entropy. For simplicity, we restrict our discussion to groups acting on graphs, which is enough for our purposes. Suppose first that $\mu$ is a probability measure on a finite or countable set, $X$. The [**entropy**]{} of $\mu$ is $$\ent(\mu) := - \sum_{x \in X} \mu(x) \log \mu(x)
\,.$$ For example, suppose that $\mu$ is a probability measure on the Borel sets of $2^\edges := \{ 0, 1 \}^\edges$ in the product topology. As usual, we identify an element of $2^\edges$ with the subset (or “configuration") of edges where the value 1 is taken. If $H$ is a finite subgraph of $\gh$, write $\mu \restrict H$ for the restriction of $\mu$ to the $\sigma$-field generated by the restrictions to $\edges(H)$. That is, $\mu \restrict H$ is defined on the finite set $2^{\edges(H)}$ and has an entropy as above. We shall twice use the following well-known lemma (see, e.g., Lemma 6.2 of \[b\].BurPem/ and p. 11 of \[b\].Bollobas/ for the bound on the binomial coefficient sum).

l.compare Let $Y$ be a finite set and $m$ be a positive integer. Write $\alpha := m/|Y|$. Suppose that $\mu$ is a probability measure on $2^Y \times 2^Y$ that is supported on the set of pairs 