$(\omega_1, \omega_2)$ with $|\omega_1 \xor
\omega_2| \le m$. Let $\mu_1$ and $\mu_2$ be the coordinate marginals of $\mu$. Then $$|\ent(\mu_1) - \ent(\mu_2)| \le \log \sum_{k=0}^m {|Y| \choose k}
\le |Y| \big(-\alpha \log \alpha - (1 - \alpha) \log (1 - \alpha)\big)
\,.$$

Suppose that the finite group $\gp$ acts on $X$ and preserves the measure $\mu$. Then the [**entropy**]{} of the pair $(\mu,\gp)$ is $$\ent(\mu,\gp) := |\gp|^{-1} \ent(\mu)
\,.$$

Finally, suppose that $\gp$ is a countable amenable finitely generated subgroup of $\Aut(\gh)$. Let $\mu$ be a probability measure on $2^\edges$ that is preserved by $\gp$. Let $\Seq{\gp_n}$ be a Følner sequence in $\gp$ and $H$ be a finite subgraph of $\gh$ such that $\gp H =\gh$, provided such an $H$ exists. Then the [**(metric) entropy**]{} of the pair $(\mu,\gp)$, also called the [**$\gp$-entropy**]{} of $\mu$, is $$\ent(\mu,\gp) :=
\lim_{n \to\infty} |\gp_n|^{-1} \ent\big(\mu \restrict (\gp_n H) \big)
\,.$$ This does not depend on the choice of $H$. See \[b\].OrnWeiss/ for more details on entropy.

Recall that $\cp(\gh)$ denotes the complexity of the graph $\gh$, i.e., the number of spanning trees of $\gh$. Let $p_k(x;\gh)$ denote the probability that simple random walk on $\gh$ started at $x$ is back at $x$ after 