Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices

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.