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 (CLT) for the fluctuations of linear statistics in dimension 2, valid for the first time down to microscales and for temperatures possibly tending to 0 or ∞ as the number of points diverges. 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 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 ρ_β introduced in our previous work (Ann. Probab. 49 (2021) 46–121).