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We obtain concentration estimates for the fluctuations of Coulomb gases inany dimension and in a broad temperature regime, including very small and verylarge temperature regimes which may depend on the number of points. We obtain afull Central Limit Theorem for the fluctuations of linear statistics indimension 2, valid for the first time down to microscales and for possibly verysmall or very large temperatures.  We show that a similar CLT can also be obtained in any larger dimensionconditional on a "no phase-transition" assumption, as soon as one can obtain aprecise enough error rate for the expansion of the free energy -- an expansionis obtained in any dimension, but the obtained rate is so far not good enoughto 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 largerscale than the temperature-dependent minimal scale $\rho_\beta$ introduced inour previous work \cite{as}.

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