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

TL;DR

This paper establishes an optimal local law and central limit theorem for $eta$-ensembles, revealing precise fluctuation scales and rigidity properties, with new results for Gaussian $eta$-ensembles independent of comparison methods.

Contribution

It proves a local law with optimal error for $eta$-ensembles and derives new fluctuation results, including a CLT for the electric potential, without relying on comparison techniques.

Findings

Optimal rigidity scale of order (log N)/N in the bulk

Particles fluctuate on scale √(log N)/N

Logarithm of electric potential satisfies a CLT

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.