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We consider proper 5-colourings of the kagome lattice. Proper q-colouringscorrespond to configurations in the zero-temperature q-state anti-ferromagneticPotts model. Salas and Sokal have given a computer assisted proof of strongspatial mixing on the kagome lattice for q>=6 under any temperature, includingzero temperature. It is believed that there is strong spatial mixing for q>=4.Here we give a computer assisted proof of strong spatial mixing for q=5 andzero temperature. It is commonly known that strong spatial mixing implies thatthere is a unique infinite-volume Gibbs measure and that the Glauber dynamicsis rapidly mixing. We give a proof of rapid mixing of the Glauber dynamics onany finite subset of the vertices of the kagome lattice, provided that theboundary is free (not coloured). The Glauber dynamics is not necessarilyirreducible if the boundary is chosen arbitrarily for q=5 colours. The Glauberdynamics can be used to uniformly sample proper 5-colourings. Thus, aconsequence of rapidly mixing Glauber dynamics is that there is fullypolynomial randomised approximation scheme for counting the number of proper5-colourings.

Strong Spatial Mixing and Rapid Mixing with Five Colours for the Kagome Lattice