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

2014/03/23 by Christof Kuelske, C. Kuelske, Kuelske, C. +2 · 2 citations
Mathematics · Physics and Astronomy · #60K35 #82B26 #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #msc:60K35 #msc:82B26

paper · pdf · doi:10.48550/arxiv.1403.5775

44 pages. To appear in Random Structures and Algorithms

openalex publication_date 2014/03/23 · arxiv created 2016/04/09 · arxiv updated 2016/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \citeKRK 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.

Cited by

• Extremality of translation-invariant phases for a finite-state SOS-model on the binary tree
• Boundary conditions for translation-invariant Gibbs measures of the Potts model on Cayley trees

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