Gibbs measures for hardcore-solid-on-solid models on Cayley trees

We investigate the finite-state p-solid-on-solid (p-SOS) model for on Cayley trees of order and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as . This phase diagram is qualitatively similar to the one observed for three-state p-SOS models with p > 0 and, in the case of k = 2, we demonstrate that, on the level of the functional equations, the transition is continuous.