Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes

Abstract

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and sample covariance matrices \cite{Baik:2005}. We consider the case when the $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. The source is small in the sense that $r$ is finite or $r=\mathcal O(n^\gamma)$, for $0< \gamma<1$. For a Gaussian potential, P\'ech\'e \cite{Peche:2006} showed that for $|a|$ sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for $|a|$ sufficiently large (the supercritical regime) $r$ eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of $r\times r$ Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.