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

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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.

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  1. 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)

  2. 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)

  3. 3

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

  4. 4

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

  5. 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. 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. 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)

  8. 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)

  9. 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)

  10. 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. 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. 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)

  13. 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)

  14. 14

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

  15. 15

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

  16. 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. 17

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

  18. 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. 19

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

  20. 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. 21

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

  22. 22

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

  23. 23

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

  24. 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)

  25. 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)

  26. 26

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

  27. 27

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

  28. 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)

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Correspondence to M. Bertola.

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

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Bertola, M. Boutroux curves with external field: equilibrium measures without a variational problem. Anal.Math.Phys. 1, 167–211 (2011).

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