Effects of random and deterministic discrete scale invariance on the critical behavior of the Potts model.

The effects of disorder on the critical behavior of the q-state Potts model in noninteger dimensions are studied by comparison of deterministic and random fractals sharing the same dimensions in the framework of a discrete scale invariance. We carried out intensive Monte Carlo simulations. In the case of a fractal dimension slightly smaller than two d(f) ~/= 1.974636, we give evidence that the disorder structured by discrete scale invariance does not change the first order transition associated with the deterministic case when q = 7. Furthermore the study of the high value q = 14 shows that the transition is a second order one both for deterministic and random scale invariance, but that their behavior belongs to different university classes.