Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Supercritical and Subcritical Regimes

Abstract

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers and sample covariance matrices. We consider the case when the n×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 = O(n^γ), for 0<γ<1. For a Gaussian potential, Péché (Probab. Theory Relat. Fields 134:127–173, 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 the r×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.

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