Translation-invariant Gibbs measures for a model with logarithmic potential on a Cayley tree

In this paper, we consider a model with logarithmical potential and with the set [0, 1] of spin values, on a Cayley tree k of the order k. In the case k = 2, 3, we shall prove that, there is a unique translation-invariant splitting Gibbs measure for this model. For the case k = 4, we show that there are three translation-invariant Gibbs measures for this model.

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