Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree1. Issue 4 (26th August 2016)

• Record Type: Journal Article
• Authors: Külske, Christof; Rozikov, Utkir A.
• Is Part Of: Random structures & algorithms. Volume 50:Issue 4(2017)
• Page Start: 636
• Page End: 678
• Publication Date: 2016-08-26
• Subjects: Potts model -- Critical temperature -- Cayley tree -- Gibbs measure -- extreme measure -- reconstruction problem; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20671
• Languages: English
• ISSNs: 1042-9832

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 q ‐state Potts model on a Cayley tree. In our previous work (Külske et al., J Stat Phys 156 (2014), 189–200) 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. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 636–678, 2017