Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime
We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert--Schmidt regime, we prove $N$-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman--Fredholm determinant of the centered interaction operator. We show that the Hilbert--Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two $U$-statistics. As consequences, we establish entropic commutator estimates with the sharp $O(N^{-1})$ additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert--Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.