Phase Transition in Long-Range $q-$state Models via Contours. Clock and Potts Models with Fields

Using the group structure of the state space of $q-$state models, a new definition of contour for long-range spin-systems in ${\mathbb{Z}}^d$ ($d\ge 2$), and a multidimensional version of Fröhlich-Spencer contours, we prove phase transition for a class of ferromagnetic long-range systems which includes the Clock and Potts models. Our arguments work for the entire region of exponents of regular power-law interactions, namely $\alpha >d$, and for any $q\ge 2$. As an application, we prove phase transition for Potts models with decaying fields when the field decays fast enough and in the presence of a random external field.