Gibbs States of Graphical Representations of the Potts Model with External Fields

We consider the ferromagnetic q-state Potts model, with each of the q spin values coupled to an external field. We also introduce a generalized random cluster model, which includes both the Potts model in arbitrary homogeneous external fields and the non-integer q random cluster model as special cases. We establish the FKG property, the nite energy condition, uniqueness of the innite cluster, and Gibbsianness of limit states for this generalized model. Furthermore, we develop the theory of Gibbs states for the Edwards-Sokal representation of the Potts model in a field, and relate the phase structure in this representation to those in the spin and random cluster representations. Finally, we characterize the possible color(s) of the innite cluster(s) and show that the correspondence between Edwards-Sokal Gibbs states and their random cluster marginals is bijective, once the color of the innite cluster is fixed.