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).
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Botirov, G.I. A model with uncountable set of spin values on a Cayley tree: phase transitions. Positivity 21, 955–961 (2017). https://doi.org/10.1007/s11117-016-0445-x
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DOI: https://doi.org/10.1007/s11117-016-0445-x