Benjamin Fahs: Maximal scaled spacings of eigenvalues of large Hermitian random matrices

Abstract

The spacings between consecutive eigenvalues of large Hermitian random matrices are a classical object of study. When scaled under the equilibrium measure, it is a classical theorem that the typical behaviour is governed by sine-kernel universality, for large classes of random matrices. More recently, progress has been made on the distribution of the maximal scaled spacing in the setting of the CUE and GUE by Arous/Bourgade and Feng/Wei (and results have been extended to other settings by numerous authors). We continue the study of maximal spacings, however we scale our spacings under the equilibrium measure which places the scaled spacings of the GUE and the CUE in the same universality class, which is new. More generally we consider one-cut regular Hermitian random matrices.