Lecturer: Izabella Stuhl (Penn State)
The hard-core model has attracted attention for quite a long time; the first rigorous results about the phase transition on a lattice were obtained by Dobrushin in late 1960s. Since then, various aspects of the model gained importance in a number of applications. We propose a solution for the high-density hard-core model on a triangular lattice. The high-density phase diagram (i.e., the collection of pure phases) depends on arithmetic properties of the exclusion distance D; a convenient classification of possible cases can be given in terms of Eisenstein primes. For two classes of values of D the phase diagram is completely described: (I) when either D or D/3–√ is a positive integer whose prime decomposition does not contain factors of the form 6k+1, (II) when D2 is an integer whose prime decomposition contains (i) a single prime of the form 6k+1, and (ii) other primes, if any, in even powers, except for the prime 3. For the remaining values of D we offer some partial results. The main method of proof is the Pirogov-Sinai theory with an addition of Zahradnik's argument. The theory of dominant ground states is also extensively used, complemented by a computer-assisted argument.
This is a joint work with A. Mazel and Y. Suhov.