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 \(\mathbb {A}_2\) and a unit honeycomb graph \(\mathbb {H}_2\), for any value of the (Euclidean) repulsion diameter \(D>0\). Only attainable values of D are relevant, for which \(D^2=a^2+b^2+ab\), \(a, b \in \mathbb {Z}\) (Löschian numbers). Depending on arithmetic properties of \(D^2\), 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 \(\mathbb {A}_2\) or \(\mathbb {H}_2\). On \(\mathbb {A}_2\), our approach works for all attainable \(D^2\); on \(\mathbb {H}_2\) we have to exclude \(D^2 = 4, 7, 31, 133\), where a sliding phenomenon occurs, similar to that on a unit square lattice \(\mathbb {Z}^2\). For all values \(D^2\) apart from the excluded ones, we prove the coexistence of multiple high-density pure phases. Their number grows at least as \(O(D^2)\); 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 \(\mathbb {A}_2\) and \(\mathbb {H}_2\) for any value of the disk diameter D.