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

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 $\hat{I}\text{c◯}$ of the spectrum is much smaller than the volume of $\hat{I}\text{c◯}$. 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 $z_1,z_2$.

58 pages + 41 page of supplemental materials, 1 figure. v1->v2: references are updated