High-density hard-core model on a triangular lattice

2018. 06. 15. 14:15
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 $D/{\sqrt 3}$ is a positive integer whose prime decomposition does not contain factors of the form $6k+1$, (II) when $D^2$ 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.