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We present several results on the mixing time of the Glauber dynamics forsampling from the Gibbs distribution in the ferromagnetic Potts model. At afixed temperature and interaction strength, we study the interplay between themaximum degree ($\Delta$) of the underlying graph and the number of colours orspins ($q$) in determining whether the dynamics mixes rapidly or not. We find alower bound $L$ on the number of colours such that Glauber dynamics is rapidlymixing if at least $L$ colours are used. We give a closely-matching upper bound$U$ on the number of colours such that with probability that tends to 1, theGlauber dynamics mixes slowly on random $\Delta$-regular graphs when at most$U$ colours are used. We show that our bounds can be improved if we restrictattention to certain types of graphs of maximum degree $\Delta$, e.g. toroidalgrids for $\Delta = 4$.

Mixing of the Glauber dynamics for the ferromagnetic Potts model