Solutions of the variational problem in the Curie--Weiss--Potts model

The variational problem for the Curie--Weiss--Potts model is solved completely. The results extend those of Ellis and Wang (1990, 1992), in which we study limit theorems and parameter estimations for the model and consider only the case of zero external field. In contrast to the Curie--Weiss model, this model has phase transitions in non-zero external field. All the solutions of the variational problem are non-degenerate points, so all the results in Ellis and Wang (1990, 1992) can be easily extended to the case considered here. We will also point out that simultaneous parameter estimation is impossible.