The three-state Potts antiferromagnet on plane quadrangulations

We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations Γ. Any quadrangulation can be built from a dual pair (G,G*). Based on the duality properties of G, we propose a new criterion to predict the phase diagram of this model. If Γ is of self-dual type (i.e. if G is isomorphic to its dual G*), the model has a zero-temperature critical point with central charge c = 1, and it is disordered at all positive temperatures. If Γ is of non-self-dual type (i.e. if G is not isomorphic to G*), three ordered phases coexist at low temperature, and the model is disordered at high temperature. In addition, there is a finite-temperature critical point (separating these two phases) which belongs to the universality class of the ferromagnetic 3-state Potts model with central charge c = 4/5. We have checked these conjectures by studying four (resp. seven) quadrangulations of self-dual (resp. non-self-dual) type, and using three complementary high-precision techniques: Monte-Carlo simulations, transfer matrices, and critical polynomials. In all cases, we find agreement with the conjecture. We have also found that the Wang-Swendsen-Kotecký Monte Carlo algorithm does not have (resp. does have) critical slowing down at the corresponding critical point on quadrangulations of self-dual (resp. non-self-dual) type.