Positivity

, Volume 21, Issue 3, pp 955–961 | Cite as

A model with uncountable set of spin values on a Cayley tree: phase transitions

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Abstract

In this paper we consider a model with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order two. This model depends on two parameters \(n\in \mathbb N\) and \(\theta \in [0,1)\). We prove that if \( 0 \le \theta \le \frac{2n+3}{2(2n+1)}\), then for the model there exists a unique translational-invariant Gibbs measure; If \(\frac{2n+3}{2(2n+1)}< \theta <1\), then there are three translational-invariant Gibbs measures (i.e. phase transition occurs).

Keywords

Cayley tree Configuration Gibbs measures Phase transitions 

Mathematics Subject Classification

Primary 82B05 82B20 Secondary 60K35 
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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institute of MathematicsNational University of UzbekistanTashkentUzbekistan

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