Gibbs states of graphical representations of the Potts model with externalfields

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 homogeneou s external fields and the noninteger q random cluster model as special case s. We establish the FKG property, the finite energy condition, uniqueness o f the infinite cluster, and Gibbsianness of limit states for this generaliz ed model. Furthermore, we develop the theory of Gibbs states for the Edward s-Sokal representation of the Potts model in a field, and relate the phase structure in this representation to those in the spin and random cluster re presentations. Finally, we characterize the possible color(s) of the infini te cluster(s) and show that the correspondence between Edwards-Sokal Gibbs states and their random cluster marginals is bijective, once the color of t he infinite cluster is fixed. (C) 2000 American Institute of Physics. [S002 2-2488(00)00803-3].