closed form formula for $\beta_0$, we show in the appendix that $\beta_0 = \frac{\log q}{\Delta - 1} + O(1)$. Thus $\beta_0$ approximately matches the rapid mixing bound of (b).

We note that, in a recent related work, Galanis *et al.* [@GSVY] give a very detailed picture of the phase transitions of the ferromagnetic Potts model on the infinite $\Delta$-regular tree. Using this analysis they also show that show that the Swendsen-Wang process (a MCMC process different to Glauber dynamics) mixes slowly at a specific phase transition point on almost all random regular graphs of degree $\Delta$.

As mentioned earlier, result (d) is only illustrative since sharper bounds for the grid are known. It is known that for the infinite 2-dimensional grid, the phase transition occurs at $q=(\lambda-1)^2$ [@wu82] (i.e. $\beta = \log(1 + q^\frac{1}{2})$) and that rapid mixing occurs for finite grids when $\beta$ is below this threshold; see [@mar99] and Theorem 2.10 of [@ull]. It is conjectured that the Glauber dynamics mixes slowly when $\beta$ is above this threshold; see Remark 2.11 of [@ull]). Borgs, Chayes and Tetali [@BCT] proved that for $q$ sufficiently large and for $\beta > \frac{\log\left(q\right)}{2}+O(q^{-1/2})$, the heat bath Glauber dynamics is slowly mixing on 