# High-density hard-core configurations on a triangular lattice

The high-density hard-core configuration model has attracted
attention for quite a long time. The first rigorous
results about the phase transition on a lattice with a
Dobrushin in 1968. In 1979, Baxter calculated the free energy
and specified the critical point on a triangular lattice
with a nearest-neighbor exclusion; in 1980 Andrews gave
a rigorous proof of Baxter’s calculation with the help of Ramanujan’s
identities. We analyze the hard-core model on a triangular lattice
and identify the extreme Gibbs measures (pure phases) for high
densities. Depending
on arithmtic properties of the hard-core diameter $D$, the number of
pure phases equals either $D2$ or $2D2$. A classification
of possible cases can be given in terms of Eisenstein primes.

If the time allows, I will mention 3D analogs of some of these
results.

This is a joint work with A Mazel and I Stuhl. No special knowledge will
be assumed from the audience.
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