We consider the Jacobi polynomial ensemble of random matrices. We show that the probability of finding no eigenvalues in the interval for a random matrix chosen from the ensemble, viewed as a function of z, satisfies a second-order differential equation. After a simple change of variable, this equation can be reduced to the Okamoto–Jimbo–Miwa form of the Painlevé VI equation. The result is achieved by a comparison of the Tracy–Widom and the Virasoro approaches to the problem, which both lead to different third-order differential equations. The Virasoro constraints satisfied by the tau functions are obtained by a systematic use of the moments, which drastically simplifies the computations.
REFERENCES
1.
M. L. Mehta, Random Matrices, 2nd ed. (Academic, San Diego, 1991).
2.
C. A. Tracy and H. Widom, “Introduction to random matrices,” in Geometric and Quantum Aspects of Integrable Systems, Lecture Notes in Physics, edited by G. F. Helminck (Springer-Verlag, Berlin, 1993), Vol. 424, pp. 103–130.
3.
C. A.
Tracy
and H.
Widom
, “Fredholm determinants, differential equations and matrix models
,” Commun. Math. Phys.
163
, 33
–72
(1994
).4.
M.
Jimbo
, T.
Miwa
, Y.
Môri
, and M.
Sato
, “Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent
,” Physica D
1
, 80
–158
(1980
).5.
M.
Adler
, T.
Shiota
, and P.
van Moerbeke
, “Random matrices, vertex operators and the Virasoro algebra
,” Phys. Lett. A
208
, 67
–78
(1995
).6.
L.
Haine
and E.
Horozov
, “Toda orbits of Laguerre polynomials and representations of the Virasoro algebra
,” Bull. Sci. Math.
117
, 485
–518
(1993
).7.
F. A. Grünbaum and L. Haine, “A theorem of Bochner, revisited,” in Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman, Progress in Nonlinear Differential Equations, edited by A. S. Fokas and I. M. Gelfand (Birkhäuser, Boston, 1997), Vol. 26, pp. 143–172.
8.
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis (Hafner, New York, 1965).
9.
T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications (Gordon and Breach, New York, 1978), Vol. 13.
10.
L. Haine, “Beyond the classical orthogonal polynomials,” in The Bispectral Problem, CRM Proc. Lecture Notes, edited by J. Harnad and A. Kasman (American Mathematical Society, Providence, RI, 1998), Vol. 14, pp. 47–65.
11.
S.
Bochner
, “Über Sturm–Liouvillesche polynomsysteme
,” Math. Z.
29
, 730
–736
(1929
).12.
M. Sato and Y. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds,” in Nonlinear Partial Differential Equations in Applied Science, North-Holland Math. Stud. 81, edited by H. Fujita, P. D. Lax, and G. Strang (Kinokuniya/North-Holland, Tokyo, 1983), pp. 259–271.
13.
J. Moser, “Finitely many mass points on the line under the influence of an exponential potential—An integrable system,” in Dynamical Systems, Theory and Applications, Lecture Notes in Physics, edited by J. Moser (Springer-Verlag, Berlin, 1975), Vol. 38, pp. 467–497.
14.
E. Witten, “Two-dimensional gravity and intersection theory on moduli space,” Surveys in Differential Geometry 1, A Supplement to the Journal of Differential Geometry, edited by C. C. Hsiung and S. T. Yau (1991), pp. 243–310.
15.
V. G. Kac and A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, Adv. Series in Math. Phys. (World Scientific, Singapore, 1987), Vol. 2.
16.
M.
Adler
and P.
van Moerbeke
, “Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials
,” Duke Math. J.
80
, 863
–911
(1995
).17.
J. P. Semengue, Ph.D. thesis, Université catholique de Louvain, 1998.
18.
P.
Damianou
, “Master symmetries and R-matrices for the Toda lattice
,” Lett. Math. Phys.
20
, 101
–112
(1990
).19.
R. L.
Fernandes
, “On the master symmetries and bi-Hamiltonian structure of the Toda lattice
,” J. Phys. A
26
, 3797
–3803
(1993
).20.
L. Haine and E. Horozov, “Tau-functions and modules over the Virasoro algebra,” in Abelian Varieties, Proceedings of the International Conference held in Egloffstein, Germany, 3–8 October 1993, edited by W. Barth, K. Hulek, and H. Lange (Walter de Gruyter, Berlin, 1995), pp. 85–104.
21.
M.
Jimbo
and T.
Miwa
, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II
,” Physica D
2
, 407
–448
(1981
).
This content is only available via PDF.
© 1999 American Institute of Physics.
1999
American Institute of Physics
You do not currently have access to this content.
