Journal of Statistical Physics

, Volume 153, Issue 4, pp 654–697 | Cite as

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

Article

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 nr. 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 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • M. Bertola
    • 1
    • 2
  • R. Buckingham
    • 3
  • S. Y. Lee
    • 4
  • V. Pierce
    • 5
  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada
  2. 2.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada
  3. 3.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  4. 4.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA
  5. 5.Department of MathematicsUniversity of Texas—Pan AmericanEdinburgUSA

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