The hard-core model in discrete 2D

Abstract: 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 vertices of a graph.

In this talk, I will report on progress achieved for the models on a unit triangular lattice $\mathbb{A}_2$, square lattice $\mathbb{Z}^2$ and a honeycomb graph $\mathbb{H}_2$ 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. It connects extreme Gibbs measures with dominant ground states.

On $\mathbb{A}_2$ we give a complete description of the set of extreme Gibbs measures; the answer is provided in terms of the prime decomposition of the Löschian number $D^2$ in the Eisenstein integer ring. On $\mathbb{Z}^2$, we work with Gaussian numbers. Here we have to exclude a finite collection of 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 $\mathbb{Z}[\zeta]$, where $\zeta$ is a primitive 12th root of unity. On $\mathbb{H}_2$, we employ connections with the model on $\mathbb{A}_2$, although there are some exceptional values requiring a special approach.

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.

dynamical systemsprobability

Audience: researchers in the topic

( slides )

Horowitz seminar on probability, ergodic theory and dynamical systems