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Level-spacing distributions and the Airy kernel

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Abstract

Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry kernel [Ai(x)Ai(y)−Ai′(x)Ai(y)]/(x−y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Môri, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlevé transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.

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References

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Bowick, M.J., Brézin, E.: Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Lett. B268, 21–28 (1991)

    Article  Google Scholar 

  4. Erdélyi, A. (ed.): Higher transcendental functions, Vol. II. New York: McGraw-Hill 1953

    Google Scholar 

  5. Clarkson, P.A., McLeod, J.B.: A connection formula for the second Painlevé transcendent. Arch. Rat. Mech. Anal.103, 97–138 (1988)

    Google Scholar 

  6. Clarkson, P.A., McLeod, J.B.: Integral equations and connection formulae for the Painlevé equations. In: Painlevé transcendents: their asymptotics and physical applications. Levi, D., Winternitz, P. (eds.), New York: Plenum Press 1992, pp. 1–31

    Google Scholar 

  7. Dyson, F.J.: Statistical theory of energy levels of complex systems, I, II, and III. J. Math. Phys.3, 140–156, 157–165, 166–175 (1962)

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Dyson, F.J.: The Coulomb fluid and the fifth Painlevé transcendent. IASSNSS-HEP-92/43 preprint, to appear in the proceedings of a conference in honor of Yang, C.N., Yau, S.-T. (eds.)

  10. Forrester, P.J.: The spectrum edge of random matrix ensembles, to appear in Nucl. Phys. B

  11. Fuchs, W.H.J.: On the eigenvalues of an integral equation arising in the theory of band-limited signals. J. Math. Anal. and Applic.9, 317–330 (1964)

    Article  Google Scholar 

  12. Harnad, J., Tracy, C.A., Widom, H.: Hamiltonian structure of equations appearing in random matrices. To appear in the NATO ARW: Low dimensional topology and quantum field theory

  13. 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  Google Scholar 

  14. Ince, E.L.: Ordinary differential equations. New York: Dover 1956

    Google Scholar 

  15. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. Int. J. Mod. Physics B4, 1003–1037 (1990)

    Article  Google Scholar 

  16. Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé: a modern theory of special functions. Braunschweig: Vieweg 1991

    Google Scholar 

  17. Jimbo, M., Miwa, T., Môri, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica1D, 80–158 (1980)

    Google Scholar 

  18. McCoy, B.M., Tracy, C.A., Wu, T.T.: Connection between the KdV equation and the twodimensional Ising model. Phys. Lett.61A, 283–284 (1977)

    Google Scholar 

  19. McLeod, J.B.: Private communication

  20. Mehta, M.L.: Random matrices. 2nd edition, San Diego: Academic 1991

    Google Scholar 

  21. Mehta, M.L.: A non-linear differential equation and a Fredholm determinant. J. de Phys. I France,2, 1721–1729 (1992)

    Article  Google Scholar 

  22. Mehta, M.L., Mahoux, G.: Level spacing functions and non-linear differential equations. Preprint

  23. Moore, G.: Matrix models of 2D gravity and isomonodromic deformation. Prog. Theor. Physics Suppl. No.102, 255–285 (1990)

    Google Scholar 

  24. Moser, J.: Geometry of quadrics and spectral theory. In: Chern Symposium 1979, Berlin, Heidelberg, New York: Springer 1980, pp. 147–188

    Google Scholar 

  25. Painlevé, P.: Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme. Acta Math.25, 1–85 (1902)

    Google Scholar 

  26. Porter, C.E.: Statistical theory of spectra: fluctuations. New York: Academic 1965

    Google Scholar 

  27. Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty-I. Bell Systems Tech. J.40, 43–64 (1961)

    Google Scholar 

  28. Tracy, C.A., Widom, H.: Introduction to random matrices. To appear in the proceedings of the 8th Scheveningen Conference, Springer Lecture Notes in Physics

  29. Tracy, C.A., Widom, H.: Level spacing distributions and the Airy kernel. Phys. Lett. B305, 115–118 (1993)

    Article  Google Scholar 

  30. Widom, H.: The strong Szegő limit theorem for circular arcs. Indiana Univ. Math. J.21, 277–283 (1971)

    Article  Google Scholar 

  31. Widom, H.: The asymptotics of a continuous analogue of orthogonal polynomials. To appear in J. Approx. Th.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Institute of Theoretical Dynamics, University of California, 95616, Davis, CA, USA

    Craig A. Tracy

  2. Department of Mathematics, University of California, 95064, Santa Cruz, CA, USA

    Harold Widom

Authors
  1. Craig A. Tracy
  2. Harold Widom

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Communicated by N. Yu. Reshetikhin

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