maximum degree $\Delta$.

Theorem \[th:main1\] will be proved in Section \[se:GD\]; see Proposition \[le:vertex\] for a more detailed statement.

In Theorem \[th:main2\] we improve the exponent of $\lambda$ in the bound, but at the expense of a larger constant. We also show that the exponent achieved is close to the best possible, by proving a corresponding slow-mixing bound for almost all regular graphs of degree $\Delta$.

\[th:main2\] Fix an integer $\Delta \geq 2$. For any $\eta \in (0,1)$ there are constants $c_1$ and $c_2$ (depending on $\eta$ and $\Delta$), such that for any integer $q \geq 2$ and any $\lambda>1$

1.  if $q>c_1\lambda^{\Delta-1+\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes rapidly for the class of connected graphs of maximum degree $\Delta$;

2.  if $q<c_2\lambda^{\Delta-1-\tfrac{1}{\Delta-1}-\eta}$ then the Glauber dynamics of the $q$-state Potts model at activity $\lambda$ mixes slowly for almost all regular graphs of degree $\Delta \geq 3$.

Theorem \[th:main2\] is proved at the end of the paper: a more detailed statement of Theorem \[th:main2\](i) can be found in Theorem \[th:glaubercompare\], while a more detailed statement of Theorem \[th:main2\](ii) can be found in Theorem \[th:torpid-random\].

Theorem \[th:main2\](ii) is proved using a conductance argument. It 