On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

The paper concerns the q-state Potts model (i.e. with spin values in {1, . . . , q}) on a Cayley tree T^k of degree k ⩾ 2 (i.e. with k + 1 edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invariant SGMs and prove a novel criterion of translation invariance. Assuming a ferromagnetic nearest-neighbour spin–spin interaction, we obtain various sufficient conditions for uniqueness. For a model with constant external field, we provide in-depth analysis of uniqueness versus non-uniqueness in the subclass of completely homogeneous SGMs by identifying the phase diagrams on the ‘temperature–field’ plane for different values of the parameters q and k. In a few particular cases (e.g. q = 2 or k = 2), the maximal number of completely homogeneous SGMs in this model is shown to be 2q − 1, and we make a conjecture (supported by computer calculations) that this bound is valid for all q ⩾ 2 and k ⩾ 2.