given lattice (or a graph). We focus on two problems: (i) specification of periodic ground states (PGSs), i.e., configurations of the maximum density which cannot be ‘improved’ by a local change, and (ii) identification of the dominant GSs generating extreme Gibbs/DLR measures for large values of fugacity by means of the Pirogov-Sinai theory. This presentation will be focused mainly (but not exclusively) on the case of a unit square lattice Z2. We plan to touch upon a number of arising topics, including sliding, tesselating, counting PGSs and dominant PGSs and the Peierls condition.
This is a joint work with A. Mazel and Y. Suhov