Extreme Gibbs measures for a Hard-Core-SOS model on Cayley trees

Abstract

We investigate splitting Gibbs measures (SGMs) of a three-state (wand-graph) hardcore SOS model on Cayley trees of order $k\ge 2$. Recently, this model was studied for the hinge-graph with $k=2,3$, while the case $k\ge 4$ remains unresolved. It was shown that as the coupling strength $\theta$ increases, the number of translation-invariant SGMs (TISGMs) evolves through the sequence $1\to 3\to 5\to 6\to 7$. In this paper, for wand-graph we demonstrate that for arbitrary $k\ge 2$, the number of TISGMs is at most three, denoted by ${\mu }_i$, $i=0,1,2$. We derive the exact critical value ${\theta }_{\text{cr}}(k)$ at which the non-uniqueness of TISGMs begins. The measure ${\mu }_0$ exists for any $\theta >0$. Next, we investigate whether ${\mu }_i$, $i=0,1,2$ is extreme or non-extreme in the set of all Gibbs measures. The results are quite intriguing: 1) For ${\mu }_0$: - For $k=2$ and $k=3$, there exist critical values ${\theta }_1(k)$ and ${\theta }_2(k)$ such that ${\mu }_0$ is extreme if and only if $\theta \in ({\theta }_1,{\theta }_2)$, excluding the boundary values ${\theta }_1$ and ${\theta }_2$, where the extremality remains undetermined. - Moreover, for $k\ge 4$, ${\mu }_0$ is never extreme. 2) For ${\mu }_1$ and ${\mu }_2$ at $k=2$ there is ${\theta }_5<{\theta }_{\text{cr}}(2)=1$ such that these measures are extreme if $\theta \in ({\theta }_5,1)$.

14 pages