Rapid mixing for lattice colourings with fewer colours

We provide an optimally mixing Markov chain for 6-colourings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colourings has strong spatial mixing, so the six-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colourings of the square lattice.