The hard-core model attracted attention since an early stage of the rigorous Statistical Mechanics, which has been enhanced by recent advances on dense-packing hard-sphere configurations in $\mathbb{R}^d$ ($d = 2, 3, 8, 24$) and in view of new applications, e.g., in Computer Science and Biology. A discrete version of the model emerges when the sphere centers are placed at sites of a lattice (or a graph). This talk focuses mainly (but not exclusively) on the hard-core model on a unit square lattice $\mathbb{Z}^2$. We analyze high-density ground states and Gibbs measures for a general exclusion diameter by means of the Pirogov-Sinai theory and its modifications. We will discuss a number of arising topics, e.g., (i) sliding, (ii) counting and comparing periodic ground states, and (iii) the Peierls condition. We also propose a quantitative analysis of hard-circle configurations in $\mathbb{R}^2$ by assessing "deviations" from a triangular dense-packing arrangement.

No preliminary knowledge of the pre-requisite material will be assumed from the audience. Joint with A. Mazel (AMC Health) and I. Stuhl (PSU)