Gaussian Fluctuations and Free Energy Expansion for Coulomb Gases at Any Temperature

We obtain concentration estimates for the fluctuations of Coulomb gases in any dimension and in a broad temperature regime, including very small and very large temperature regimes which may depend on the number of points. We obtain a full Central Limit Theorem for the fluctuations of linear statistics in dimension 2, valid for the first time down to microscales and for possibly very small or very large temperatures. We show that a similar CLT can also be obtained in any larger dimension conditional on a "no phase-transition" assumption, as soon as one can obtain a precise enough error rate for the expansion of the free energy -- an expansion is obtained in any dimension, but the obtained rate is so far not good enough to conclude. These CLTs can be interpreted as a convergence to the Gaussian Free Field. All the results are valid as soon as the test-function lives on a larger scale than the temperature-dependent minimal scale $\rho_\beta$ introduced in our previous work \cite{as}.