  We consider proper 5-colourings of the kagome lattice. Proper q-colourings
correspond to configurations in the zero-temperature q-state anti-ferromagnetic
Potts model. Salas and Sokal have given a computer assisted proof of strong
spatial mixing on the kagome lattice for q>=6 under any temperature, including
zero 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 and
zero temperature. It is commonly known that strong spatial mixing implies that
there is a unique infinite-volume Gibbs measure and that the Glauber dynamics
is rapidly mixing. We give a proof of rapid mixing of the Glauber dynamics on
any finite subset of the vertices of the kagome lattice, provided that the
boundary is free (not coloured). The Glauber dynamics is not necessarily
irreducible if the boundary is chosen arbitrarily for q=5 colours. The Glauber
dynamics can be used to uniformly sample proper 5-colourings. Thus, a
consequence of rapidly mixing Glauber dynamics is that there is fully
polynomial randomised approximation scheme for counting the number of proper
5-colourings.
