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

We investigate the finite-state p -solid-on-solid ( p -SOS) 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.