proved independently in the physics community by Salas and Sokal [@sal97]. This result has been followed by numerous refinements gradually reducing the ratio of colours to degree required for rapid mixing: see [@fri07] for a recent survey. In this paper we shall investigate the interplay of the maximum degree $\Delta$ of the graph $G$ and the number of colours $q$ in determining whether the convergence of Glauber dynamics for the *ferromagnetic* Potts model is fast (rapid mixing) or slow.

Definitions
-----------

Throughout we shall be concerned with discrete-time, reversible, ergodic Markov chains with finite state space $\Omega$. Let $\mathcal{M}$ be such a Markov chain with transition matrix $P$ and (unique) stationary distribution $\pi$. For $\varepsilon > 0$ and $x \in \Omega$, we define $$\tau_x(\mathcal{M}, \varepsilon) = \min \{ t: \, \|P^t(x, \cdot) - \pi(\cdot)\|_{TV} \leq \varepsilon \},$$ where $\| \cdot \|_{TV}$ denotes total variation distance between two distributions: that is, $$\| \phi - \phi' \|_{TV} := \frac12 \sum_{x \in \Omega}|\phi(x) - \phi'(x)|.$$ for any two probability distributions $\phi$, $\phi'$ on $\Omega$. We define $\tau(\mathcal{M},\varepsilon)= \max_x \tau_x(\mathcal{M}, \varepsilon)$.

Let $G=(V,E)$ be a graph with $n:=|V|$, and let $[q]=\{ 1, \ldots, q \}$ be a set of colours (or spins). We write 