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

Time: Tue 2020-11-03 15.15

Lecturer: Giorgio Cipolloni, IST Austria

Location:

### 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.

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