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(nγ), 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.

Spectra of random Hermitian matrices with a small-rank external source: the supercritical and subcritical regimes / Bertola, M.; Buckingham, R.; Lee, S. Y.; Pierce, V.. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 153:4(2013), pp. 654-697. [10.1007/s10955-013-0845-2]

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

Bertola, M.;
2013-01-01

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(nγ), 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.
2013
153
4
654
697
http://dx.doi.org/10.1007/s10955-013-0845-2
https://arxiv.org/abs/1009.3894
https://link.springer.com/article/10.1007%2Fs10955-013-0845-2
Bertola, M.; Buckingham, R.; Lee, S. Y.; Pierce, V.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/17353
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