Thermodynamic versus Topological Phase Transitions: Cusp in the   Kert\'esz Line
  We present a study of phase transitions of the Curie--Weiss Potts model at (inverse) temperature $\beta$, in presence of an external field $h$. Both thermodynamic and topological aspects of these transitions are considered. For the first aspect we complement previous results and give an explicit equation of the thermodynamic transition line in the $\beta$--$h$ plane as well as the magnitude of the jump of the magnetization (for $q \geqslant 3)$. The signature of the latter aspect is characterized here by the presence or not of a giant component in the clusters of a Fortuin--Kasteleyn type representation of the model. We give the equation of the Kert\'esz line separating (in the $\beta$--$h$ plane) the two behaviours. As a result, we get that this line exhibits, as soon as $q \geqslant 3$, a very interesting cusp where it separates from the thermodynamic transition line. 
Philippe Blanchard (BIBOS), Daniel Gandolfo (CPT), Jean Ruiz (CPT),   Marc Wouts (PMA, MODAL'X)