0 50 70 67 v 2 2 8 O ct 2 00 5 Improved mixing bounds for the Anti - Ferromagnetic Potts Model on Z 2 ∗

We consider the anti-ferromagnetic Potts model on the the integer lattice Z. The model has two parameters, q, the number of spins, and λ = exp(−β), where β is “inverse temperature”. It is known that the model has strong spatial mixing if q > 7, or if q = 7 and λ = 0 or λ > 1/8, or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colourings of Z. We show that the system has strong spatial mixing for q ≥ 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function) and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λ than was previously known for q = 3, 4 and 5.