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

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## 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 \(r=\mathcal{O}(n^{\gamma})\), 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.

## Keywords

Random matrices Universality Asymptotics External sources Eigenvalues Riemann-Hilbert problems## Notes

### Acknowledgements

The authors thank Jinho Baik, Ken McLaughlin, Sandrine Péché, and Dong Wang for several illuminating discussions. We thank Baik and Wang for sharing their unpublished results. We also thank the anonymous referees for their helpful comments. M. Bertola was supported by NSERC. R. Buckingham was supported by NSF grant DMS-1312458, Simons Foundation Collaboration Grant for Mathematicians #245775, and the Charles Phelps Taft Research Center. V. Pierce was supported by NSF grant DMS-0806219.

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