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Random matrices, Virasoro algebras, and noncommutative KP

M. Adler, T. Shiota, and P. van Moerbeke
Source: Duke Math. J. Volume 94, Number 2 (1998), 379-431.
First Page: Show Hide
Primary Subjects: 58F07
Secondary Subjects: 17B68, 60H25
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077230277
Mathematical Reviews number (MathSciNet): MR1638599
Zentralblatt MATH identifier: 01425145
Digital Object Identifier: doi:10.1215/S0012-7094-98-09417-0

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.
Mathematical Reviews (MathSciNet): MR93g:58054
Zentralblatt MATH: 0756.35074
Digital Object Identifier: doi:10.1007/BF02099527
Project Euclid: euclid.cmp/1104250525
[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.
Mathematical Reviews (MathSciNet): MR96k:58101
Zentralblatt MATH: 0848.17027
Digital Object Identifier: doi:10.1215/S0012-7094-95-08029-6
Project Euclid: euclid.dmj/1077246296
[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.
Mathematical Reviews (MathSciNet): MR98a:58076
Zentralblatt MATH: 0880.58012
[4] M. Adler and P. van Moerbeke, Finite random Hermitian ensembles, to appear.
[5] M. Adler and P. van Moerbeke, The spectrum of coupled random matrices, to appear in Ann. of Math. (2), 1999.
Mathematical Reviews (MathSciNet): MR1709307
Zentralblatt MATH: 0936.15018
Digital Object Identifier: 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.
Mathematical Reviews (MathSciNet): MR95k:58067
Zentralblatt MATH: 0961.37514
Digital Object Identifier: 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.
Mathematical Reviews (MathSciNet): MR96m:82026
Zentralblatt MATH: 0925.58032
Digital Object Identifier: 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.
Mathematical Reviews (MathSciNet): MR97a:58072
Zentralblatt MATH: 0839.35116
Digital Object Identifier: doi:10.1007/BF02104678
Project Euclid: euclid.cmp/1104273764
[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.
Mathematical Reviews (MathSciNet): MR97e:58105
Zentralblatt MATH: 0858.35109
Digital Object Identifier: doi:10.1007/BF02101187
Project Euclid: euclid.cmp/1104287240
[10] L. Alvarez-Gaumé, C. Gomez, and J. Lacki, Integrability in random matrix models, Phys. Lett. B 253 (1991), no. 1-2, 56–62.
Mathematical Reviews (MathSciNet): MR92a:81162
Digital Object Identifier: 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.
Mathematical Reviews (MathSciNet): MR83j:81067
Zentralblatt MATH: 0453.05035
Digital Object Identifier: 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.
Mathematical Reviews (MathSciNet): MR86a:58093
Zentralblatt MATH: 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.
Mathematical Reviews (MathSciNet): MR98k:47097
Zentralblatt MATH: 0936.47028
Digital Object Identifier: 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.
Mathematical Reviews (MathSciNet): MR93d:58067
Zentralblatt MATH: 0753.35075
[15]1 F. Dyson, Statistical theory of the energy levels of complex systems, I, J. Math. Phys. 3 (1962), 140–156.
Mathematical Reviews (MathSciNet): MR26:1111
Zentralblatt MATH: 0105.41604
Digital Object Identifier: doi:10.1063/1.1703773
[15]2 F. Dyson, Statistical theory of the energy levels of complex systems, II, J. Math. Phys. 3 (1962), 157–165.
Mathematical Reviews (MathSciNet): MR26:1112
Zentralblatt MATH: 0105.41604
Digital Object Identifier: doi:10.1063/1.1703774
[15]3 F. Dyson, Statistical theory of the energy levels of complex systems, III, J. Math. Phys. 3 (1962), 166–175.
Mathematical Reviews (MathSciNet): MR26:1113
Zentralblatt MATH: 0105.41604
Digital Object Identifier: doi:10.1063/1.1703775
[16] F. Dyson, Fredholm determinants and inverse scattering problems, Comm. Math. Phys. 47 (1976), no. 2, 171–183.
Mathematical Reviews (MathSciNet): MR53:9993
Zentralblatt MATH: 0323.33008
Digital Object Identifier: doi:10.1007/BF01608375
Project Euclid: euclid.cmp/1103899727
[17] 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.
Mathematical Reviews (MathSciNet): MR91k:82009
Zentralblatt MATH: 0719.35091
Digital Object Identifier: doi:10.1142/S0217979290000504
[18] 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.
Mathematical Reviews (MathSciNet): MR84k:82037
Digital Object Identifier: doi:10.1016/0167-2789(80)90006-8
[19] V. G. Kac, Infinite-Dimensional Lie Algebras, 3d ed., Cambridge Univ. Press, Cambridge, 1990.
Mathematical Reviews (MathSciNet): MR92k:17038
Zentralblatt MATH: 0716.17022
[20] 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.
Mathematical Reviews (MathSciNet): MR94j:82018
Zentralblatt MATH: 0875.35080
[21] 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.
Mathematical Reviews (MathSciNet): MR81g:58017
Zentralblatt MATH: 0449.35080
[22] M. L. Mehta, Random Matrices, 2d ed., Academic Press, Boston, 1991.
Mathematical Reviews (MathSciNet): MR92f:82002
Zentralblatt MATH: 0780.60014
[23] M. L. Mehta, A nonlinear differential equation and a Fredholm determinant, J. Physique I 2 (1992), no. 9, 1721–1729.
Mathematical Reviews (MathSciNet): MR93i:58163
Digital Object Identifier: doi:10.1051/jp1:1992240
[24] H. Peng, The spectrum of random matrices for symmetric ensembles, dissertation, Brandeis Univ., Waltham, Mass., 1997.
[25] C. E. Porter and N. Rosenzweig, Statistical properties of atomic and nuclear spectra, Ann. Acad. Sci. Fenn. Ser. A VI No. 44 (1960), 66.
Mathematical Reviews (MathSciNet): MR22:9150
Zentralblatt MATH: 0092.23304
[26] C. E. Porter and N. Rosenzweig, Repulsion of energy levels in complex atomic spectra, Phys. Rev. 120 (1960), 1698–1714.
[27] 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.
Mathematical Reviews (MathSciNet): MR96d:11059
Zentralblatt MATH: 0831.58045
[28] M. Sato, Soliton equations and the universal Grassmann manifold (by Noumi in Japanese), Math. Lect. Note Ser., vol. 18, Sophia University, Tokyo, 1984.
Zentralblatt MATH: 0541.58001
[29] T. Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83 (1986), no. 2, 333–382.
Mathematical Reviews (MathSciNet): MR87j:14047
Zentralblatt MATH: 0621.35097
Digital Object Identifier: doi:10.1007/BF01388967
[30] N. A. Slavnov, Fredholm determinants and $\tau$-functions, Teoret. Mat. Fiz. 109 (1996), no. 3, 357–371.
Mathematical Reviews (MathSciNet): MR99b:58127
Zentralblatt MATH: 0935.35162
[31] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., New York, 1939.
Mathematical Reviews (MathSciNet): MR1,14b
Zentralblatt MATH: 0023.21505
[32] 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.
Mathematical Reviews (MathSciNet): MR95a:82050
Zentralblatt MATH: 0791.15017
Digital Object Identifier: doi:10.1007/BFb0021444
[33] C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), no. 1, 151–174.
Mathematical Reviews (MathSciNet): MR95e:82003
Zentralblatt MATH: 0789.35152
Digital Object Identifier: doi:10.1007/BF02100489
Project Euclid: euclid.cmp/1104254495
[34] C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Comm. Math. Phys. 161 (1994), no. 2, 289–309.
Mathematical Reviews (MathSciNet): MR95e:82004
Zentralblatt MATH: 0808.35145
Digital Object Identifier: doi:10.1007/BF02099779
Project Euclid: euclid.cmp/1104269903
[35] C. A. Tracy and H. Widom, Fredholm determinants, differential equations and matrix models, Comm. Math. Phys. 163 (1994), no. 1, 33–72.
Mathematical Reviews (MathSciNet): MR95e:82005
Zentralblatt MATH: 0813.35110
Digital Object Identifier: doi:10.1007/BF02101734
Project Euclid: euclid.cmp/1104270379
[36] J. W. van de Leur, The Adler-Shiota-van Moerbeke formula for the BKP hierarchy, J. Math. Phys. 36 (1995), no. 9, 4940–4951.
Mathematical Reviews (MathSciNet): MR96m:58115
Zentralblatt MATH: 0844.35109
Digital Object Identifier: doi:10.1063/1.531352
[37] 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.
Mathematical Reviews (MathSciNet): MR98f:58109
Zentralblatt MATH: 0850.81049
[38] E. P. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambridge Philos. Soc. 47 (1951), 790–798.
Zentralblatt MATH: 0044.44203
[39] 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.
Mathematical Reviews (MathSciNet): MR93e:32028
Zentralblatt MATH: 0757.53049
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