$\lambda>1$

1.  if $q>c_3\lambda^{3+\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the class of connected graphs of maximum degree 4;

2.  if $q>c_4\lambda^{2+\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the toroidal grid;

3.  if $q<c_5\lambda^{\tfrac{8}{3}-\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes slowly for almost all regular graphs of degree $4$.

In particular, for sufficiently large $\lambda$ there is a positive integer $q$ such that the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the toroidal grid, but slowly for almost all regular graphs of degree $4$.

The purpose of Theorem \[th:main3\] is illustrative and it is proved at the end of the paper. Theorem \[th:main3\](i) and (iii) are immediate consequence of Theorem \[th:main2\] (by substituting $\Delta=4$), while Theorem \[th:main3\](ii) is a useful illustration of our general technique applied to the grid. A more detailed statement of Theorem \[th:main3\](ii) is given as part of Theorem \[th:glaubergrid\].

Section \[se:rapid\] contains our results on rapid mixing of Glauber dynamics. Section \[se:extremal\] is devoted to an extremal problem whose solution allows us to obtain improved 