Random matrices: Universality of local spectral statistics of non-Hermitian matrices

TL;DR

This paper proves the universality of local spectral statistics for non-Hermitian random matrices, extending results from Gaussian ensembles to more general distributions with matching moments, and addresses spectral instability issues.

Contribution

It establishes universality of local spectral statistics for non-Hermitian matrices with independent entries matching Gaussian moments up to fourth order, using novel log-determinant techniques.

Findings

Universality of local spectral statistics for non-Hermitian matrices.

Extension of CLT for eigenvalue counts to broader ensembles.

Asymptotic count of real eigenvalues in real matrices.

Abstract

It is a result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on $\mathbb{C}$ with kernel $K_{\infty}(z,w):=\frac{1}{\pi}e^{-|z|^2/2-|w|^2/2+z\bar{w}}$ in the limit $n\to\infty$. We show that this asymptotic law is universal among all random $n\times n$ matrices $M_n$ whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles. These are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this, we work with the log-determinants $\log|\det(M_n-z_0)|$ rather than with the Stieltjes transform $\frac{1}{n}\operatorname {tr}(M_n-z_0)^{-1}$. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real $n\times n$ matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has $\sqrt{\frac{2n}{\pi}}+o(\sqrt{n})$ real eigenvalues asymptotically almost surely.