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 $D^2=a^2+b^2+ab$, $a, b \in\mathbb{Z}$ (L\"oschian 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 existence of a first-order phase transition where the number of co-existing pure phases grows at least as $O(D^2)$. The proof is based on the Pirogov--Sinai theory which requires non-trivial verifications 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 re-distributed 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 $D$.