Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices

TL;DR

This paper establishes that the linear eigenvalue statistics of large non-Hermitian random matrices follow a Gaussian distribution on mesoscopic scales, extending previous macroscopic results with a new local law for resolvent products.

Contribution

It introduces a local law for resolvent products at mesoscopic scales, extending Gaussian fluctuation results from macroscopic to mesoscopic regimes for non-Hermitian matrices.

Findings

Eigenvalue statistics are asymptotically Gaussian on mesoscopic scales.

A new local law for resolvent products improves error estimates.

Extension of previous macroscopic results to mesoscopic regimes.

Abstract

We prove that the mesoscopic linear statistics $\sum_i f(n^a(\sigma_i-z_0))$ of the eigenvalues $\{\sigma_i\}_i$ of large $n\times n$ non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any $H^{2}_0$-functions $f$ around any point $z_0$ in the bulk of the spectrum on any mesoscopic scale $0<a<1/2$. This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, $a=0$, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of $X$ at spectral parameters $z_1, z_2$ with an improved error term in the entire mesoscopic regime $|z_1-z_2|\gg n^{-1/2}$. The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator.