\Lambda$ we have $${\mathbb{E}}(d(A',B')\mid (A,B)) \leq \beta .$$ Then by path coupling we may conclude that $$\tau(\mathcal{M},\varepsilon) \leq \frac{\log(n\, \varepsilon^{-1})}{1-\beta}.$$

Glauber dynamics {#se:GD}
----------------

Our goal in this subsection is to prove Theorem \[th:main1\]. In the subsections that follow, we shall see how we can improve Proposition \[le:vertex\] in some special cases, but in Section \[se:tormix\], we shall see that the bound given below is close to best possible, at least in terms of the exponent of $\lambda$.

We actually prove the following proposition, which immediately implies Theorem \[th:main1\] but also provides a bound on the mixing time. The proof is a standard coupling calculation.

\[le:vertex\] Let $G$ be a graph with maximum degree $\Delta$, and fix an activity $\lambda>1$. Suppose that $q$ is an integer which satisfies $ q \geq \Delta\lambda^\Delta + 1$. Recall that $\mathcal{M}_{\mathrm{GD}} = 
\mathcal{M}_{\mathrm{GD}}(G, \lambda, q)$ denotes the Glauber dynamics for the $q$-state Potts model on $G$ at activity $\lambda$. Then $$\tau(\mathcal{M}_{\mathrm{GD}},\varepsilon) \leq 
    (\Delta+1)\, n\log(n\, \varepsilon^{-1}).$$

Fix $(A,B)\in\Lambda$ and let $u$ be the (unique) vertex which is coloured differently by $A$ and $B$. We define a coupling $(A,B)\mapsto (A',B')$ as follows: let ${\bm{v}}$ be a uniformly random vertex of $G$, and given that ${\bm{v}}=v$, obtain 