Optimal local law and central limit theorem for $\beta$-ensembles

Abstract

In the setting of generic $\beta$-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 $\sqrt{\log N}/N$. (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 $\beta$-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.

arXiv comment47 pages, to appear in Comm. Math. Phys

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• Edge Statistics for Lozenge Tilings of Polygons, I: Concentration of Height Function on Strip Domains
• Multi-Point Functional Central Limit Theorem for Wigner Matrices
• A convergence framework for Airy${}_{\beta }$ line ensemble via pole evolution