The eigenvalues of i.i.d. matrices are hyperuniform

TL;DR

This paper proves that the eigenvalues of i.i.d. matrices form a hyperuniform point process, with significantly reduced variance in eigenvalue counts within subdomains, through precise covariance computations of resolvents.

Contribution

It introduces a novel method for analyzing eigenvalue distributions of i.i.d. matrices by computing resolvent covariances, establishing hyperuniformity.

Findings

Eigenvalues form a hyperuniform point process

Variance of eigenvalue counts is much smaller than domain volume

Precise covariance calculations underpin the main results

Abstract

We prove that the point process of the eigenvalues of real or complex non-Hermitian matrices $X$ with independent, identically distributed entries is hyperuniform: the variance of the number of eigenvalues in a subdomain $\Omega$ of the spectrum is much smaller than the volume of $\Omega$. Our main technical novelty is a very precise computation of the covariance between the resolvents of the Hermitization of $X-z_1, X-z_2$, for two distinct complex parameters $z_1,z_2$.