A Reverse Hard-Core Model in $\mathbb{A}_2$: an Application of the Pirogov-Sinai Theory
In this paper we use the Pirogov--Sinai theory to analyze a class of particle models of Statistical Mechanics on the unit triangular lattice $\mathbb{A}_2$. The models are specified by two parameters: the particle activity $u\in(0,1)$ and a positive L\"oschian number $t$ interpreted as a maximal squared clearing radius. Admissible configurations are those in which every empty lattice site has an occupied site within squared distance at most $t$, thereby forbidding empty disks of radius exceeding or equal $t$. The Hamiltonian favors configurations with as few occupied sites as possible, while the admissibility condition enforces a positive density of particles. We prove that the periodic ground states of the model, i.e., the configurations achieving an optimal balance of two tendencies, are the ones whose occupied sites form triangular sublattices of explicitly determined squared side-length $d_*(t)$. Furthermore, for sufficiently small values of the activity parameter $u$, some (but not necessarily all) periodic ground states generate extreme Gibbs (DLR) measures. We describe the resulting phase structure and characterize the pure phases associated with stable ground states.