Constructive Approximation

, Volume 39, Issue 1, pp 173–196 | Cite as

Painlevé Kernels in Hermitian Matrix Models

  • Maurice Duits


After reviewing the Hermitian one-matrix model, we will give a brief introduction to the Hermitian two-matrix model and present a summary of some recent results on the asymptotic behavior of the two-matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a 4×4 Riemann–Hilbert problem related to the Painlevé II equation. Also an open problem will be presented.


Hermitian matrix models Eigenvalue distribution Correlation kernel Critical phenomena Painlevé transcendents Biorthogonal polynomials Riemann–Hilbert problems 

Mathematics Subject Classification

30E25 60B20 82B26 15B52 30F10 31A05 42C05 



Research supported by the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation.


  1. 1.
    Adler, M., Ferrari, P., van Moerbeke, P.: Non-intersecting random walks in the neighborhood of a symmetric tacnode. Ann. Probab. (to appear). arXiv:1007.1163
  2. 2.
    Aptekarev, A., Bleher, P., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, part II. Commun. Math. Phys. 259(2), 367–389 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bertola, M., Eynard, B.: The PDEs of biorthogonal polynomials arising in the two-matrix model. Math. Phys. Anal. Geom. 9, 162–212 (2006) MathSciNetGoogle Scholar
  4. 4.
    Bertola, M., Lee, S.Y.: First colonization of a spectral outpost in random matrix theory. Constr. Approx. 30, 225–263 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  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., Eynard, B., Harnad, J.: Duality, biorthogonal polynomials and multi-matrix models. Commun. Math. Phys. 229, 73–120 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bertola, M., Eynard, B., Harnad, J.: Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann–Hilbert problem. Commun. Math. Phys. 243, 193–240 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bleher, P., Its, A.: Double scaling limit in the random matrix model: the Riemann–Hilbert approach. Commun. Pure Appl. Math. 56, 433–516 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bleher, P., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, part III: double scaling limit. Commun. Math. Phys. 270, 481–517 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Borodin, A.: Determinantal point processes. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford Handbook on Random Matrix Theory. Oxford University Press, Oxford (2011). arXiv:0911.1153 Google Scholar
  11. 11.
    Brézin, E., Hikami, S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E. 57(4), 7176–7185 (1998) CrossRefGoogle Scholar
  12. 12.
    Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phys. Rev. E. 58(6), 4140–4149 (1998) CrossRefGoogle Scholar
  13. 13.
    Claeys, T.: Birth of a cut in unitary random matrix ensembles. Int. Math. Res. Not. 2008, rnm166 (2008). 40 pp. MathSciNetGoogle Scholar
  14. 14.
    Claeys, T., Kuijlaars, A.B.J.: Universality of the double scaling limit in random matrix models. Commun. Pure Appl. Math. 59, 1573–1603 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Claeys, T., Vanlessen, M.: Universality of a double scaling limit near singular edge points in random matrix models. Commun. Math. Phys. 273, 499–532 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Claeys, T., Its, A., Krasovsky, I.: Higher order analogues of the Tracy-Widom distribution and the Painlevé II hierarchy. Commun. Pure Appl. Math. 63, 362–412 (2010) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Daul, J.M., Kazakov, V., Kostov, I.K.: Rational theories of 2D gravity from the two-matrix model. Nucl. Phys. B 409, 311–338 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3. Amer. Math. Soc., Providence (1999) Google Scholar
  19. 19.
    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. Commun. Pure Appl. Math. 52, 1335–1425 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics for polynomials orthogonal with respect to varying exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Delvaux, S., Kuijlaars, A.B.J., Zhang, L.: Critical behavior of non-intersecting Brownian motions at a tacnode. Commun. Pure Appl. Math. 64, 1305–1383 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Duits, M., Geudens, D.: A critical phenomenon in the two matrix model in the quartic/quadratic case. Duke Math. J. (to appear) Google Scholar
  23. 23.
    Duits, M., Kuijilaars, A.B.J.: Painlevé I asymptotic for orthogonal polynomials with respect to a varying quartic weight. Nonlinearity 19, 2211–2245 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Duits, M., Kuijlaars, A.B.J.: Universality in the two-matrix model: a Riemann–Hilbert steepest descent analysis. Commun. Pure Appl. Math. 62, 1076–1153 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Duits, M., Geudens, D., Kuijlaars, A.B.J.: A vector equilibrium problem for the two-matrix model in the quartic/quadratic case. Nonlinearity 24(3), 951–993 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Duits, M., Kuijlaars, A.B.J., Mo, M.Y.: The Hermitian two-matrix model with an even quartic potential. Mem. Am. Math. Soc. 217(1022), 105 (2012) MathSciNetGoogle Scholar
  27. 27.
    Duits, M., Kuijlaars, A.B.J., Mo, M.Y.: Asymptotic analysis of the two matrix model with a quartic potential. arXiv:1210.0097
  28. 28.
    Ercolani, N.M., McLaughlin, K.T.-R.: Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model. Physica D 152/153, 232–268 (2001) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Eynard, B.: Large-N expansion of the 2 matrix model. J. High Energy Phys. 1, 051 (2003), 38 p. CrossRefMathSciNetGoogle Scholar
  30. 30.
    Eynard, B., Mehta, M.L.: Matrices coupled in a chain: eigenvalue correlations. J. Phys. A 31, 4449–4456 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Flaschka, H., Newell, A.C.: Monodromy and spectrum-preserving deformations I. Commun. Math. Phys. 76, 65–116 (1980) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147, 395–430 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Fokas, A.S., Its, A.R., Kapaev, A.A., Novokshenov, V.Y.: Painlevé Transcendents: A Riemann–Hilbert Approach. Mathematical Surveys and Monographs, vol. 128. Amer. Math. Soc., Providence (2006) CrossRefGoogle Scholar
  34. 34.
    Geudens, D., Zhang, L.: Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel. arXiv:1208.0762
  35. 35.
    Guionnet, A.: First order asymptotics of matrix integrals; a rigorous approach towards the understanding of matrix models. Commun. Math. Phys. 244, 527–569 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Hardy, A., Kuijlaars, A.B.J.: Weakly admissible vector equilibrium problems. J. Approx. Theory 164, 854–868 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Hastings, S.P., McLeod, J.B.: A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Ration. Mech. Anal. 73, 31–51 (1980) CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Johansson, K.: Random Matrices and Determinantal Processes. Mathematical Statistical Physics, pp. 1–55. Elsevier, Amsterdam (2006) Google Scholar
  41. 41.
    Johansson, K.: Non-colliding Brownian Motions and the extended tacnode process. Commun. Math. Phys. (to appear) Google Scholar
  42. 42.
    Kapaev, A.A.: Riemann–Hilbert problem for bi-orthogonal polynomials. J. Phys. A 36, 4629–4640 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    König, W.: Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2, 385–447 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Kuijlaars, A.B.J.: Universality. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford Handbook on Random Matrix Theory. Oxford University Press, Oxford (2011) Google Scholar
  45. 45.
    Kuijlaars, A.B.J., McLaughlin, K.T.-R.: A Riemann–Hilbert problem for biorthogonal polynomials. J. Comput. Appl. Math. 178, 313–320 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Lubinsky, D.S.: Some recent methods for establishing universality limits. Nonlinear Anal. 71, e2750–e2765 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Lyons, R.: Determinantal probability measures. Publ. Math. IHÉS 98, 167–212 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    McLaughlin, K.T.-R., Miller, P.D.: The \(\bar{\partial}\) steepest descent method for orthogonal polynomials on the real line with vary in weights. Int. Math. Res. Not. 2008, rnn075 (2008), 66 pp. MathSciNetGoogle Scholar
  49. 49.
    Mo, M.Y.: The Riemann–Hilbert approach to double scaling limit of random matrix eigenvalues near the “birth of a cut” transition. Int. Math. Res. Not. 2008, rnm042 (2008), 51 pp. Google Scholar
  50. 50.
    Mo, M.Y.: Universality in the two matrix model with a monomial quartic and a general even polynomial potential. Commun. Math. Phys. 291, 863–894 (2009) CrossRefzbMATHGoogle Scholar
  51. 51.
    Okounkov, A., Reshetikhin, N.: Random skew plane partitions and the Pearcey process. Commun. Math. Phys. 269(3), 571–609 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Orantin, N.: Chain of matrices, loop equations, and topological recursion. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford Handbook on Random Matrix Theory. Oxford University Press, Oxford (2011) Google Scholar
  53. 53.
    Pastur, L., Shcherbina, M.: Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130, 205–250 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Field. Grundlehren der Mathematischen Wissenschaften, vol. 316. Springer, Berlin (1997) CrossRefGoogle Scholar
  55. 55.
    Shcherbina, M.: Double scaling limit for matrix models with non analytic potentials. J. Math. Phys. 49, 033401 (2008). 34 pp. MathSciNetGoogle Scholar
  56. 56.
    Soshnikov, A.: Determinantal random point fields. Usp. Mat. Nauk 55(5(335)), 107–160 (2000). Translation in Russ. Math. Surv. 55(5), 923–975 (2000) CrossRefMathSciNetGoogle Scholar
  57. 57.
    Tracy, C., Widom, H.: The Pearcey process. Commun. Math. Phys. 263, 381–400 (2006) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of Technology (KTH)StockholmSweden

Personalised recommendations