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 $$p = \infty$$ on Cayley trees of order $$k \geq 2$$ . 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 $$k \geq 2$$ . 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 $$\bullet$$ For $$m=2$$ , $$2\leq k\leq 9$$ the number of TISGMs at most 7. $$\bullet$$ For $$m=2$$ , $$ k\geq 10$$ , and some parameter values, the number of TISGMs is at least 9. $$\bullet$$ For $$m=3$$ , $$k\geq 2$$ , and some parameter values, the number of TISGMs is at least 14.

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