On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external fieldTo the memory of H-O Georgii

• Published in: Journal of statistical mechanics Vol. 2019; no. 7
• Main Authors: Bogachev, Leonid V, Rozikov, Utkir A
• Format: Journal Article
• Language: English
• Published: IOP Publishing and SISSA 24.07.2019
• ISSN: 1742-5468
• DOI: 10.1088/1742-5468/ab270b

The paper concerns the q-state Potts model (i.e. with spin values in ) on a Cayley tree of degree (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 and .