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Dominik Schröder

Mesoscopic central limit theorem for non-Hermitian random matrices

Giorgio Cipolloni, László Erdős, Dominik Schröder

Probab. Theory Relat. Fields(2023)


We extend our previous result on the CLT for the linear statistics of IID random matrices to the entire mesoscopic regime.


We prove that the mesoscopic linear statistics if(na(σiz0))\sum_i f(n^a(\sigma_i-z_0)) of the eigenvalues {σi}i\{\sigma_i\}_i of large n×nn\times n non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H02H^{2}_0-functions ff around any point z0z_0 in the bulk of the spectrum on any mesoscopic scale 0<a<1/20<a<1/2. This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, a=0a=0, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of XX at spectral parameters z1,z2z_1, z_2 with an improved error term in the entire mesoscopic regime z1z2n1/2|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.