February 11: Iza Stuhl

Title: Hard-core disks on 2D lattices

Abstract:

It is well-known that in a plane, 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 or a unit square lattice, then we get a discrete analog of this problem, with the Euclidean exclusion distance. (An alternative is to use the graph distance.)

I will discuss high-density Gibbs/DLR measures for the hard-core model of these lattices, 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 periodic ground states. For the hard-core model the ground states are associated with maximally dense configurations, and dominance is determined by counting defects in local excitations.

On a triangular lattice 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 a square lattice, the situation is made more complicated by various (related) phenomena: sliding, non-tessellation etc. Here, some results are available; conjectures of various generality and precision can also be proposed.

A number of our results are computer-assisted (and require a large memory, which is a restriction).

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