Mesoscopic central limit theorem for non-Hermitian random matrices

article Mesoscopic central limit theorem for non-Hermitian random matrices published yes Giorgio Cipolloni author 42198EFA-F248-11E8-B48F-1D18A9856A870000-0002-4901-7992 László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Dominik J Schröder author 408ED176-F248-11E8-B48F-1D18A9856A870000-0002-2904-1856 LaEr department Random matrices beyond Wigner-Dyson-Mehta project We prove that the mesoscopic linear statistics ∑if(na(σi−z0)) of the eigenvalues {σi}i of large n×n non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H20-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 z1,z2 with an improved error term in the entire mesoscopic regime |z1−z2|≫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. Springer Nature2024 eng Probability Theory and Related Fields 0178-8051 1432-2064 2210.12060 00111897250000110.1007/s00440-023-01229-1 1881131-1182 G. Cipolloni, L. Erdös, D.J. Schröder, Probability Theory and Related Fields 188 (2024) 1131–1182. Cipolloni, G., Erdös, L., & Schröder, D. J. (2024). Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields. Springer Nature. <a href="https://doi.org/10.1007/s00440-023-01229-1">https://doi.org/10.1007/s00440-023-01229-1</a> Cipolloni G, Erdös L, Schröder DJ. Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields. 2024;188:1131-1182. doi:<a href="https://doi.org/10.1007/s00440-023-01229-1">10.1007/s00440-023-01229-1</a> Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Mesoscopic Central Limit Theorem for Non-Hermitian Random Matrices.” Probability Theory and Related Fields. Springer Nature, 2024. <a href="https://doi.org/10.1007/s00440-023-01229-1">https://doi.org/10.1007/s00440-023-01229-1</a>. Cipolloni, Giorgio, et al. “Mesoscopic Central Limit Theorem for Non-Hermitian Random Matrices.” Probability Theory and Related Fields, vol. 188, Springer Nature, 2024, pp. 1131–82, doi:<a href="https://doi.org/10.1007/s00440-023-01229-1">10.1007/s00440-023-01229-1</a>. Cipolloni G, Erdös L, Schröder DJ. 2024. Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields. 188, 1131–1182. G. Cipolloni, L. Erdös, and D. J. Schröder, “Mesoscopic central limit theorem for non-Hermitian random matrices,” Probability Theory and Related Fields, vol. 188. Springer Nature, pp. 1131–1182, 2024. 144082023-10-08T22:01:17Z2025-08-05T13:28:15Z