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In this section we begin the analysis of *block dynamics* in which, at each step, the colours of several vertices (or a *block* of vertices) are updated. We first present the framework and show general results on block dynamics. In the next subsection we discuss suitable choices of blocks and, in Theorem \[pr:3items\], show rapid mixing of block dynamics for certain block systems.

As before, let $G=(V,E)$ be a graph, fix $\lambda >1$ and let $\Omega = [q]^V$, where $[q] = \{1, \ldots, q\}$. Let $\mathscr{S} = \{ S_1,\ldots, S_R\}$ be a collection of subsets of $V$ such that $\cup_{S\in\mathscr{S}} S = V$. Each element of $\mathscr{S}$ is called a *block*, and we call $\mathscr{S}$ a *block system* for $G$. Fix a probability distribution $\psi$ on $\mathscr{S}$. We define a Markov chain $\mathcal{M}_{\mathrm{BD}}=\mathcal{M}_{\mathrm{BD}}^{\mathscr{S}, \psi}(G, \lambda, q)$ with state space $\Omega$, which we call the $(\mathscr{S}, \psi)$-*block dynamics*. We ensure that the new chain also has the Gibbs distribution as its stationary distribution. First we need some more notation.

Given $S\in \mathscr{S}$, for $c\in [q]^{S}$ and $X\in\Omega$ we let $X^{(S,c)}\in \Omega$ be the configuration defined by $$X^{(S,c)}(u) = \begin{cases} X(u) & \text{ if $u\not\in S$,}\\
         c(u) & \text{ if $u\in 