First-order transition in Potts models with "invisible' states: Rigorous proofs

TL;DR

This paper rigorously proves the existence of a first-order phase transition in a class of Potts models with 'invisible' states, complementing previous numerical and mean-field analyses.

Contribution

It provides a rigorous proof of the first-order transition in these models using adapted existing methods and introduces a random-cluster representation for the model.

Findings

First-order transition rigorously established

Random-cluster representation introduced

Supports previous numerical and mean-field results

Abstract

In some recent papers by Tamura, Tanaka and Kawashima [arXiv:1102.5475, 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 be of independent interest.