Workshop: Universality of extremal eigenvalues of large non-Hermitian random matrices

Speaker
Yuanyuan Xu, Chinese Academy of Sciences

Abstract
We will report recent progress on the universality of extremal eigenvalues of a large random matrix with i.i.d. entries. Beyond the radius of the celebrated circular law, we will establish a precise three-term asymptotic expansion for the largest eigenvalue (in modulus) with an optimal error term. Based on this result, we will further show that the properly normalized largest eigenvalue converges to a Gumbel distribution as the dimension goes to infinity. We also prove that the argument of the largest eigenvalue is uniform on the unit circle and that the extremal eigenvalues form a Poisson point process. Similar results also apply to the rightmost eigenvalues. Based on several joint works with Giorgio Cipolloni, Laszlo Erdos, and Dominik Schroder.