Phase Transition and Critical Values of a Nearest-Neighbor System with Uncountable Local State Space on Cayley Trees

We consider a ferromagnetic nearest-neighbor model on a Cayley tree of degree $k⩾2$ with uncountable local state space [0,1] where the energy function depends on a parameter 𝜃 ∈[0, 1). We show that for $0⩽\theta ⩽\frac{5}{3k}$ the model has a unique translation-invariant Gibbs measure. If $\frac{5}{3k}<\theta <1$ there is a phase transition, in particular there are three translation-invariant Gibbs measures.