Mesoscopic central limit theorem for non-Hermitian random matrices

We prove that the mesoscopic linear statistics Sigma(i)f (n(a)(sigma(i) - z(0))) of the eigenvalues {sigma(i)}(i) of large nxn non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H-0(2) -functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0 < a < 1/2. This extends our previous result (Cipolloni et al. in Commun Pure Appl Math, 2019. 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)| >> 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.