Positive Solutions of a Nonlinear System Characterizing Gibbs Measures in Hard-Core SOS Models on Cayley Trees

We study the m -state p -Solid-on-Solid model with p = ∞ on Cayley trees of order k ≥ 2 . We derive a system of functional equations whose solutions correspond to (splitting) Gibbs measures for the model. For cases $$m=2,3$$ we identify conditions under which the number of translation-invariant Gibbs measures (TIGMs) is exactly $$1$$ , $$2$$ , or $$3$$ , depending on the order k ≥ 2 . Furthermore, we demonstrate that two of these measures are always nonextreme, while the third may be extreme only within a bounded temperature interval—specifically for the Cayley tree of order $$k=2$$ . Moreover, using numerical analysis we conjecture that • For $$m=2$$ , 2≤ k≤ 9 the number of TISGMs at most 7. • For $$m=2$$ , k≥ 10 , and some parameter values, the number of TISGMs is at least 9. • For $$m=3$$ , k≥ 2 , and some parameter values, the number of TISGMs is at least 14.