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 \(\lambda   9/4\) (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.
gibbs measure fertile hard-core model cayley tree critical temperature extreme measure reconstruction problem 82b26 (primary) 60k35 (secondary)