First-Order Transition in Potts Models with "Invisible" States : Rigorous Proofs

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In some recent papers by Tamura, Tanaka and Kawashima [R. Tamura, S. Tanaka and N. Kawashima, Prog. Theor. Phys. 124 (2010), 381; S. Tanaka, R. Tamura and N. Kawashima, J. Phys.: Conf. Ser. 297 (2011), 012022; S. Tanaka and R. Tamura, arXiv:1012.4254] a class of Potts models with "invisible" states was introduced, for which the authors argued, by numerical arguments and by a mean-field analysis, that a first-order transition occurs. Here we show that the existence of this first-order transition can be proven rigorously, by relatively minor adaptations of existing proofs for ordinary Potts models. In our argument, we present a random-cluster representation for the model, which might also be of interest for general parameter values of the model.
2011-11-25