Gibbs measures for hardcore-SOS models on Cayley trees

We investigate the finite-state p-solid-on-solid model, for p=∞, on Cayley trees of order k≥ 2 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, $k=2,3$ and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as 1→3→5→6→7. 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 p→∞ is continuous.