$\Omega = [q]^V$ for the set of configurations of $G$ (*i.e.* not-necessarily proper $q$-colourings). Fix a constant $\lambda > 1$, which is called the *activity*. The *Gibbs distribution* $\pi = \pi(G, \lambda, q)$ on $\Omega$ is given by $$\pi(\sigma) \propto \lambda^{ {\mu}(\sigma)}$$ for all $\sigma\in\Omega$, where ${\mu}(\sigma)$ denotes the number of monochromatic edges of $G$ in the configuration $\sigma$. More precisely, $\pi(\sigma) = \lambda^{ {\mu}(\sigma)}/Z$, where $Z$ is the partition function $$Z = Z(G, \lambda, q) = \sum_{\sigma \in \Omega} \lambda^{{\mu}(\sigma)}.$$

The *Glauber dynamics* is a very simple Markov chain on $\Omega$, with stationary distribution given by the Gibbs distribution. Given a configuration $X\in\Omega$, a vertex $v\in V$, and a colour $c \in [q]$, let $n(X,v,c)$ denote the number of neighbours of $v$ with colour $c$ in $X$. Define the probability distribution $\phi_X^v$ on $[q]$ by $$\phi_X^v(c) \propto \lambda^{n(X,v,c)}.$$ The transition procedure of the Glauber dynamics from current state $X_t\in\Omega$ is as follows:

-   choose a vertex ${\bm{v}}$ of $G$ uniformly at random;

-   given that ${\bm{v}}=v$ (here ${\bm{v}}$ is random and $v$ is fixed), choose a colour $c \in [q]$ according to the distribution $\phi = \phi_{X_t}^v$;

-   for each $u \in V$ let $X_{t+1}(u) = 
    \begin{cases} 
    X_t(u)  &\text{if 