×

zbMATH — the first resource for mathematics

Level-spacing distributions and the Airy kernel. (English) Zbl 0789.35152
Summary: Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models of \(N\times N\) hermitian matrices and then going to the limit \(N\to\infty\) leads to the Fredholm determinant of the sine kernel \(\sin \pi(x-y)/ \pi(x-y)\). Similarly a scaling limit at the “edge of the spectrum” leads to the Airy kernel \([\text{Ai}(x) \text{Ai}(y)- \text{Ai}'(x) \text{Ai}(y)]/ (x-y)\). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of PDE’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 general \(n\), of the probability that an interval contains precisely \(n\) eigenvalues.

MSC:
35Q58 Other completely integrable PDE (MSC2000)
15B52 Random matrices (algebraic aspects)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ablowitz, M.J., Segur, H.: Exact linearization of a Painlevé transcendent. Phys. Rev. Lett.38, 1103–1106 (1977) · doi:10.1103/PhysRevLett.38.1103
[2] Basor, E.L., Tracy, C.A., Widom, H.: Asymptotics of level spacing distributions for random matrices. Phys. Rev. Lett.69, 5–8 (1992) · Zbl 0968.82501 · doi:10.1103/PhysRevLett.69.5
[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) · doi:10.1016/0370-2693(91)90916-E
[4] Erdélyi, A. (ed.): Higher transcendental functions, Vol. II. New York: McGraw-Hill 1953 · Zbl 0051.30303
[5] Clarkson, P.A., McLeod, J.B.: A connection formula for the second Painlevé transcendent. Arch. Rat. Mech. Anal.103, 97–138 (1988) · Zbl 0653.34020 · doi:10.1007/BF00251504
[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 · Zbl 0856.34009
[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) · Zbl 0105.41604 · doi:10.1063/1.1703773
[8] Dyson, F.J.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys.47, 171–183 (1976) · Zbl 0323.33008 · doi:10.1007/BF01608375
[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) · Zbl 0178.47403 · doi:10.1016/0022-247X(64)90017-4
[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) · Zbl 0426.34019 · doi:10.1007/BF00283254
[14] Ince, E.L.: Ordinary differential equations. New York: Dover 1956 · Zbl 0063.02971
[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) · Zbl 0719.35091 · doi:10.1142/S0217979290000504
[16] Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé: a modern theory of special functions. Braunschweig: Vieweg 1991 · Zbl 0743.34014
[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) · Zbl 1194.82007
[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)
[19] McLeod, J.B.: Private communication
[20] Mehta, M.L.: Random matrices. 2nd edition, San Diego: Academic 1991 · Zbl 0780.60014
[21] Mehta, M.L.: A non-linear differential equation and a Fredholm determinant. J. de Phys. I France,2, 1721–1729 (1992) · doi:10.1051/jp1:1992240
[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) · Zbl 0875.33006 · doi:10.1143/PTPS.102.255
[24] Moser, J.: Geometry of quadrics and spectral theory. In: Chern Symposium 1979, Berlin, Heidelberg, New York: Springer 1980, pp. 147–188
[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) · JFM 32.0340.01 · doi:10.1007/BF02419020
[26] Porter, C.E.: Statistical theory of spectra: fluctuations. New York: Academic 1965 · Zbl 0144.22603
[27] Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty-I. Bell Systems Tech. J.40, 43–64 (1961) · Zbl 0184.08601
[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 · Zbl 0791.15017
[29] Tracy, C.A., Widom, H.: Level spacing distributions and the Airy kernel. Phys. Lett. B305, 115–118 (1993) · Zbl 0789.35152 · doi:10.1016/0370-2693(93)91114-3
[30] Widom, H.: The strong Szego limit theorem for circular arcs. Indiana Univ. Math. J.21, 277–283 (1971) · Zbl 0223.33015 · doi:10.1512/iumj.1971.21.21022
[31] Widom, H.: The asymptotics of a continuous analogue of orthogonal polynomials. To appear in J. Approx. Th. · Zbl 0801.42017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.