Optimal Local Law and Central Limit Theorem for $$\beta $$-Ensembles

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