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

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. [ABSTRACT FROM AUTHOR]