High-Density Hard-Core Model on Triangular and Hexagonal Lattices

We perform a rigorous study of the Gibbs statistics of high-density hard-core random configurations on a unit triangular lattice A2 and a unit honeycomb graph H2, for any value of the (Euclidean) repulsion diameter D>0. Only attainable values of D are relevant, for which D2=a2+b2+ab, a,b∈Z (Löschian numbers). Depending on arithmetic properties of D2, we identify, for large fugacities, the pure phases (extreme Gibbs measures) and specify their symmetries. The answers depend on the way(s) an equilateral triangle of side-length D can be inscribed in A2 or H2. On A2, our approach works for all attainable D2; on H2 we have to exclude D2=4,7,31,133, where a sliding phenomenon occurs, similar to that on a unit square lattice Z2. For all values D2 apart from the excluded ones, we prove the coexistence of multiple high-density pure phases. Their number grows at least as O(D2); this establishes the existence of a phase transition. The proof is based on the Pirogov–Sinai theory which, in its original form, requires the verification of key assumptions: finiteness of the set of periodic ground states and the Peierls bound. To establish the Peierls bound, we develop a general method based on the concept of a redistributed area for Delaunay triangles. Some of the presented proofs are computer-assisted. As a by-product of the ground state identification, we solve the disk-packing problem on A2 and H2 for any value of the disk diameter D.