Potts model with invisible colours: Random-cluster representation and Pirogov-Sinai analysis

We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among $q$ “visible” colours along with the presence of $r$ non-interacting “invisible” colours. We introduce a random-cluster representation for the model, for which we prove the existence of a first-order transition for any $q>0$, as long as $r$ is large enough. When $q>1$, the low-temperature regime displays a $q$-fold symmetry breaking. The proof involves a Pirogov-Sinai analysis applied to this random-cluster representation of the model.