Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices

Cipolloni G, Erdös L, Schröder DJ. 2021. Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. Communications on Pure and Applied Mathematics.


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Abstract
We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32].
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Date Published
2021-12-01
Journal Title
Communications on Pure and Applied Mathematics
Acknowledgement
L.E. would like to thank Nathanaël Berestycki and D.S.would like to thank Nina Holden for valuable discussions on the Gaussian freefield.G.C. and L.E. are partially supported by ERC Advanced Grant No. 338804.G.C. received funding from the European Union’s Horizon 2020 research and in-novation programme under the Marie Skłodowska-Curie Grant Agreement No.665385. D.S. is supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation.
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Cipolloni G, Erdös L, Schröder DJ. Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. Communications on Pure and Applied Mathematics. 2021. doi:10.1002/cpa.22028
Cipolloni, G., Erdös, L., & Schröder, D. J. (2021). Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. Communications on Pure and Applied Mathematics. Wiley. https://doi.org/10.1002/cpa.22028
Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Central Limit Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” Communications on Pure and Applied Mathematics. Wiley, 2021. https://doi.org/10.1002/cpa.22028.
G. Cipolloni, L. Erdös, and D. J. Schröder, “Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices,” Communications on Pure and Applied Mathematics. Wiley, 2021.
Cipolloni G, Erdös L, Schröder DJ. 2021. Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. Communications on Pure and Applied Mathematics.
Cipolloni, Giorgio, et al. “Central Limit Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” Communications on Pure and Applied Mathematics, Wiley, 2021, doi:10.1002/cpa.22028.
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