Constructive Approximation

, Volume 30, Issue 2, pp 225–263 | Cite as

First Colonization of a Spectral Outpost in Random Matrix Theory



We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N−2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature.

The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate Nγ, whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.


Orthogonal polynomials Random matrix theory Schlesinger transformations Riemann–Hilbert problems 

Mathematics Subject Classification (2000)

05E35 15A52 


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  1. 1.
    Bertola, M.: Boutroux curves with external field: equilibrium measures without a minimization problem. arXiv:0705.1283, 2007
  2. 2.
    Bertola, M., Mo, M.Y.: Commuting difference operators, spinor bundles and the asymptotics of pseudo-orthogonal polynomials with respect to varying complex weights. ArXiv Math. Phys. e-prints, May 2006 Google Scholar
  3. 3.
    Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and universality in the matrix model. Ann. Math. (2) 150(1), 185–266 (1999) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Claeys, T.: The birth of a cut in unitary random matrix ensembles. ArXiv Math. Phys. e-prints, arXiv:0711.2609, 2007
  5. 5.
    Deift, P.A.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3. New York University Courant Institute of Mathematical Sciences, New York (1999) Google Scholar
  6. 6.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R.: New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95(3), 388–475 (1998) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Deift, P., Kriecherbauer, T., McLaughlin, T.K.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52(12), 1491–1552 (1999) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52(11), 1335–1425 (1999) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eynard, B.: Universal distribution of random matrix eigenvalues near the “birth of a cut” transition. J. Stat. Mech.: Theory Exp. 2006(07), P07005 (2006) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Farkas, H.M., Kra, I.: Riemann Surfaces, 2nd edn. Graduate Texts in Mathematics, vol. 71. Springer, New York (1992) MATHGoogle Scholar
  11. 11.
    Fay, J.D.: Theta Functions on Riemann Surfaces. Lecture Notes in Mathematics, vol. 352. Springer, Berlin (1973) MATHGoogle Scholar
  12. 12.
    Fokas, A.S., Its, A.R., Kitaev, A.V.: Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys. 142(2), 313–344 (1991) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147(2), 395–430 (1992) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Freud, G.: On polynomial approximation with the weight \(\exp \{-{1\over 2}x^{2k}\}\) . Acta Math. Acad. Sci. Hung. 24, 363–371 (1973) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Its, A.R., Kitaev, A.V., Fokas, A.S.: An isomonodromy approach to the theory of two-dimensional quantum gravity. Usp. Mat. Nauk 45(6)(276), 135–136 (1990) MathSciNetGoogle Scholar
  16. 16.
    Its, A.R., Kitaev, A.V., Fokas, A.S.: Matrix models of two-dimensional quantum gravity, and isomonodromic solutions of Painlevé “discrete equations”. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187 (Differentsialnaya Geom. Gruppy Li i Mekh. 12), 3–30, 171, 174 (1991) Google Scholar
  17. 17.
    Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory. Physica D 2(2), 306–352 (1981) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kuijlaars, A.B.J., McLaughlin, K.T.-R.: Asymptotic zero behavior of Laguerre polynomials with negative parameter. Constr. Approx. 20(4), 497–523 (2004) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mehta, M.L.: Random Matrices. 3rd edn., Pure and Applied Mathematics, vol. 142, Elsevier/Academic, Amsterdam (2004) MATHGoogle Scholar
  20. 20.
    Mo, M.Y.: The Riemann–Hilbert approach to double scaling limit of random matrix eigenvalues near the “birth of a cut” transition. arxiv:0711.3208, 2007
  21. 21.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer, Berlin (1997). Appendix B by Thomas Bloom MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Centre de recherches mathématiquesUniversité de MontréalMontréalCanada

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