On models with uncountable set of spin values on a Cayley tree: integral equations (Q430022)

Starting with a previous investigation of the Potts model with a countable set of spin values on the Cayley tree [\textit{N. N. Ganikhodjaev} and \textit{U. A. Rozikov}, Lett. Math. Phys. 75, No. 2, 99--109 (2006; Zbl 1101.82003)], the authors address models with an uncountable set of spin values. The major technical problem in the theory of Gibbs measures is to describe an infinite volume (limiting) measure corresponding to a given Hamiltonian. Although the Doborushin-Lanford-Ruelle theory gives various existence proofs, a complete analysis of the set of limiting Gibbs measures for a specific Hamiltonian is often a diffficult problem. In the present paper, models with nearest neighbor interaction and uncountably many spin values in \([0,1]\) are investigated. The main object of study are translation invariant Gibbs measures for the pertinent homogeneous Hamiltonian lattice model on the Cayley tree. A concept of the splitting Gibbs measure is introduced, depending on the order \(k\geq 1\) of the Cayley tree. Its existence and uniqueness problem is reduced to the solvability of a nonlinear integral equation that is analogous to procedures adopted for systems with a finite and countable set of spin values. For the Potts model with an uncountable set of spin values, the uniqueness of the splitting Gibbs measure is established.