Gibbs measures for models with uncountable set of spin values

We consider models with the set $[0,1]$ of spin values on a Cayley tree of order $k\ge 2$. We give criterion on parameters of such models for non-uniqueness of Gibbs measure (the phase-transition behavior) on Cayley trees with arbitrary degree $k\ge 2$. Moreover, using ART (Akin, Rozikov, Temir), Bleher-Ganikhodjaev, Zachary constructions continuum sets of new Gibbs measures are constructed.