High-density hard-core model on triangular and hexagonal lattices

Abstract

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 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 $D$.

Analysis of the model on the hexagonal lattice has been added. Assisting programs are in Ancillary files