Leírás
Előadó: Iza Stuhl
Cím: Hard-core model on Z^2
Absztrakt: It is well-known that in R^2 the maximum-density configuration of hardcore (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 A_2 or a unit square lattice 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 Z^2 for a large value of fugacity z. Comparisons will be made with the case of A_2 as well. 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 configurations, and dominance is determined by counting defects in local excitations.
On A_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 Z^2, the situation is made more complicated by various (related) phenomena: sliding, non-tessellation etc. Here, some results are available; conjectures of various generality can also be proposed.
This is a joint work with A. Mazel and Y. Suhov.