2019/12/09 by Giorgio Cipolloni, László Erdős, Cipolloni, Giorgio +3 · 1 citation
Computer Science · Mathematics · #15B52 #60B20 #Advanced Algebra and Geometry #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1912.04100
openalex publication_date 2019/12/09 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We consider large non-Hermitian random matrices X with complex,\nindependent, identically distributed centred entries and show that the linear\nstatistics of their eigenvalues are asymptotically Gaussian for test functions\nhaving 2+\ε derivatives. Previously this result was known only for a\nfew special cases; either the test functions were required to be analytic\n[Rider, Silverstein 2006], or the distribution of the matrix elements needed to\nbe Gaussian [Rider, Vir 'ag 2007], or at least match the Gaussian up to the\nfirst four moments [Tao, Vu 2016; Kopel 2015]. We find the exact dependence of\nthe limiting variance on the fourth cumulant that was not known before. The\nproof relies on two novel ingredients: (i) a local law for a product of two\nresolvents of the Hermitisation of X with different spectral parameters and\n(ii) a coupling of several weakly dependent Dyson Brownian Motions. These\nmethods are also the key inputs for our analogous results on the linear\neigenvalue statistics of real matrices X that are presented in the companion\npaper [Cipolloni, Erd Hos, Schr "oder 2019].\n