Optimal Local Law and Central Limit Theorem for β -Ensembles

In the setting of generic β-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of order (log N) / N in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale logN/N. (iii) The logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem. Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian β-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.