Skip to main content
SpringerLink
Log in

Dyson’s Constants in the Asymptotics of the Determinants of Wiener-Hopf-Hankel Operators with the Sine Kernel

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let \(K_\alpha^\pm \) stand for the integral operators with the sine kernels \(\frac{\sin(x-y)}{\pi(x-y)} \pm \frac{\sin(x+y)}{\pi(x+y)}\) acting on L 2[0,α]. Dyson conjectured that the asymptotics of the Fredholm determinants of \(I-K_\alpha^\pm\) are given by

$$ \log\det(I-K_{\alpha}^\pm) = -\frac{\alpha^2}{4}\mp \frac{\alpha}{2}-\frac{\log\alpha}{8}+\frac{\log 2}{24}\pm \frac{\log 2}{4} +\frac{3}{2} \zeta'(-1)+o(1),$$

as α→∞. In this paper we are going to give a proof of these two asymptotic formulas.

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. Barnes, E.B.: The theory of the G-function. Quart. J. Pure Appl. Math. XXXI, 264–313 (1900)

  2. Basor E.L. and Ehrhardt T. (2002). Some identities for determinants of structured matrices. Linear Algebra Appl. 343/344: 5–19

    Article  MathSciNet  Google Scholar 

  3. Basor E.L. and Ehrhardt T. (2005). On the asymptotics of certain Wiener-Hopf-plus-Hankel determinants. New York J. Math. 11: 171–203

    MATH  MathSciNet  Google Scholar 

  4. Basor E.L., Tracy C.A. and Widom H. (1992). Asymptotics of level-spacing distributions for random matrices. Phys. Rev. Lett. 69(1): 5–8

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Böttcher A. and Silbermann B. (2006). Analysis of Toeplitz Operators 2nd ed. Springer, Berlin

    Google Scholar 

  6. Deift, P., Its, A., Krasovksy, I., Zhou, X.: The Widom-Dyson constant for the gap probability in random matrix theory. http://arxiv.org/list/math.FA/0601535, 2006

  7. Deift, P., Its, A., Zhou, X.: The Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. of Math. (2) 146(1), 149–235 (1997)

    Google Scholar 

  8. des Cloiseau J. and Mehta M.L. (1973). Asymptotic behavior of spacing distributions for the eigenvalues of random matrices. J. Math. Phys. 14: 1648–1650

    Article  Google Scholar 

  9. Dyson F.J. (1976). Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47: 171–183

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Ehrhardt T. (2006). Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel. Commun. Math. Phys. 262: 317–341

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Ehrhardt T. and Silbermann B. (1999). Approximate identities and stability of discrete convolution operators with flip. Operator Theory: Adv. Appl. 110: 103–132

    MathSciNet  Google Scholar 

  12. Gohberg, I., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators in Hilbert space. Trans. Math. Monographs 18, Providence, R.I.: Amer. Math. 1969

  13. Jimbo M., Miwa T., Môri Y. and Sato M. (1980). Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. 1(D)(1): 80–158

    Google Scholar 

  14. Krasovsky, I.V.: Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. Int. Math. Res. Not. (25), 1249–1272 (2004)

  15. Mehta, M.L.: Random Matrices. 6th edition, San Diego: Academic Press, 1994

  16. Suleimanov, B.: On asymptotics of regular solutions for a special kind of Painlevé V equation. In: Appendix 1 of: Its, A.R., Novokshenov, V.Yu.: The isomonodromic deformation method in the theory of Painlevé equations. Lect. Notes in Math. 1191, Berlin: Springer, 1986, pp. 230–260

  17. Tracy, C.A., Widom, H.: Introduction to Random Matrices. In: Geometric and quantum aspects of integrable systems (Scheveningen, 1992), Lecture Notes in Physics 424, Berlin: Springer, 1993, pp. 103–130

  18. Tracy C.A. and Widom H. (1994). Level-spacing distributions and the Bessel kernel. Commun. Math. Phys. 161: 289–310

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. Tracy C.A. and Widom H. (1994). Fredholm Determinants, Differential Equations and Matrix Models. Commun. Math. Phys. 163: 33–72

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Widom H. (1994). The asymptotics of a continuous analogue of orthogonal polynomials. J. Appr. Theory 77: 51–64

    Article  MATH  MathSciNet  Google Scholar 

  21. Widom H. (1995). Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals. Commun. Math. Phys. 171: 159–180

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Cruz, CA, 95065, USA

    Torsten Ehrhardt

Authors
  1. Torsten Ehrhardt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Torsten Ehrhardt.

Additional information

Communicated by J.L. Lebowitz.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ehrhardt, T. Dyson’s Constants in the Asymptotics of the Determinants of Wiener-Hopf-Hankel Operators with the Sine Kernel. Commun. Math. Phys. 272, 683–698 (2007). https://doi.org/10.1007/s00220-007-0239-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0239-x

Keywords

Search

Navigation