Phase transition and thermodynamic properties of the Hard-Core-Potts model

We investigate translation-invariant Gibbs measures of the Hard-Core–Potts (HC–Potts) model on the Cayley tree. The model combines Potts-type ferromagnetic interactions with a hard-core exclusion rule, leading to a nontrivial interplay between magnetic ordering and occupancy constraints. For the hinge-type fertile graph, we analyze the cases k = 2 and k = 3, and determine explicit critical values of θ that mark the transition from uniqueness to multiplicity of Gibbs measures. The model exhibits up to five translationinvariant phases depending on the interaction strength. Thermodynamic quantities such as magnetization and quadrupolar moment are computed, revealing ordered phases at low temperatures and a paramagnetic phase at high temperatures.