Changeover in the transition nature of local-interaction Potts models: a new perspective

We present a novel combinatorial approach which allows the determination of the critical temperature and the phase transition order of Potts models with a round-the-face interaction. Using this approach, it is demonstrated that for some two dimensional ferromagnetic Potts models with completely local interaction there is a changeover in the transition order at a critical integer $q_c\leq 3$, where a first order transition is observed for $q>q_c$ and a second order transition is assumed at least for $q= q_c$. This stands in contrast to the standard two-spin interaction Potts model where the critical integer value for which the transition is continuous is $q_c=4$.