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

TL;DR

This paper introduces a random-cluster representation for a Potts model variant with invisible colours, proving a first-order phase transition and symmetry breaking at low temperatures using Pirogov-Sinai analysis.

Contribution

It develops a novel random-cluster framework for the Potts model with invisible colours and establishes phase transition properties through Pirogov-Sinai theory.

Findings

Existence of a first-order transition for large r

Low-temperature q-fold symmetry breaking

Random-cluster representation applicable to the model

Abstract

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.