Multilevel limits of spiked random matrix minors

Abstract

We consider the largest eigenvalues of minors of the classical Gaussian random matrices with a finite rank spike in the critical BBP regime. We show that they converge to a multi-level interlacing particle system whose lowest level is the Airy${}_{\hat{I}}^2$ point process. Our construction and scaling limits extend in a natural way to non-classical ${\hat{I}}^2>0$.