Journal of Statistical Physics

, Volume 146, Issue 3, pp 475–518 | Cite as

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

Article

Abstract

Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the energy levels of the physical system. 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. For a Gaussian potential, it was shown by Péché (Probab. Theory Relat. Fields 134:127–173, 2006) that when r is fixed or grows sufficiently slowly with n (a small-rank source), r eigenvalues are expected to exit the main bulk for |a| large enough. Furthermore, at the critical value of a when the outliers are at the edge of a band, the eigenvalues at the edge are described by the r-Airy kernel. We establish the universality of the r-Airy kernel for a general class of analytic potentials for \(r=\mathcal{O}(n^{\gamma})\) for 0≤γ<1/12.

Keywords

Riemann-Hilbert problem Asymptotic analysis Nonlinear steepest descent analysis r-Airy kernel Critical phenomena 

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Copyright information

© Springer Science+Business Media, LLC 2011

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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