bounds for our slow-mixing results in Section \[se:tormix\].

Comparison with related results and phase transitions
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We write $o(1)$ for an expression that tends to $0$ as $q \to \infty$. (The most interesting setting for our results is when $q$ is large.) We now restate our results in terms of the inverse temperature $\beta$, under the assumption that $J=1$, so that $\lambda = e^\beta$.

The results of Theorems \[th:main1\], \[th:main2\](i), \[th:main2\](ii), \[th:main3\](ii) say respectively:

-   if $\beta \leq \frac{1 + o(1))}{\Delta}\, \log q$ then the Glauber dynamics of the $q$-state Potts model mixes rapidly on graphs of maximum degree $\Delta$;

-   if $\beta \leq \frac{1 + o(1)}{\Delta-1}\, \log q$ then the Glauber dynamics of the $q$-state Potts mixes rapidly on graphs of maximum degree $\Delta$;

-   if $\beta > \frac{1 + o(1)}{\Delta - 1 - \nfrac{1}{\Delta-1}}\, \log q $ then the Glauber dynamics of the $q$-state Potts model mixes slowly for almost all regular graphs of degree $\Delta \geq 3$;

-   if $\beta < \frac{1+o(1)}{2}\, \log q$ then the Glauber dynamics of the $q$-state Potts model mixes rapidly for toroidal grids.

There is some overlap between Theorem \[th:main1\] and a result of Hayes [@hayes Proposition 14] for $q=2$, which was 