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 $$\lambda >0$$?>0 on the Cayley tree of order $$k\ge 1$$k?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 $$\forall \lambda >0$$??>0 and $$k\ge 1$$k?1, there exists a unique TISGM; (ii) for hinge (resp. wand) case at $$k=2$$k=2 if $$\lambda 9/4$$?>9/4 (resp. $$\lambda >1$$?>1), there exist three TISGMs. In this paper we generalize the result (ii) for any $$k\ge 2$$k?2, i.e., for hinge and wand cases we find the exact critical value $$\lambda _\mathrm{cr}(k)$$?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.