Eigenvalues for the Minors of Wigner Matrices

Abstract

The eigenvalues for the minors of real symmetric ($\beta=1$) and complex Hermitian ($\beta=2$) Wigner matrices form the Wigner corner process, which is a multilevel interlacing particle system. In this paper, we study the microscopic scaling limit of the Wigner corner process both near the spectral edge and in the bulk, and prove they are universal. We show: (i) Near the spectral edge, the corner process exhibit a decoupling phenomenon, as first observed in [24]. Individual extreme particles have Tracy-Widom$_{\beta}$ distribution; the spacings between the extremal particles on adjacent levels converge to independent Gamma distributions in a much smaller scale. (ii) In the bulk, the microscopic scaling limit of the Wigner corner process is given by the bead process for general Sine$_\beta$ process, as constructed recently in [34].