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

Abstract

Abstract. We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the qstate Potts model on a Cayley tree. In our previous work [14] 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.Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

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