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

摘要

We study translation-invariant splitting Gibbs measures (TISGMs, tree-indexed Markov chains) for the fertile three-state hard-core models with activity λ ＞ 0 on the Cayley tree of order 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 (A)λ ＞ 0 and k ≥ 1, there exists a unique TISGM; (ⅱ) for hinge (resp. wand) case at k = 2 if λ < λ_(cr) = 9/4 (resp. λ ＜ λ_(cr) = 1), there exists a unique TISGM, and for λ ＞ 9/4 (resp. λ ＞ 1), there exist three TISGMs. In this paper we generalize the result (ⅱ) for any k ≥ 2, i.e., for hinge and wand cases we find the exact critical value λ_(cr)(k) with properties mentioned in (ⅱ). Moreover, we find some regions for the λ 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.