An outstanding open problem in statistical mechanics is to determine if the
system of identical non-overlapping hard disks in the plane admits a unique
Gibbs measure at high-densities. In physical terms, the question is about the
existence of a phase transition in this system. A discretization of this problem
is a system of hard disks of a given Euclidean diameter D on a unit triangular
lattice $A_2$ and a unit square lattice $Z^2$. A natural view is that, as $D$ tends
to infinity, the discrete model (under an appropriate scaling) approaches the
continuous one.
However, our results show that the situation is more subtle. We give a
complete solution, for a general value of $D$, of both discrete versions of model,
on $A_2$ and $Z^2$; in the latter case – in absence of sliding. The main tool is
the Pirogov–Sinai theory: we determine the structure of high-density Gibbs
measures in the model, relating it to periodic ground states. The answers depend
on arithmetic properties of the value $D$ and are given in terms of Eisenstein
primes for $A_2$ and solutions to norm equations in the cyclotomic integer
ring $Z[ζ]$ for $Z^2$, where $ζ$ is a primitive 12th root of unity. The number of
high-density Gibbs measures in both cases grows indefinitely with $D$ but non-
monotonically.

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