Phase Transition in Ferromagnetic \(q-\)state Models: Contours, Long-Range Interactions and Decaying Fields

Using the group structure of the state space of \(q-\)state models and a new definition of contour for long-range spin-systems in \(\mathbb{Z}^d\), with \(d\geq 2\), a multidimensional version of Fr\"{o}hlich-Spencer contours, we prove the 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 \(q \geq 2\). As an application, we prove the phase transition for Potts models with decaying fields when the field decays fast enough.