The hard-core model in discrete 2D: ground states, dominance, Gibbs measures (Izabella Stuhl, Penn State)

25.11.2019 16:15

The hard-core model describes a system of non-overlapping identical hard spheres in a space or on a lattice (more generally, on a graph). An interesting open problem is: do hard disks in a plane admit a unique Gibbs measure at high density? It seems natural to approach this question by possible discrete approximations where disks must have the centers at sites of a lattice or vertexes of a graph.
In this talk, I will report on a progress achieved for the models on a unit triangular lattice A_2 and a unit square lattice Z^2 for a general value of disk diameter D (in the Euclidean metric). We analyze the structure of Gibbs measures for large fugacities (i.e., high densities) by means of the Pirogov-Sinai theory and its modifications, including dominance among periodic ground states.
On A_2 we give a complete description of the set of extreme Gibbs measures; the answer is provided in terms of prime decomposition of the Löschian number D^2 in the Eisenstein integer ring. Here, the extreme Gibbs measures are generated by D-sub-lattices, their shifts and reflections.
On Z^2, we have to exclude the values of D with sliding; for the remaining exclusion distances the answer is given in terms of solutions to a discrete minimization problem. The latter is connected to Norm equations in the cyclotomic integer ring Z[zeta], where zeta is a primitive 12th root of unity.
Parts of our argument contain computer-assisted proofs: identification of instances of sliding, resolution of dominance issues.
This is a joint work with A. Mazel and Y. Suhov.

Lieu

Room 17, Séminaire "Mathématique Physique"

entrée libre