Positive fixed points of cubic operators on ${\mathbb{R}}^2$ and Gibbs Measures

Abstract

In this paper we consider one model with nearest-neighbor interactions and with the set $[0,1]$ of spin values on the Cayley tree of order three. Translation-invariant Gibbs measures for the model are studied. Results are proved by using properties of the positive fixed points of a cubic operator in the cone ${\mathbb{R}}_{+}^2$.

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