Universality phenomena for non-Hermitian random matrices

Giorgio Cipolloni (Princeton)
Fine Hall 401

We consider random matrices X with i.i.d. entries, and we analyse its spectrum using Girko's formula. First, we show that the local spectral statistics at the edge is universal (the non Hermitian analogue of the celebrated Tracy-Widom law). Second, we prove the Gaussian fluctuation of linear eigenvalue statistics for generic test functions. The proofs rely on a combination of three ingredients: (i) weakly correlated Dyson Brownian motions, (ii) multi resolvent local laws for the Hermitisation of X, (iii) rigorous supersymmetric (SUSY) approach.