Skip to main content
SpringerLink
Log in

Boutroux curves with external field: equilibrium measures without a variational problem

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

The nonlinear steepest descent method for rank-two systems relies on the notion of g-function. The applicability of the method ranges from orthogonal polynomials (and generalizations) to Painlevé transcendents, and integrable wave equations (KdV, NonLinear Schrödinger, etc.). For the case of asymptotics of generalized orthogonal polynomials with respect to varying complex weights we can recast the requirements for the Cauchy-transform of the equilibrium measure into a problem of algebraic geometry and harmonic analysis and completely solve the existence and uniqueness issue without relying on the minimization of a functional. This addresses and solves also the issue of the “free boundary problem", determining implicitly the curves where the zeroes of the orthogonal polynomials accumulate in the limit of large degrees and the support of the measure. The relevance to the quasi-linear Stokes phenomenon for Painlevé equations is indicated. A numerical algorithm to find these curves in some cases is also explained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bertola M., Mo M.Y.: Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights. Adv. Math. 220(1), 154–218 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertola M., Eynard B., Harnad J.: Partition functions for matrix models and isomonodromic tau functions. Random matrix theory. J. Phys. A 36(12), 3067–3083 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertola M., Eynard B., Harnad J.: Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions. Commun. Math. Phys 263(2), 401–437 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertola M., Gekhtman M.: Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions. Constr. Approx. 26(3), 383–430 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bertola, M., Tovbis, A.: Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painlevé equation. arXiv:1108.0321

  6. Bertola, M., Tovbis, A.: Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the tritronquee solution to Painlevé I. arXiv:1004.1828

  7. Boutroux P.: Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre (French). Ann. Sci. École Norm. Sup. (3) 30, 255–375 (1913)

    MathSciNet  Google Scholar 

  8. Boutroux P.: Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre (suite) (French). Ann. Sci. École Norm. Sup. (3) 31, 99–159 (1914)

    MathSciNet  Google Scholar 

  9. Bleher P., Its A.R.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. (2) 150(1), 185–266 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bleher, P., Its, A.R.: On asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert method. In: Symmetries and Integrability of Difference Equations (Canterbury, 1996), pp. 165–177. London Math. Soc. Lecture Note Ser., vol. 255

  11. 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; American Mathematical Society, Providence (1999)

  12. Deift P.A., Kriecherbauer T., McLaughlin K.T., Venakides S., Zhou X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52(12), 1491–1552 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Duits M., Kuijlaars A.: Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight. Nonlinearity 19, 2211–2245 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eynard, B.: Universal distribution of random matrix eigenvalues near the “birth of a cut” transition. J. Stat. Mech. P07005 (2006)

  16. Farkas, H.M., Kra, I.: Riemann surfaces. In: Graduate Texts in Mathematics, vol. 71. Springer, New York-Berlin. ISBN:0-387-90465-4 (1980)

  17. Fay, J.: Theta Functions on Riemann Surfaces. In: Lect. Notes in Math., vol. 352. Springer, Berlin (1973)

  18. Fokas, A.S., Its, A.R., Kapaev, A.A., Yu V.: Novokshenov, Painlevé trascendents. The Riemann–Hilbert approach. Math. Surveys and Monographs, vol. 128. AMS (2006)

  19. Fokas A., Its A., Kitaev A.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147, 395–430 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Its, A.R., Kapaev, A.A.: The nonlinear steepest descent approach to the asymptotics of the second Painlevè transcendent in the complex domain. MathPhys Odyssey, pp. 273–311 (2001)

  21. Jenkins J.A., Spencer D.C.: Hyperelliptic trajectories. Ann. Math. 53(1), 4–35 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kamvissis S., Rakhmanov E.A.: Existence and regularity for an energy maximization problem in two dimensions. J. Math. Phys. 46(8), 083505 (2005)

    Article  MathSciNet  Google Scholar 

  23. Marcellán F., Rocha I.A.: Complex path integral representation for semiclassical linear functionals. J. Approx. Theory 94, 107–127 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. McLaughlin K.T.-R.: Asymptotic analysis of random matrices with external source and a family of algebraic curves. Nonlinearity 20(7), 1547–1571 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jimbo M., Miwa T., Ueno K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients I. Physica D 2, 306–352 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  26. Saff, E.B., Totik, V.: Logarithmic potentials with external fields. In: Comprehensive Studies in Mathematics, vol. 316. Springer, Berlin (1997)

  27. Strebel K.: Quadratic Differentials. Modern Surveys in Mathematics. Springer, Berlin (1984)

    Google Scholar 

  28. Tovbis A., Venakides S.: Nonlinear steepest descent asymptotics for semiclassical limit of integrable systems: continuation in the parameter space. Commun. Math. Phys. 295(1), 139–160 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montreal, QC, H3G 1M8, Canada

    M. Bertola

  2. Centre de recherches mathématiques, Université de Montréal, Montreal, Canada

    M. Bertola

Authors
  1. M. Bertola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Bertola.

Additional information

Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Electronic Supplementary Material

The Below are the Electronic Supplementary Material.

ESM 1 (AVI 972 kb)

ESM 2 (AVI 1688 kb)

ESM 3 (AVI 911 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bertola, M. Boutroux curves with external field: equilibrium measures without a variational problem. Anal.Math.Phys. 1, 167–211 (2011). https://doi.org/10.1007/s13324-011-0012-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-011-0012-3

Keywords

Search

Navigation