We consider three models of statistical mechanics: the classical XY model in arbitrary dimension, the lattice Coulomb gas in dimension two, and the square well model in arbitrary dimension. For each of these three models, we prove that the free energy is analytic in the disordered regime (the square well model is disordered at any positive temperature). In order to prove these results, we prove that the Gibbs measures of these models are factors of i.i.d. with information clusters of exponentially decaying size (volume). In the case of the Coulomb gas, we obtain a strong version of Debye screening with an arbitrary number of arbitrary local observables of the Coulomb gas, and we prove that the Debye phase contains the complement of the Berezinskii-Kosterlitz-Thouless phase.