Giorgio Cipolloni: Fluctuation around the circular law for non-Hermitian i.i.d. random matrices

Abstract

We consider a large non-Hermitian i.i.d. matrix X with real or complex entries and show that the linear statistics of the eigenvalues are asymptotically Gaussian for test function having \(2+\epsilon\) derivatives. Previously this result was known only for the Ginibre ensemble, where explicit formulas for the correlation functions are available, and ensembles close to Ginibre in the sense of moment matching; our result holds for general distribution of the matrix entries. The proof relies on two main novel ingredients: (i) local law for product of resolvents of the Hermitisation of X at two different spectral parameters, (ii) coupling of several dependent Dyson Brownian motions.