Communications in Mathematical Physics

, Volume 263, Issue 2, pp 401–437 | Cite as

Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions



The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, M., van Moerbeke, P.: Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems. Commun. Math. Phys. 207, 589–620 (1999)CrossRefADSMATHGoogle Scholar
  2. 2.
    Adler, M., van Moerbeke, P.: Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum. Ann. of Math. 153(2), 149–189 (2001)MathSciNetGoogle Scholar
  3. 3.
    Bertola, M.: Bilinear semi–classical moment functionals and their integral representation. J. App. Theory 121, 71–99 (2003)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bertola, M., Eynard, B., Harnad, J.: Partition functions for matrix models and isomonodromic tau functions. J. Phys. A: Math. Gen. 36, 3067–3083 (2003)CrossRefADSMATHMathSciNetGoogle Scholar
  5. 5.
    Bertola, M., Harnad, J., Hurtubise, J., Pusztai, G.: R-matrix approach to the general rational isomonodromic deformation equations.Google Scholar
  6. 6.
    Borodin, A., Soshnikov, A.: Janossy densities I. Determinantal ensembles. J. Stat. Phys 113, 611–622 (2003)MathSciNetGoogle Scholar
  7. 7.
    Chihara, T.S.: An introduction to orthogonal polynomials. Mathematics and its Applications, Vol. 13, New York-London-Paris: Gordon and Breach Science Publishers, (1978)Google Scholar
  8. 8.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.R., Venakides, S., Zhou, Z.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52, 1491–1552 (1999)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D gravity and random matrices. Phys. Rep. 254, 1–131 (1995).CrossRefADSGoogle Scholar
  10. 10.
    Its, A.R., Kitaev, A.V., Fokas, A.S.: An isomonodromic Approach in the Theory of Two-Dimensional Quantum Gravity. Usp. Matem. Nauk 45, 6 (276), 135–136 (1990), (Russian), translation in Russ. Math.Surv. 45(6), 155–157 (1990)Google Scholar
  11. 11.
    Jimbo, M., Miwa, T., Ueno, K.: Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients I. Physica 2D, 306–352 (1981)MathSciNetGoogle Scholar
  12. 12.
    Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica 2D, 407–448 (1981)ADSMathSciNetGoogle Scholar
  13. 13.
    Ismail, M.E.H., Chen, Y.: Ladder operators and differential equations for orthogonal polynomials. J. Phys. A. 30, 7818–7829 (1997)Google Scholar
  14. 14.
    Marcellán, F., Rocha, I. A. Complex Path Integral Representation for Semiclassical Linear Functionals. J. Appr. Theory 94, 107–127 (1998)Google Scholar
  15. 15.
    Marcellán, F., Rocha, I.A.: On semiclassical linear functionals: Integral representations. J. Comput. Appl. Math. 57, 239–249 (1995)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Miller, K.S., Shapiro, H.S.: On the Linear Independence of Laplace Integral Solutions of Certain Differential Equations. Comm. Pure Appl. Math. 14, 125–135 (1961)MATHMathSciNetGoogle Scholar
  17. 17.
    Tracy, C.A., Widom, H.: Correlation functions,cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92, 809–835 (1998)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    van Moerbeke, P.: Integrable lattices: random matrices and random permutations. In: Random matrix models and their applications , Math. Sci. Res. Inst. Publ. 40, Cambridge: Cambridge Univ. Press, pp. 321–406 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Centre de recherches mathématiquesUniversité de MontréalMontréalCanada
  2. 2.Department of Mathematics and StatisticsConcordia UniversityMontréal
  3. 3.Service de Physique ThéoriqueGif-sur-Yvette CedexFrance

Personalised recommendations