Do hard disks in the plane admit a unique Gibbs measure at high density?
This is one of the outstanding open problems of statistical mechanics,
and it seems natural to approach it by requiring the centers to
lie in a fine lattice; equivalently, we may fix the lattice, but let the
Euclidean diameter $D$ of the hard disks tend to infinity. Unlike most
models in statistical physics, we find non-universality and connections to
number theory, with different new phenomena arising in the triangular
lattice $\mathbb{A}_2$, the square lattice $\mathbb{Z}^2$ and the hexagonal
tiling $\mathbb{H}_2$.

In particular, number-theoretic properties of the exclusion diameter $D$ turn^M
out to be important.

We analyze high-density hard-core Gibbs measures via Pirogov-Sinai
theory. The first step is to identify periodic ground states, i.e.,
maximal density disk configurations which cannot be locally 'improved'. A
key finding is that only certain `dominant' ground states, which we
determine, generate nearby Gibbs measures. Another important ingredient
is the Peierls bound separating ground states from other admissible
configurations.

Answers are provided in terms of Eisenstein primes for $\mathbb{A}_2$ and
norm equations in the ring $\mathbb{Z}[\sqrt{3}]$
for $\mathbb{Z}^2$. The number of high-density hard-core Gibbs measures
grows indefinitely with $D$ but non-monotonically. In $\mathbb{Z}^2$ we analyze
the phenomenon of 'sliding'.