  We consider Gibbs measures on the configuration space $S^{\mathbb{Z}^d}$,
where mostly $d\geq 2$ and $S$ is a finite set. We start by a short review on
concentration inequalities for Gibbs measures. In the Dobrushin uniqueness
regime, we have a Gaussian concentration bound, whereas in the Ising model (and
related models) at sufficiently low temperature, we control all moments and
have a stretched-exponential concentration bound. We then give several
applications of these inequalities whereby we obtain various new results.
Amongst these applications, we get bounds on the speed of convergence of the
empirical measure in the sense of Kantorovich distance, fluctuation bounds in
the Shannon-McMillan-Breiman theorem, fluctuation bounds for the first
occurrence of a pattern, as well as almost-sure central limit theorems.
