The Hard-Core Model on 2D Graphs

We study extreme Gibbs measures for the Hard-Core model on a unit triangular/square/honeycomb lattice. We use the Pirogov-Sinai theory and a number of complementary results and methods developed by Zahradnik, Dobrushin-Shlosman and Bricmont-Slawny. The model is determined by the fugacity $u$ and the exclusion diameter $D$ (the minimal attained distance between occupied sites). On the square lattice for large enough $u$, Dobrushin (1968) proved that for the smallest non-trivial value $D={\sqrt 2}$ there are two extreme Gibbs measures generated by the checker-board ground states via high-fugacity polymer expansions. However, for the next value $D=2$ he discovered that the ground states exhibit a phenomenon of sliding, which leads to countably many periodic states with no Peierls bound between them. The model on lattice can be viewed as a natural discretization of the Hard-Core model, where there is no sliding, periodic ground states are triangular $D$-sub-lattices, and the Peierls bound is conveniently written in terms of Voronoi cells. Hence, the description of extreme Gibbs measures is reduced to a count of dominant periodic ground states. The model on lattice is more difficult and requires additional constructions. Here we give a complete list of values $D$ with sliding, and then identify periodic ground states and prove the Peierls bound in terms of Delaunay triangulations. Throughout the whole work we use fruitful connections with algebraic number theory, in particular with Eisenstein primes and the cyclotomic ring. A number of our results are computer-assisted. This is a joint work with A. Mazel and I. Stuhl.
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