Skip to main content
SpringerLink
Log in

Painlevé II in Random Matrix Theory and Related Fields

  • Published:
Constructive Approximation Aims and scope

Abstract

We review some occurrences of Painlevé II transcendents in the study of two-dimensional Yang–Mills theory and fluctuation formulas for growth models, and as distribution functions within random matrix theory. We first discuss settings in which the parameter \(\alpha \) in the Painlevé equation is zero, and the boundary condition is that of the Hasting–MacLeod solution. As well as expressions involving the Painlevé transcendent itself, one encounters the sigma form of the Painlevé II equation, and Lax pair equations in which the Painlevé transcendent occurs as coefficients. We then consider settings which give rise to general \(\alpha \) Painlevé II transcendents. In a particular random matrix setting, new results for the corresponding boundary conditions in the cases \(\alpha = \pm 1/2\), 1, and 2 are presented.

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. Ablowitz, M.J., Segur, H.: Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38, 1103–1106 (1977)

    Article  MathSciNet  Google Scholar 

  2. Adler, M., Shiota, T., van Moerbeke, P.: Random matrices, vertex operators and the Virasoro algebra. Phys. Lett. A 208, 67–78 (1995)

    Article  MathSciNet  Google Scholar 

  3. Akemann, G., Damgaard, P.H., Magnea, U., Nishigaki, S.M.: Multicritical microscopic spectral correlators of Hermitian and complex matrices. Nucl. Phys. B 519, 682–714 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12, 1119–1178 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Baik, J., Liechty, K., Schehr, G.: On the joint distribution of the maximum and its position of the Airy2 process minus a parabola. J. Math. Phys. 53, 083303 (2012)

  6. Baik, J., Rains, E.M.: Algebraic aspects of increasing subsequences. Duke Math. J. 109, 1–65 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Baik, J., Rains, E.M.: Symmetrized random permutations, Random matrix models and their applications. In: Bleher, P.M., Its, A.R. (eds.) Mathematical Sciences Research Institute Publications, vol. 40, pp. 171–208. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  8. Bleher, P.M.: Lectures on random matrix models, Random Matrices, random processes and integrable systems. In: Harnard, J. (ed.) CRM Series in Mathematical Physics, vol. 28, pp. 251–349. Springer, New York (2011)

    Google Scholar 

  9. Bleher, P.M., Its, A.: Double scaling limit in the random matrix model: the Riemann–Hilbert approach. Comm. Pure Appl. Math. 56, 433–516 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bloemendal, A., Virág, B.: Limits of spiked random matrices II (2011). arXiv:1109.3704

  11. Bohigas, O., Pato, M.P.: Missing levels in correlated spectra. Phys. Lett. B 595, 171–176 (2004)

    Article  Google Scholar 

  12. Borodin, A., Deift, P.: Fredholm determinants, Jimbo-Miwa-Ueno tau-functions and representation theory. Commun. Pur. Appl. Math. 55, 1160–1230 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Borodin, A., Forrester, P.J.: Increasing subsequences and the hard-to-soft transition in matrix ensembles. J. Phys. A 36, 2963–2981 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen, Y., Feigin, M.V.: Painlevé IV and degenerate Gaussian unitary ensembles. J. Phys. A 39, 12381–12393 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Claeys, T., Kuijlaars, A.B.J., Vanlessen, M.: Multi-critical unitary random matrix ensembles and the general Painlevé II equation. Ann. Math. 168, 601–641 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Clarkson, P. A.: Asymptotics of the second Painlevé transcendent, Special functions and orthogonal polynomials, Contemp. Math. vol. 471, Amer. Math. Soc., Providence, RI, 2008, pp. 69–83 (2008)

  17. Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 01, 1130001 (2012)

    Article  MathSciNet  Google Scholar 

  18. Corwin, I., Quastel, J., Remenik, D.: Continuum statistics of the Airy2 process. Commun. Math. Phys. 317, 347–362 (2013)

  19. Crescimanno, M., Naculich, S.G., Schnitzer, H.J.: Evaluation of the free energy of two-dimensional Yang–Mills theory. Phys. Rev. D 54, 1809–1813 (1993)

    Article  MathSciNet  Google Scholar 

  20. Crnković, C., Douglas, M., Moore, G.: Loop equations and topological phase of multi-cut matrix models. Int. J. Mod. Phys. A 7, 7693–7711 (1992)

    Article  MATH  Google Scholar 

  21. Douglas, M., Kazakov, V.: Large \(N\) phase transition in continuum \(QCD_2\). Phys. Lett. B 319, 219–230 (1993)

    Article  Google Scholar 

  22. Feierl, T.: The height of watermelons with wall. J. Phys. A 45, 095003 (2012)

    Article  MathSciNet  Google Scholar 

  23. Flaschka, H., Newell, A.C.: Monodromy and spectrum-preserving deformations. Comm. Math. Phys. 76, 65–116 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  24. Fokas, A.S., Its, A.R., Kapaev, A.A., Novokshenov, V.Y.: Painlevé Transcendents: The Riemann–Hilbert Approach. American Math. Society, Providence, RI (2006)

    Book  Google Scholar 

  25. Forrester, P.J.: The spectrum edge of random matrix ensembles. Nucl. Phys. B 402, 709–728 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  26. Forrester, P.J.: Evenness symmetry and inter-relationships between gap probabilities in random matrix theory. Forum Math. 18, 711–743 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton, NJ (2010)

    Book  MATH  Google Scholar 

  28. Forrester, P.J.: Probabilites densities and distributions for spiked and general variance Wishart \(\beta \)-ensembles. Random Matrices Theory Appl. 02, 1350011 (2013)

  29. Forrester, P.J.: Asymptotics of spacing distributions 50 years later (2012). arXiv:1204.3225

  30. Forrester, P.J., Majumdar, S.N., Schehr, G.: Non-intersecting Brownian walkers and Yang–Mills theory on the sphere. Nucl. Phys. B 844, 500–526 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. Forrester, P.J., Odlyzko, A.M.: Gaussian unitary ensemble eigenvalues and Riemann \(\zeta \) function zeros: a non-linear equation for a new statistic. Phys. Rev. E 54, R4493–R4495 (1996)

    Article  MathSciNet  Google Scholar 

  32. Forrester, P.J., Rains, E.M.: Inter-relationships between orthogonal, unitary and symplectic matrix ensembles, Random matrix models and their applications. In: Bleher, P.M., Its, A.R. (eds.) Mathematical Sciences Research Institute Publications, vol. 40, pp. 171–208. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  33. Forrester, P.J., Rains, E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Prob. Theory Related Fields 131, 1–61 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Forrester, P.J., Witte, N.S.: Application of the \(\tau \) -function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Commun. Math. Phys. 219, 357–398 (2000)

    Article  MathSciNet  Google Scholar 

  35. Forrester, P.J., Witte, N.S.: \(\tau \)-function evaluations of gap probabilities in orthogonal and symplectic matrix ensembles. Nonlinearity 15, 937–954 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Forrester, P.J., Witte, N.S.: Discrete Painlevé equations and random matrix averages. Nonlinearity 16, 1919–1944 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Fulton, W.: Young Tableaux. London Mathematical Society Student Texts, CUP, Cambridge (1997)

    MATH  Google Scholar 

  38. Gambier, B.: Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33, 1–55 (1909)

    Article  MATH  MathSciNet  Google Scholar 

  39. Gessel, I.M.: Symmetric functions and \(p\) -recursiveness. J. Comb. Theory A 53, 257–285 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  40. Gross, D.J., Matytsin, A.: Instanton induced large \(n\) phase transitions in two and four dimensions. Nucl. Phys. B 429, 50–74 (1994)

    Article  Google Scholar 

  41. Gross, D.J., Witten, E.: Possible third-order phase transition in the large-\(n\) lattice gauge limit. Phys. Rev. D 21, 446–453 (1980)

    Article  Google Scholar 

  42. Hastings, S.P., McLeod, J.B.: A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rat. Mech. Anal. 73, 31–51 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  43. Its, A.R., Kuijlaars, A.B.J., Östensson, J.: Critical edge behavior in unitary random matrix ensembles and the thirty-fourth Painlevé transcendent. IMRN: International Mathematics Research Notices (2008)

  44. Johansson, K.: The longest increasing subsequence in a random permutation and a unitary random matrix model. Math. Res. Lett. Phys. 5, 63–82 (1998)

    MATH  Google Scholar 

  45. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)

    Article  MATH  Google Scholar 

  46. Jones, R.C., Kosterlitz, J.M., Thouless, D.J.: The eigenvalue spectrum of a large symmetric random matrix with a finite mean. J. Phys. A 11, L45–L48 (1978)

    Article  MathSciNet  Google Scholar 

  47. Katori, M., Izumi, M., Kobayashi, N.: Two Bessel bridges conditioned never to collide, double Dirichlet series, and Jacobi theta functions. J. Stat. Phys. 131, 1067–1083 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  48. Liechty, K.: Non-intersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials. J. Stat. Phys. 147, 582–622 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  49. Migdal, A.A.: Recursion equations in gauge field theories. Sov. Phys. JETP 42, 413–418 (1975)

    Google Scholar 

  50. Minahan, J.A.: Matrix models with boundary terms and the generalized Painlevé ii equation. Phys. Lett. B 268, 29–34 (1991)

    Article  MathSciNet  Google Scholar 

  51. Mo, M.Y.: The rank 1 real Wishart spiked model. Comm. Pure. Appl. Math. 65, 1528–1638 (2012)

  52. Myers, R.C., Periwal, V.: Orientability of random surfaces. Phys. Rev. D 42, 3600–3603 (1990)

    Article  MathSciNet  Google Scholar 

  53. Nagao, T., Slevin, K.: Laguerre ensembles of random matrices: nonuniversal correlation functions. J. Math. Phys. 34, 2317–2330 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  54. Okamoto, K.: Studies of the Painlevé equations III. Second and fourth Painlevé equations, \(P_{II}\) and \(P_{IV}\). Math. Ann. 275, 221–255 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  55. Pasquetti, S., Schiappa, R.: Borel and Stokes nonperturbative phenomena in topological string theory and \(c=1\) matrix models. Ann. Henri Poincaré 11, 351–431 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  56. Pastur, L., Shcherbina, M.: Eigenvalue Distribution of Large Random Matrices. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  57. Periwal, V., Shevitz, D.: Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett. 64, 1326–1329 (1990)

    Article  Google Scholar 

  58. Prähofer, M., Spohn, H.: Exact scaling functions for one-dimensional stationary KPZ growth. J. Stat. Phys. 108, 1071–1106 (2004)

    Article  Google Scholar 

  59. Quastel, J., Remenik, D.: Tails of the endpoint distribution of directed polymers (2012). arXiv:1203.2907

  60. Rains, E.M.: Increasing subsequences and the classical groups. Elect. J. Comb. 5, #R12 (1998)

    MathSciNet  Google Scholar 

  61. Rusakov, B.: Loop averages and partition functions in \(U(N)\) gauge theory on two-dimensional manifolds. Mod. Phys. Lett. A 5, 693–703 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  62. Schehr, G.: Extremes of \(n\) vicious walkers for large \(n\): application to the directed polymer and KPZ, interface. J. Stat. Phys. 149, 385–410 (2012)

  63. Schehr, G., Majumdar, S.N., Comtet, A., Forrester, P.J.: Reunion probability of \(n\) vicious walkers: typical and large fluctuations for large \(n\). J. Stat, Phys (2012)

  64. Schehr, G., Majumdar, S.N., Comtet, A., Randon-Furling, J.: Exact distribution of the maximal height of \(p\) vicious walkers. Phys. Rev. Lett. 101, 150601 (2008)

    Article  MathSciNet  Google Scholar 

  65. Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  66. Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  67. Wang, D.: The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I. J. Stat. Phys. 146, 719–761 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  68. Wilson, K.: Confinement of quarks. Phys. Rev. D 10, 2445–2459 (1974)

    Article  Google Scholar 

  69. Witte, N.S., Bornemann, F., Forrester, P.J.: Joint distribution of the first and second eigenvalues at the soft edge. Nonlinearlity 26, 1799–1822 (2013)

Download references

Acknowledgments

This work was supported by the Australian Research Council.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Victoria, 3010, Australia

    Peter J. Forrester & Nicholas S. Witte

Authors
  1. Peter J. Forrester
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicholas S. Witte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Peter J. Forrester or Nicholas S. Witte.

Additional information

Communicated by Percy Deift and Alexander Its.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Forrester, P.J., Witte, N.S. Painlevé II in Random Matrix Theory and Related Fields. Constr Approx 41, 589–613 (2015). https://doi.org/10.1007/s00365-014-9243-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-014-9243-5

Keywords

Mathematics Subject Classification

Search

Navigation