Description of translation-invariant Gibbs measures on a Cayley tree of arbitrary order

A class of translation-invariant Gibbs measures for models with a continuum of spin values on a Cayley tree was examined. The results show that the problem of describing such measures reduces to studying positive solutions of a nonlinear integral equation of Hammerstein type. A connection between the solutions of the integral equation and the properties of a corresponding polynomial was established. A theorem on the existence and uniqueness of a positive solution of the Hammerstein equation for an arbitrary tree order 𝑘 ≥ 2 was proved. Explicit formulas for the solutions were obtained via the roots of a polynomial of degree 𝑘 + 1, constructively describing the set of translation-invariant Gibbs measures.

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