Translation-invariant Gibbs measures of a model on Cayley tree

We consider a model where the spin takes values in the set [0,1]d, and is assigned to the vertexes of the Cayley tree. We reduce the problem of describing the “splitting Gibbs measures” of the model to the description of the solutions of some non-linear integral equation. For a concrete form of the Kernel of the integral equation we show the uniqueness of solution.

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