It is well-known that in $\mathbb{R}^2$ the maximum-density configuration of hard-core (non-overlapping) disks of diameter $D$ is given by a triangular/hexagonal arrangement (Fejes Tóth, Hsiang). If the disk centers are placed at sites of a lattice, say, a unit triangular lattice $\mathbb{L}^2$ or a unit square lattice $\mathbb{Z}^2$, then we get a discrete analog of this problem, with the Euclidean exclusion distance.

I will discuss high-density Gibbs/DLR measures for the hard-core model on $\mathbb{L}^2$ and $\mathbb{Z}^2$ for a large value of fugacity $z$. According to the Pirogov-Sinai theory, the extreme Gibbs measures are obtained via a polymer expansion from dominating ground states. For the hard-core model the ground states are associated with maximally dense sublattices, and dominance is determined by counting defects in local excitations.  

On $\mathbb{L}^2$ we have a complete description of the extreme Gibbs measures for a large $z$ and any $D$; a convenient tool here is the Eisenstein integer ring. For $\mathbb{Z}^2$, the situation is made more complicated by various (related) phenomena: sliding, non-tessellation and shapes of Voronoi cells. Nevertheless, some theorems are available; conjectures of various generality can also be proposed. A number of our results are computer-assisted.

This is a joint work with A. Mazel and Y. Suhov.