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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 nearest-neighbor exclusion where published by 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. On a square lattice the nearest-neighbor exclusion critical point has been estimated from above and below in a series by a number of authors.
We analyze the hard-core model on a triangular lattice and identify the extreme Gibbs measures (pure phases) for high densities. Depending on arithmetic properties of the hard-core diameter $D$, the number of pure phases equals either $D^2$ or $2D^2$. 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; cf. arXiv:1803.04041. No special knowledge will be assumed from the audience.
