follows: at high temperature in the ferromagnetic region, $\beta$ is small and so $\lambda$ is close to 1; thus all configurations are weighted roughly equally and the Glauber dynamics walks freely over the state space without getting ‘stuck’. At low temperatures in the ferromagnetic region, $\beta$ is large and so $\lambda$ is also large; thus configurations consisting of predominantly one colour are far more heavily weighted than configurations with a balance of colours, so the Glauber dynamics will become trapped in configurations of the former type. However, determining the exact range of temperature in which Glauber dynamics mixes rapidly is, in general, open.

In the [anti-ferromagnetic]{} case, where it is known that there can be no FPRAS in general, the MCMC technique has still yielded many results approximating the partition function for restricted classes of graph, notably bounded-degree graphs. In the zero temperature limit of the anti-ferromagnetic Potts model only proper vertex colourings have non-zero weight. Thus approximating the partition function is equivalent to approximately counting [proper]{} $q$-colourings of the underlying graph. Jerrum [@jer95] first showed that provided the number of colours is more than twice the maximum degree of the graph then the Glauber dynamics will mix rapidly, also 