Gibbs measures for the fertile three-state hard-core models on a Cayley tree

Abstract

We study translation-invariant splitting Gibbs measures (TISGMs, tree-indexed Markov chains) for the fertile three-state hard-core models with activity $\lambda >0$ on the Cayley tree of order $k\ge 1$. There are four such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. It is known that (i) for the wrench and pipe cases $\forall \lambda >0$ and $k\ge 1$, there exists a unique TISGM; (ii) for hinge (resp. wand) case at $k=2$ if $\lambda <\lambda _\mathrm{cr}=9/4$ (resp. $\lambda <\lambda _\mathrm{cr}=1$), there exists a unique TISGM, and for $\lambda > 9/4$ (resp. $\lambda >1$), there exist three TISGMs. In this paper we generalize the result (ii) for any $k\ge 2$, i.e., for hinge and wand cases we find the exact critical value $\lambda _\mathrm{cr}(k)$ with properties mentioned in (ii). Moreover, we find some regions for the $\lambda$ parameter ensuring that a given TISGM is extreme or non-extreme in the set of all Gibbs measures. For the Cayley tree of order two, we give explicit formulae and some numerical values.