Description
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.