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Spectra of random Hermitian matrices with a small-rank external source: the supercritical and subcritical regimes. (English) Zbl 1302.82049
Summary: 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 \(n\times n\) 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, S. Péché [Probab. Theory Relat. Fields 134, No. 1, 127–173 (2006; Zbl 1088.15025)] 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 \(r\times r\) Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
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