Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 43–76 | Cite as

Large n Limit of Gaussian Random Matrices with External Source, Part I

  • Pavel BleherEmail author
  • Arno B. J. Kuijlaars


We consider the random matrix ensemble with an external source

Open image in new window

defined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a>1, we establish the universal behavior of local eigenvalue correlations in the limit n→∞, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3×3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.


Sine Matrix Model External Source Quantum Computing Steep Descent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aptekarev, A.I.: Multiple orthogonal polynomials. J. Comput. Appl. Math. 99, 423–447 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aptekarev, A.I., Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source part II. math-ph/0408041Google Scholar
  3. 3.
    Aptekarev, A.I., Branquinho, A., Van Assche, W.: Multiple orthogonal polynomials for classical weights. Trans. Amer. Math. Soc. 355, 3887–3914 (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Aptekarev, A.I., Stahl, H.: Asymptotics of Hermite-Padé polynomials. In: Progress in Approximation Theory. A.A. Gonchar, E.B. Saff (eds), New York: Springer-Verlag, 1992, pp. 27–167Google Scholar
  5. 5.
    Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and the universality in the matrix model. Ann. Math. 150, 185–266 (1999)zbMATHGoogle Scholar
  6. 6.
    Bleher, P., Its, A.: Double scaling limit in the random matrix model. The Riemann-Hilbert approach. Commun. Pure Appl. Math. 56, 433–516 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bleher, P.M., Kuijlaars, A.B.J.: Random matrices with external source and multiple orthogonal polynomials. Int. Math. Research Notices 2004(3), 109–129 (2004)CrossRefGoogle Scholar
  8. 8.
    Bleher, P.M., Kuijlaars, A.B.J.: Integral representations for multiple Hermite and multiple Laguerre polynomials. math.CA/0406616Google Scholar
  9. 9.
    Brézin, E., Hikami, S.: Spectral form factor in a random matrix theory. Phys. Rev. E 55, 4067–4083 (1997)CrossRefGoogle Scholar
  10. 10.
    Brézin, E., Hikami, S.: Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479, 697–706 (1996)CrossRefGoogle Scholar
  11. 11.
    Brézin, E., Hikami, S.: Extension of level spacing universality. Phys. Rev. E 56, 264–269 (1997)CrossRefGoogle Scholar
  12. 12.
    Brézin, E., Hikami, S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E 57, 4140–4149 (1998)CrossRefGoogle Scholar
  13. 13.
    Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phys. Rev. E 58, 7176–7185 (1998)CrossRefGoogle Scholar
  14. 14.
    Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, Vol. 3, Providence R.I.: Amer. Math. Soc., 1999Google Scholar
  15. 15.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics of polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335–1425 (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math 52, 1491–1552 (1999)CrossRefzbMATHGoogle Scholar
  17. 17.
    Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295–368 (1993)zbMATHGoogle Scholar
  18. 18.
    Dyson, F.J.: Correlations between eigenvalues of a random matrix. Commun. Math. Phys 19, 235–250 (1970)zbMATHGoogle Scholar
  19. 19.
    Gonchar, A.A., Rakhmanov, E.A.: On the convergence of simultaneous Padé approximants for systems of functions of Markov type (Russian). Trudy Mat. Inst. Steklov 157, 31–48 (1981); English transl. in Proc. Steklov Math. Inst. 3, 31–50 (1983)zbMATHGoogle Scholar
  20. 20.
    Kuijlaars, A.B.J.: Riemann-Hilbert analysis for orthogonal polynomials. In: Orthogonal Polynomials and Special Functions, E. Koelink, W. Van Assche, (eds), Lecture Notes in Mathematics, Vol. 1817, Berlin-Heidelberg-New York: Springer-Verlag, 2003, pp. 167–210Google Scholar
  21. 21.
    Kuijlaars, A.B.J., McLaughlin, K.T.-R.: Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields. Commun. Pure Appl. Math. 53, 736–785 (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kuijlaars, A.B.J., Van Assche, W., Wielonsky, F.: Quadratic Hermite-Padé approximation to the exponential function: a Riemann-Hilbert approach. math.CA/0302357, to appear in Constr. ApproxGoogle Scholar
  23. 23.
    Mehta, M.L.: Random Matrices. 2nd edition, Boston: Academic Press, 1991Google Scholar
  24. 24.
    Nikishin, E.M., Sorokin, V.N.: Rational Approximations and Orthogonality. Transl Math. Monogr.: Providence R.I. 92, Amer. Math. Soc., 1991Google Scholar
  25. 25.
    Nuttall, J.: Asymptotics of diagonal Hermite-Padé polynomials. J. Approx. Theory 42, 299–386 (1984)zbMATHGoogle Scholar
  26. 26.
    Pastur, L.A.: The spectrum of random matrices (Russian). Teoret. Mat. Fiz. 10, 102–112 (1972)Google Scholar
  27. 27.
    Van Assche, W., Coussement, E.: Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127, 317–347 (2001)CrossRefzbMATHGoogle Scholar
  28. 28.
    Van Assche, W., Geronimo, J., Kuijlaars, A.B.J.: Riemann-Hilbert problems for multiple orthogonal polynomials. In: Special Functions 2000: Current Perspectives and Future Directions, J. Bustoz, et al. (eds), Dordrecht, Kluwer, 2001, pp. 23–59Google Scholar
  29. 29.
    Zinn-Justin, P.: Random Hermitian matrices in an external field. Nucl. Phys. B 497, 725–732 (1997)CrossRefzbMATHGoogle Scholar
  30. 30.
    Zinn-Justin, P.: Universality of correlation functions of Hermitian random matrices in an external field. Commun. Math. Phys. 194, 631–650 (1998)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndiana University-Purdue University IndianapolisIndianapolisU.S.A
  2. 2.Department of MathematicsKatholieke Universiteit LeuvenLeuvenBelgium

Personalised recommendations