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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.
Reviewer: Reviewer (Berlin)

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