Abstract
Orthogonal polynomial random matrix models ofN×N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (ϕ(x)ψ(y)−ψ(x)ϕ(y))/x−y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is the union of intervals\(J = \cup _{j = 1}^m (a_{2j - 1 ,{\text{ }}} a_{2j} )\). The emphasis is on the determinants thought of as functions of the end-pointsa k.
We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as ϕ and ψ satisfy a certain type of differentiation formula. The (ϕ, ψ) pairs for the sine, Airy, and Bessel kernels satisfy such relations, as do the pairs which arise in the finiteN Hermite, Laguerre and Jacobi ensembles and in matrix models of 2D quantum gravity. Therefore we shall be able to write down the systems of PDE's for these ensembles as special cases of the general system.
An analysis of these equations will lead to explicit representations in terms of Painlevé transcendents for the distribution functions of the largest and smallest eigenvalues in the finiteN Hermite and Laguerre ensembles, and for the distribution functions of the largest and smallest singular values of rectangular matrices (of arbitrary dimensions) whose entries are independent identically distributed complex Gaussian variables.
There is also an exponential variant of the kernel in which the denominator is replaced bye bx−eby, whereb is an arbitrary complex number. We shall find an analogous system of differential equations in this setting. Ifb=i then we can interpret our operator as acting on (a subset of) the unit circle in the complex plane. As an application of this we shall write down a system of PDE's for Dyson's circular ensemble ofN×N unitary matrices, and then an ODE ifJ is an arc of the circle.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barouch, E., McCoy, B.M., Wu, T.T.: Zero-field susceptibility of the two-dimensional Ising model nearT c. Phys. Rev. Letts.31, 1409–1411 (1973); Tracy, C., McCoy, B.M.: Neutron scattering and the correlation functions of the Ising model nearT c. Phys. Rev. Letts.31, 1500–1504 (1973)
Basor, E.L., Tracy, C.A., Widom, H.: Asymptotics of level spacing distributions for random matrices. Phys. Rev. Letts.69, 5–8 (1992)
Bassom, A.P., Clarkson, P.A., Hicks, A.C., McLeod, J.B.: Integral equations and exact solutions for the fourth Painlevé transcendent. Proc. R. Soc. Lond. A437, 1–24 (1992)
Bauldry, W.C.: Estimates of asymmetric Freud polynomials. J. Approx. Th.63, 225–237 (1990)
Bonan, S.S., Clark, D.S.: Estimates of the Hermite and the Freud polynomials. J. Approx. Th.63, 210–224 (1990)
Bowick, M.J., Brézin, E.: Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Letts.B268, 21–28 (1991)
Brézin, E.: LargeN limit and discretized two-dimensional quantum gravity. In: Gross, D.J., Piran, T., Weinberg, S. (eds.) Two Dimensional Quantum Gravity and Random Surfaces, Singapore: World Scientific Publ., 1992 pp. 1–40
Brézin, E., Kazakov, V.A.: Exactly solvable field theories of closed strings. Phys. Letts.B236, 144–150 (1990)
Chen, Y.: Private communication
Clarkson, P.A., McLeod, J.B.: Integral equations and connection formulae for the Painlevé equations. In: Levi, D., Winternitz, P. (eds.) Painlevé Transcendents: Their Asymptotics and Physical Applications, New York: Plenum Press, 1992, pp. 1–31
David, F.: Non-perturbative effects in matrix models and vacua of two dimensional gravity. SPhT/92-159 preprint
Douglas, M.R.: Strings in less than one dimension and the generalized KdV hierarchies. Phys. Letts.B238, 176–180 (1990)
Douglas, M.R., Shenker, S.H.: Strings in less than one dimension. Nucl. Phys.B335, 635–654 (1990)
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. IASSNSSHEP-92/43 preprint, to appear in the proceedings of a conference in honor of C.N. Yang. S.-T. Yau (ed.)
Edelman, A.: Eigenvalues and condition numbers of random matrices. Ph.D. thesis, Mass. Inst. of Tech., 1989
Erdélyi, A. (ed.): Higher Transcendental Functions,Vols. I and II New York: McGraw-Hill, 1953
Forrester, P.J.: The spectrum edge of random matrix ensembles. Nucl. Phys.B402, [FS], 709–728 (1993)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products. Corrected and Enlarged Edition. San Diego: Academic, 1980
Gross, D.J., Migdal, A.A.: Nonperturbative two-dimensional quantum gravity. Phys. Rev. Letts.64, 127–130 (1990); A nonperturbative treatment of two-dimensional quantum gravity. Nucl. Phys.B340, 333–365 (1990)
Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. Int. J. Mod. PhysicsB4, 1003–1037 (1990)
Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé equations. Publ. RIMS, Kyoto Univ.18, 1137–1161 (1982)
Jimbo, M., Miwa, T.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients. II. Physica2D, 407–448 (1981)
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.: Painlevé functions of the third kind. J. Math. Phys.18, 1058–1092 (1977)
Mahoux, G., Mehta, M.L.: Level spacing functions and nonlinear differential equations. J. Phys. I. France3, 697–715 (1993)
Mehta, M.L.: Random Matrices. 2nd ed. San Diego: Academic, 1991
Mehta, M.L.: A non-linear differential equation and a Fredholm determinant. J. de Phys. I. France2, 1721–1729 (1992)
Moore, G.: Matrix models of 2D gravity and isomonodromic deformation. Prog. Theor. Physics Suppl. No.102, 255–285 (1990)
Muttalib, K.A., Chen Y., Ismail, M.E.H., Nicopoulos, V.N.: A new family of unitary random matrices. Phys. Rev. Lett.71, 471–475 (1993)
Nagao, T., Wadati, M.: Correlation functions of random matrix ensembles related to classical orthogonal polynomials. J. Phys. Soc. Japan60, 3298–3322 (1991)
Okamoto, K.: Studies on the Painlevé equations II. Fifth Painlevé equationP v. Japan. J. Math.13, 47–76 (1987)
Okamoto, K.: Studies on the Painlevé equations III. Second and Fourth Painlevé Equations,P II andP IV. Math. Ann.275, 221–255 (1986)
Okamoto, K.: Studies on the Painlevé equations IV. Third Painlevé equationP III. Funkcialaj Ekvacioj30, 305–332 (1987)
Porter, C.E.: Statistical Theory of Spectra: Fluctuations. New York: Academic, 1965
Tracy, C.A., Widom, H.: Introduction to random matrices. In: Geometric and Quantum Aspects of Integrable Systems. G. F. Helminck (ed.) Lecture Notes in Physics, Vol. 424, Berlin, Heidelberg, New York: Springer, 1993, pp. 407–424
Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys.159, 151–174 (1994)
Tracy, C.A., Widom, H.: Level-spacing distributions and the Bessel kernel. Commun. Math. Phys.161, 289–309 (1994)
Widom, H.: The asymptotics of a continuous analogue of orthogonal polynomials. To appear in J. Approx. Th.
Wu, T.T., McCoy, B.M., Tracy, C.A., Barouch, E.: Spin-spin correlation functions of the two-dimensional Ising model: Exact theory in the scaling region. Phys. Rev.B13, 316–374 (1976)
Author information
Authors and Affiliations
Additional information
Communicated by M. Jimbo
Rights and permissions
About this article
Cite this article
Tracy, C.A., Widom, H. Fredholm determinants, differential equations and matrix models. Commun.Math. Phys. 163, 33–72 (1994). https://doi.org/10.1007/BF02101734
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02101734