Central Limit Theorem for Linear Eigenvalue Statistics of non-Hermitian Random Matrices
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Date
2023-05
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Journal Article
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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 urn:x-wiley:00103640:media:cpa22028:cpa22028-math-0001 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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76 (5)
Pages / Article No.
946 - 1034
Publisher
Wiley
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02889 - ETH Institut für Theoretische Studien / ETH Institute for Theoretical Studies