Comparison theorem for some extremal eigenvalue statistics

Understand

We introduce a method for the comparison of some extremal eigenvalue statistics of random matrices.

• For example, it allows one to compare the maximal eigenvalue gap in the bulk of two generalized Wigner ensembles, provided that the first four moments of their matrix entries match.
• As an application, we extend results of Bourgade--Ben Arous and Feng--Wei that identify the limit of the maximal eigenvalue gap in the bulk of the GUE to all complex Hermitian generalized Wigner matrices.