Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree

We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the $q$-state Potts model on a Cayley tree. In our previous work \cite{KRK} we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least $2^{q-1} + q$ extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.Comment: 44 pages. To appear in Random Structures and Algorithm

Existence and extremality of periodic Gibbs measures for fertile three-state hard-core models in the case Wand

We consider fertile three-state Hard-Core (HC) models with the activity parameter $\lambda>0$ 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 $\lambda>0$ 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.Comment: in Englis