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

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 $\mathrm{\Omega }$ of the spectrum is much smaller than the volume of $\mathrm{\Omega }$. 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 $z_1,z_2$.