Potts Model with Invisible Colors: Random-Cluster Representation and Pirogov–Sinai Analysis

We study a recently introduced variant of the ferromagnetic Potts model consisting of a ferromagnetic interaction among q “visible” colors along with the presence of r non-interacting “invisible” colors. 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.