@article{10405,
  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]. },
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1097-0312},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {5},
  pages        = {946--1034},
  publisher    = {Wiley},
  title        = {{Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices}},
  doi          = {10.1002/cpa.22028},
  volume       = {76},
  year         = {2023},
}