Periodic Gibbs measures for three-state hard-core models in the case Wand
We consider fertile three-state Hard-Core (HC) models with the activity parameter $λ>0$ on a Cayley tree. It is known that there exist four types of such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. In the case wand on a Cayley tree of order $k\geq2$, exact critical values $λ>0$ are found for which two-periodic Gibbs measures are not unique. Moreover, we study the extremality of the existing two-periodic Gibbs measures on a Cayley tree of order two.
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