  We study the antiferromagnetic 3-state Potts model on general (periodic)
plane quadrangulations $\Gamma$. 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 $\Gamma$ 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 $\Gamma$ 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-Kotecky 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.
