Fluctuation around the circular law for random matrices with real entries

@article{Cipolloni2020FluctuationAT,
  title={Fluctuation around the circular law for random matrices with real entries},
  author={Giorgio Cipolloni and L'aszl'o ErdHos and Dominik Schroder},
  journal={arXiv: Probability},
  year={2020}
}
We extend our recent result [Cipolloni, Erdős, Schroder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices $X$ with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our… Expand

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