Critical Behavior of Coupled q-state Potts Models under Weak Disorder

We investigate the effect of weak disorder on different coupled $q$-state Potts models with $q\le 4$ using two loops renormalisation group. This study presents new examples of first order transitions driven by randomness. We found that weak disorder makes the models decouple. Therefore, it appears that no relations emerge, at a perturbation level, between the disordered $q_1\times q_2$-state Potts model and the two disordered $q_1$, $q_2$-state Potts models ($q_1\ne q_2$), despite their central charges are similar according to recent numerical investigations. Nevertheless, when two $q$-state Potts models are considered ($q>2$), the system remains always driven in a strong coupling regime, violating apparently the Imry-Wortis argument.