Coexistence of ordered and disordered phases in Potts models in the continuum

This is the second of two papers on a continuum version of the Potts model, where particles are points in $\mathbb R^d$, $d\ge 2$, with a spin which may take $S\ge 3$ possible values. Particles with different spins repel each other via a Kac pair potential of range $\ga^{-1}$, $\ga>0$. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist $S+1$ mutually distinct DLR measures.