Seminari del Dipartimento

Probabilita'

We consider a large non-Hermitian i.i.d. matrix X with real or complex entries, and we analyse its spectrum using Girko's formula. We will then show some applications of this approach proving: (i) edge universality (the non-Hermitian analogue of Tracy-Widom universality); (ii) Gaussian fluctuation of linear statistics of the eigenvalues for general test functions. The proof relies on three main ingredients: (i) local law for product of resolvents of the Hermitisation of X, (ii) coupling of several weakly dependent Dyson Brownian motion, (iii) rigorous supersymmetric analysis.