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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ablowitz, M.J., Segur, H.: Exact linearization of a Painlevé transcendent. Phys. Rev. Lett.38, 1103–1106 (1977)
Basor, E.L., Tracy, C.A., Widom, H.: Asymptotics of level spacing distributions for random matrices. Phys. Rev. Lett.69, 5–8 (1992)
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)
Erdélyi, A. (ed.): Higher transcendental functions, Vol. II. New York: McGraw-Hill 1953
Clarkson, P.A., McLeod, J.B.: A connection formula for the second Painlevé transcendent. Arch. Rat. Mech. Anal.103, 97–138 (1988)
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
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)
Dyson, F.J.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys.47, 171–183 (1976)
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.)
Forrester, P.J.: The spectrum edge of random matrix ensembles, to appear in Nucl. Phys. B
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)
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
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)
Ince, E.L.: Ordinary differential equations. New York: Dover 1956
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)
Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé: a modern theory of special functions. Braunschweig: Vieweg 1991
Jimbo, M., Miwa, T., Môri, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica1D, 80–158 (1980)
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)
McLeod, J.B.: Private communication
Mehta, M.L.: Random matrices. 2nd edition, San Diego: Academic 1991
Mehta, M.L.: A non-linear differential equation and a Fredholm determinant. J. de Phys. I France,2, 1721–1729 (1992)
Mehta, M.L., Mahoux, G.: Level spacing functions and non-linear differential equations. Preprint
Moore, G.: Matrix models of 2D gravity and isomonodromic deformation. Prog. Theor. Physics Suppl. No.102, 255–285 (1990)
Moser, J.: Geometry of quadrics and spectral theory. In: Chern Symposium 1979, Berlin, Heidelberg, New York: Springer 1980, pp. 147–188
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)
Porter, C.E.: Statistical theory of spectra: fluctuations. New York: Academic 1965
Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty-I. Bell Systems Tech. J.40, 43–64 (1961)
Tracy, C.A., Widom, H.: Introduction to random matrices. To appear in the proceedings of the 8th Scheveningen Conference, Springer Lecture Notes in Physics
Tracy, C.A., Widom, H.: Level spacing distributions and the Airy kernel. Phys. Lett. B305, 115–118 (1993)
Widom, H.: The strong Szegő limit theorem for circular arcs. Indiana Univ. Math. J.21, 277–283 (1971)
Widom, H.: The asymptotics of a continuous analogue of orthogonal polynomials. To appear in J. Approx. Th.
Author information
Authors and Affiliations
Additional information
Communicated by N. Yu. Reshetikhin
Rights and permissions
About this article
Cite this article
Tracy, C.A., Widom, H. Level-spacing distributions and the Airy kernel. Commun.Math. Phys. 159, 151–174 (1994). https://doi.org/10.1007/BF02100489
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02100489