Sharp total variation rates of convergence for fluctuations of linear statistics of $\beta$-ensembles

In this article, we revisit the question of fluctuations of linear statistics of beta ensembles in the single cut and non-critical regime for general potentials $V$ under mild regularity and growth assumptions. Our main objective is to establish sharp quantitative Central Limit Theorems (CLT) for strong distances, such as the total variation distance, which to the best of our knowledge, is new for general potentials, even qualitatively. Namely, setting $\mu_V$ the equilibrium measure, for a test function $\xi \in \mathscr{C}^{14}$, we establish the convergence in total variation of $X_n=\sum_{i=1}^n \xi(\lambda_i)-n\langle \xi,\mu_V\rangle$ to an explicit Gaussian variable at the sharp speed $1/n$. Under the same assumptions, we also establish multivariate CLTs for vectors of linear statistics in $p-$Wasserstein distances for any $p\ge 1$, with the optimal rate $1/n$, a result which already in dimension one sharpens the speed of convergence established in the recent contribution [26] as well as the required regularity on the test functions. A second objective of this paper, in a more qualitative direction, is to establish the so-called super-convergence of linear statistics, that is to say the convergence of all derivatives of the densities of $X_n$ uniformly on $\mathbb{R}$, provided that $\xi\in\mathscr{C}^\infty(\mathbb{R})$ and is not too degenerated in some sense.