Workshop: Extrema of non-Hermitian eigenvalue statistics

Speaker
Giorgio Cipolloni, University of Arizona

Abstract
In a seminal paper in 2006 Rider and Virag proved that the fluctuations of the eigenvalues of Gaussian non-Hermitian matrices converge to a two-dimensional log-correlated field. This convergence was later extended to general real and complex i.i.d. matrices. We now compute the leading order asymptotic of the maximum of this field. Interestingly, we discover a new connection between real matrices and inhomogeneous branching random walks. A fundamental input for this proof are recent decorrelation estimates, which we also use to show that the spectral radius of large non–Hermitian matrices is universally Gumbel distributed, verifying a conjecture by Bordenave and Chafai.