Abstract

Phase transitions in constrained spin systems on hierarchical lattices are fundamental in statistical mechanics, with applications ranging from surface adsorption to combinatorics. We characterize extreme Gibbs measures for a three-state hard-core solid-on-solid (HC SOS) model on Cayley trees under wand-graph constraints, where adjacent spins must satisfy $|\sigma (x)-\sigma (y)|\le 1$. While prior work analyzed hinge-graph constraints (yielding up to 7 translation-invariant SGMs), we prove that for wand-graph constraints and any tree order $k\ge 2$:

• At most three translation-invariant SGMs ${\mu }_0,{\mu }_1,{\mu }_2$ exist.

• The critical threshold ${𝜃}_{\text{ cr}}(k)$ for nonuniqueness is determined exactly.

• ${\mu }_0$ exists for all $𝜃>0$ but is extreme only for $k=2,3$ in $({𝜃}_1(k),{𝜃}_2(k))$, where ${𝜃}_1(k)$ and ${𝜃}_2(k)$ some determined values.

• ${\mu }_0$ is never extreme for $k\ge 4$.

• ${\mu }_1,{\mu }_2$ are extreme for $k=2$ when $𝜃\in ({𝜃}_5,1)$, where ${𝜃}_5<{𝜃}_{\text{ cr}}(2)=1$.

These results reveal how graph constraints and tree dimensionality fundamentally alter the structure of extreme Gibbs measures, with direct implications for phase coexistence in constrained systems.

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