We consider a ferromagnetic nearest-neighbor model on a Cayley tree of degree k≥ 2
k\geqslant2 with uncountable local state space 0, 1 where the energy function depends on a
parameter 𝜃∈ 0, 1). We show that for 0≤ 𝜃≤ 5 3 k 0\leqslantθ\leqslant53k the model has a
unique translation-invariant Gibbs measure. If 5 3 k< 𝜃< 1 53k<θ<1 there is a phase
transition, in particular there are three translation-invariant Gibbs measures.

… Abstract 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 ⩽ θ ⩽ 5 … In the present note we continue the
investigation from [2] and consider a model with nearest-neighbor interactions and local
state space given by the uncountable set [0,1] on a Cayley tree Ŵk of general order k ⩾ 2.
The translation-invariant Gibbs measures are studied via a non-linear functional equation …