generalised to arbitrary $q$ by Ullrich [@ull Corollary 2.14]. Ullrich showed that when the inverse temperature $\beta$ satisfies $\beta \leq 2c/\Delta$ for some $0<c<1$, then the Glauber dynamics is rapidly mixing on graphs of maximum degree $\Delta$. Hence our result (a) holds for a wider range of $\beta$ when $q$ is large. (For small values of $q$, Theorem \[th:main1\] does not apply but [@ull Corollary 2.14] is valid).

As we have mentioned, there is often a link between certain phase transitions and the critical inverse temperature of associated dynamics (i.e. an inverse temperature below which the dynamics mix rapidly and above which they mix slowly). We will not define what we mean by phase transitions here but mention only that, for Glauber dynamics of the $q$-state Potts model on a random $\Delta$-regular graph, the relevant phase transition is the transition from unique to non-unique Gibbs measure on the infinite $\Delta$-regular tree. H[ä]{}ggstr[ö]{}m [@Hag] showed that this phase transition occurs at an inverse temperature $\beta_0 = \log B$, where $B$ is the unique value for which the polynomial $$(q-1)x^{\Delta} + (2 - B - q)x^{\Delta - 1} +Bx - 1$$ has a double root in $(0,1)$. While there is no general 