Antiferromagnetic Potts Models on the Square Lattice

We study the antiferromagnetic $q$-state Potts model on the square lattice for $q=3$ and $q=4$, using the Wang-Swendsen-Koteck'y Monte Carlo algorithm and a new finite-size-scaling extrapolation method. For $q=3$ we obtain good control up to correlation length $\xi \sim 5000$; the data are consistent with $\xi (\beta )=A{e}^{2\beta }\beta (1+a_1{e}^{-\beta }+\dots )$ as $\beta \to \mathrm{\infty }$. For $q=4$ the model is disordered even at zero temperature.