Giorgio Cipolloni -- Correlated DBMs and fluctuations in the circular law
Channel: Matrices Et Graphes Aléatoires GDR CNRS
Date: 12/02/2021

Correlated Dyson Brownian motions and fluctuations in the circular law: CLT for 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.