Random matrices, Virasoro algebras, and noncommutative KP. (English) Zbl 1061.37047
It is known that probability distributions of finite random matrix ensembles are usually given by explicit matrix integrals; whereas probability measures for infinite random matrix ensembles are given by certain kernels. The paper deals with the second case which is connected with the KdV equation. The authors suggest that studying connections between random matrices and integrable Hamiltonian systems should lead to a new and unifying approach to the theory of random matrices.
Reviewer: Yuri Kifer (Jerusalem)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 17B68 | Virasoro and related algebras |
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
| 15B52 | Random matrices (algebraic aspects) |
References:
| [1] | M. Adler and P. van Moerbeke, A matrix integral solution to two-dimensional \(W_ p\)-gravity , Comm. Math. Phys. 147 (1992), no. 1, 25-56. · Zbl 0756.35074 · doi:10.1007/BF02099527 |
| [2] | M. Adler and P. van Moerbeke, Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials , Duke Math. J. 80 (1995), no. 3, 863-911. · Zbl 0848.17027 · doi:10.1215/S0012-7094-95-08029-6 |
| [3] | M. Adler and P. van Moerbeke, String-orthogonal polynomials, string equations and \(2\)-Toda symmetries , Comm. Pure Appl. Math. 50 (1997), no. 3, 241-290. · Zbl 0880.58012 · doi:10.1002/(SICI)1097-0312(199703)50:3<241::AID-CPA3>3.0.CO;2-B |
| [4] | M. Adler and P. van Moerbeke, Finite random Hermitian ensembles , · Zbl 1153.39300 |
| [5] | M. Adler and P. van Moerbeke, The spectrum of coupled random matrices , to appear in Ann. of Math. (2), 1999. JSTOR: · Zbl 0936.15018 · doi:10.2307/121077 |
| [6] | M. Adler, T. Shiota, and P. van Moerbeke, From the \(w_\infty\)-algebra to its central extension: A \(\tau\)-function approach , Phys. Lett. A 194 (1994), no. 1-2, 33-43. · Zbl 0961.37514 · doi:10.1016/0375-9601(94)00306-A |
| [7] | M. Adler, T. Shiota, and P. van Moerbeke, Random matrices, vertex operators and the Virasoro algebra , Phys. Lett. A 208 (1995), no. 1-2, 67-78. · Zbl 0925.58032 · doi:10.1016/0375-9601(95)00725-I |
| [8] | M. Adler, T. Shiota, and P. van Moerbeke, A Lax representation for the vertex operator and the central extension , Comm. Math. Phys. 171 (1995), no. 3, 547-588. · Zbl 0839.35116 · doi:10.1007/BF02104678 |
| [9] | M. Adler, T. Shiota, A. Morozov, and P. van Moerbeke, A matrix integral solution to \([P,Q]=P\) and matrix Laplace transforms , Comm. Math. Phys. 180 (1996), no. 1, 233-263. · Zbl 0858.35109 · doi:10.1007/BF02101187 |
| [10] | L. Alvarez-Gaumé, C. Gomez, and J. Lacki, Integrability in random matrix models , Phys. Lett. B 253 (1991), no. 1-2, 56-62. · doi:10.1016/0370-2693(91)91363-Z |
| [11] | D. Bessis, C. Itzykson, and J. B. Zuber, Quantum field theory techniques in graphical enumeration , Adv. in Appl. Math. 1 (1980), no. 2, 109-157. · Zbl 0453.05035 · doi:10.1016/0196-8858(80)90008-1 |
| [12] | E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Transformation groups for soliton equations , Nonlinear Integrable Systems-Classical Theory and Quantum Theory (Kyoto, 1981), World Sci., Singapore, 1983, pp. 39-119. · Zbl 0571.35098 |
| [13] | P. Deift, A. Its, and X. Zhou, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics , Ann. of Math. (2) 146 (1997), no. 1, 149-235. JSTOR: · Zbl 0936.47028 · doi:10.2307/2951834 |
| [14] | L. Dickey, Soliton Equations and Hamiltonian Systems , Adv. Ser. Math. Phys., vol. 12, World Sci., River Edge, N.J., 1991. · Zbl 0753.35075 |
| [15] | 1 F. Dyson, Statistical theory of the energy levels of complex systems, I , J. Math. Phys. 3 (1962), 140-156. · Zbl 0105.41604 · doi:10.1063/1.1703773 |
| [16] | 2 F. Dyson, Statistical theory of the energy levels of complex systems, II , J. Math. Phys. 3 (1962), 157-165. · Zbl 0105.41604 · doi:10.1063/1.1703773 |
| [17] | 3 F. Dyson, Statistical theory of the energy levels of complex systems, III , J. Math. Phys. 3 (1962), 166-175. · Zbl 0105.41604 · doi:10.1063/1.1703773 |
| [18] | F. Dyson, Fredholm determinants and inverse scattering problems , Comm. Math. Phys. 47 (1976), no. 2, 171-183. · Zbl 0323.33008 · doi:10.1007/BF01608375 |
| [19] | A. Its, A. G. Izergin, V. E. Korepin, and N. A. Slavnov, Differential equations for quantum correlation functions , Internat. J. Modern Phys. B 4 (1990), no. 5, 1003-1037. · Zbl 0719.35091 · doi:10.1142/S0217979290000504 |
| [20] | M. Jimbo, T. Miwa, Y. Môri, and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent , Physica D 1 (1980), no. 1, 80-158. · Zbl 1194.82007 · doi:10.1016/0167-2789(80)90006-8 |
| [21] | V. G. Kac, Infinite-Dimensional Lie Algebras , 3d ed., Cambridge Univ. Press, Cambridge, 1990. · Zbl 0716.17022 |
| [22] | B. M. McCoy, Spin systems, statistical mechanics and Painlevé functions , Painlevé Transcendents (Sainte-Adèle, PQ, 1990), NATO Adv. Sci. Inst. Ser. B Phys., vol. 278, Plenum, New York, 1992, pp. 377-391. · Zbl 0875.35080 |
| [23] | H. P. McKean, Integrable systems and algebraic curves , Global Analysis (Proc. Biennial Sem. Canad. Math. Congr., Univ. Calgary, Calgary, Alta., 1978), Lecture Notes in Math., vol. 755, Springer-Verlag, Berlin, 1979, pp. 83-200. · Zbl 0449.35080 |
| [24] | M. L. Mehta, Random Matrices , 2d ed., Academic Press, Boston, 1991. · Zbl 0780.60014 |
| [25] | M. L. Mehta, A nonlinear differential equation and a Fredholm determinant , J. Physique I 2 (1992), no. 9, 1721-1729. · doi:10.1051/jp1:1992240 |
| [26] | H. Peng, The spectrum of random matrices for symmetric ensembles , dissertation, Brandeis Univ., Waltham, Mass., 1997. |
| [27] | C. E. Porter and N. Rosenzweig, Statistical properties of atomic and nuclear spectra , Ann. Acad. Sci. Fenn. Ser. A VI No. 44 (1960), 66. · Zbl 0092.23304 |
| [28] | C. E. Porter and N. Rosenzweig, Repulsion of energy levels in complex atomic spectra , Phys. Rev. 120 (1960), 1698-1714. |
| [29] | P. Sarnak, Arithmetic quantum chaos , The Schur Lectures (Tel Aviv, 1992), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 183-236. · Zbl 0831.58045 |
| [30] | M. Sato, Soliton equations and the universal Grassmann manifold (by Noumi in Japanese) , Math. Lect. Note Ser., vol. 18, Sophia University, Tokyo, 1984. · Zbl 0541.58001 |
| [31] | T. Shiota, Characterization of Jacobian varieties in terms of soliton equations , Invent. Math. 83 (1986), no. 2, 333-382. · Zbl 0621.35097 · doi:10.1007/BF01388967 |
| [32] | N. A. Slavnov, Fredholm determinants and \(\tau\)-functions , Teoret. Mat. Fiz. 109 (1996), no. 3, 357-371. · Zbl 0935.35162 · doi:10.1007/BF02073869 |
| [33] | G. Szegö, Orthogonal Polynomials , Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., New York, 1939. · Zbl 0023.21505 |
| [34] | C. A. Tracy and H. Widom, Introduction to random matrices , Geometric and Quantum Aspects of Integrable Systems (Scheveningen, 1992), Lecture Notes in Phys., vol. 424, Springer-Verlag, Berlin, 1993, pp. 103-130. · Zbl 0791.15017 · doi:10.1007/BFb0021444 |
| [35] | C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel , Comm. Math. Phys. 159 (1994), no. 1, 151-174. · Zbl 0789.35152 · doi:10.1007/BF02100489 |
| [36] | C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel , Comm. Math. Phys. 161 (1994), no. 2, 289-309. · Zbl 0808.35145 · doi:10.1007/BF02099779 |
| [37] | C. A. Tracy and H. Widom, Fredholm determinants, differential equations and matrix models , Comm. Math. Phys. 163 (1994), no. 1, 33-72. · Zbl 0813.35110 · doi:10.1007/BF02101734 |
| [38] | J. W. van de Leur, The Adler-Shiota-van Moerbeke formula for the BKP hierarchy , J. Math. Phys. 36 (1995), no. 9, 4940-4951. · Zbl 0844.35109 · doi:10.1063/1.531352 |
| [39] | P. van Moerbeke, Integrable foundations of string theory , Lectures on Integrable Systems (Sophia-Antipolis, 1991), World Sci., River Edge, N.J., 1994, pp. 163-267. · Zbl 0850.81049 |
| [40] | E. P. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels , Proc. Cambridge Philos. Soc. 47 (1951), 790-798. · Zbl 0044.44203 |
| [41] | E. Witten, Two-dimensional gravity and intersection theory on moduli space , Surveys in Differential Geometry (Cambridge, Mass., 1990), Lehigh Univ., Bethlehem, Penn., 1991, pp. 243-310. · Zbl 0757.53049 |
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