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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn075, 66 pages, doi:10.1093/imrn/rnn075 published on July 21, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

The Formula Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights

K. T.-R. McLaughlin1 and P. D. Miller2

1 Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
2 Department of Mathematics, University of Michigan, East Hall, 530 Church Street, Ann Arbor, MI 48109-1109, USA

Correspondence: Correspondence to be sent to: mcl{at}math.arizona.edu

We obtain Plancherel–Rotach-type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form Formula on the real line, assuming that V has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application, we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. We develop a new technique of asymptotic analysis for matrix Riemann–Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.



References

  1. Baik J., Kriecherbauer T., McLaughlin K. T.-R., Miller P. D. Discrete Orthogonal Polynomials. Asymptotics and Applications (2007) Princeton: Princeton University Press. Annals of Mathematics Studies 164.
  2. Bleher P., Its A. Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and the universality in the matrix model. In: Annals of Mathematics (1999) 50:185–266.
  3. Deift P. Integrable Operators. In: Differential Operators and Spectral Theory: M. Sh. Birman’s 70th Anniversary Collection—Buslaev V., Solomyak M., Yafaev D., eds. (1999) Providence, RI: American Mathematical Society. American Mathematical Society Translation Series 2, 159.
  4. Deift P. Some open problems in random matrix theory and the theory of integrable systems. Contemporary Mathematics (2008) 458:419–30.
  5. Deift P., Gioev D. Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Communications on Pure and Applied Mathematics (2007) 60:867–910.[CrossRef][ISI]
  6. 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. Communications on Pure and Applied Mathematics (1999) 52:1335–1425.[CrossRef][ISI]
  7. Deift P., Kriecherbauer T., McLaughlin K. T.-R., Venakides S., Zhou X. Strong asymptotics of orthogonal polynomials with respect to exponential weights. Communications on Pure and Applied Mathematics (1999) 52:1491–1552.[CrossRef][ISI]
  8. Deift P., Venakides S., Zhou X. New results in small dispersion KdV by an extension of the steepest descent method for Riemann–Hilbert problems. International Mathematics Research Notices (1997) 6:285–99.
  9. Deift P., Zhou X. A steepest descent method for oscillatory Riemann–Hilbert problems: asymptotics for the mKdV equation. Annals of Mathematics (1993) 137:295–368.[CrossRef][ISI]
  10. Fokas A., Its A., Kitaev A. V. Discrete Painlevé equations and their appearance in quantum gravity. Communications in Mathematical Physics (1991) 142:313–44.[CrossRef][ISI]
  11. Kamvissis S., McLaughlin K. T.-R., Miller P. D. Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (2003) Princeton: Princeton University Press. Annals of Mathematics Studies 154.
  12. Kriecherbauer T., McLaughlin K. T.-R. Strong asymptotics of polynomials orthogonal with respect to Freud weights. International Mathematics Research Notices (1999) 1999:299–324.[Free Full Text]
  13. Kuijlaars A. B. J., McLaughlin K. T.-R., Van Assche W., Vanlessen M. The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on Formula . Advances in Mathematics (2004) 188:337–98.[CrossRef][ISI]
  14. Levin E., Lubinsky D. Universality limits in the bulk for varying measures. Advances in Mathematics. forthcoming.
  15. McLaughlin K. T.-R., Miller P. D. The Formula steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights. International Mathematics Research Papers (2006) 2006:1–77.
  16. Mehta M. L. Random Matrices (1991) 2nd ed. San Diego, CA: Academic Press.
  17. Mehta M. L., Gaudin M. On the density of eigenvalues of a random matrix. Nuclear Physics (1960) 18:420–27.[CrossRef][ISI]
  18. Miller P. D. Applied Asymptotic Analysis (2006) Providence, RI: American Mathematical Society. Graduate Studies in Mathematics 75.
  19. Pastur L., Shcherbina M. Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. Journal of Statistical Physics (1997) 86:109–47.[CrossRef][ISI]
  20. Pastur L., Shcherbina M. On the edge universality of the local eigenvalue statistics of matrix models. Matematicheskaya Fizika, Analiz, Geometriya (2003) 10:335–65.
  21. Pastur L., Shcherbina M. Bulk universality and related properties of Hermitian matrix models. Journal of Statistical Physics (2008) 130:205–50.[CrossRef][ISI]
  22. Plancherel M., Rotach W. Sur les valeurs asymptotiques des polynomes d’Hermite Formula . Commentarii Mathematici Helvetici (1929) 1:227–54.[CrossRef]
  23. Tracy C. A., Widom H. Level spacing distributions and the airy kernel. Communications in Mathematical Physics (1994) 159:151–74.[CrossRef][ISI]

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This Article
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Right arrow Articles by McLaughlin, K. T.-R.
Right arrow Articles by Miller, P. D.
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