Random matrices and permutations, matrix integrals and integrable systems

van Moerbeke, Pierre

Séminaire Bourbaki : volume 1999/2000, exposés 865-879, Astérisque, no. 276 (2002), Exposé no. 879, 23 p.

[1] M. Adler and P. Van Moerbeke: Symmetric random matrices and the Pfaff Lattice, to appear in: Annals of Mathematics, sept 2000 (solv-int/9903009) and The Hermitian, symmetric and symplectic ensembles and PDE's (math. -phys./009001). | MR

[2] M. Adler and P. Van Moerbeke: The spectrum of coupled random matrices, Annals of Mathematics, 149, 921-976 (1999). | MR | Zbl

[3] M. Adler and P. Van Moerbeke: Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math. (2000) (math.CO/9912143). | MR | Zbl

[4] M. Adler, T. Shiota and P. Van Moerbeke: Random matrices, vertex operators and the Virasoro algebra, Phys. Lett. A 208, 101-112, (1995). And Random matrices, Virasoro algebras and non-commutative KP, Duke Math. J. 94, 379-431 (1998). | MR | Zbl

[5] D. Aldous and P. Diaconis: Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem, Bull. Am. Math. Soc. (new series) 36 (4), 413-432 (1999). | MR | Zbl

[6] J. Baik, P. Deift and K. Johansson: On the distribution of the length of the longest increasing subsequence of random permutations, Math. Archive, Journal Amer. Math. Soc. 12, 1119-1178 (1999) (math. CO/9810105). | MR | Zbl

[7] J. Baik and E. Rains: Algebraic aspects of increasing subsequences, (1999), math.CO/9905083

[8] D. Bessis, Cl. Itzykson and J.-B. Zuber : Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1, 109-157 (1980). | MR | Zbl

[9] P. Biane: Representations of symmetric groups and free probability, preprint (1998). | MR

[10] M. Bowick and E. Brézin: Universal scaling of the tail of the density of eigenvalues in random matrix models, Phys. Letters B 268, 21-28 (1991). | MR

[11] S. Chadha, G. Mahoux, M. L. Mehta : A method of integration over matrix variables : II, J. Phys. A: Math. Gen. 14, 579-586 (1981). | MR

[12] C. M. Cosgrove: Chazy classes IX-XII of third-order differential equations, Stud. Appl. Math. 104, 3, 171-228 (2000). | MR | Zbl

[13] P. Diaconis, M. Shashahani: On the eigenvalues of random matrices J. Appl. Prob., suppl. in honour of Takàcs 31A, 49-61 (1994). | MR | Zbl

[14] P. Erdös and G. Szekeres: A combinatorial theorem in geometry, Compositio Math., 2, 463-470 (1935). | JFM | Numdam

[15] A. S. Fokas, A. R. Its, A. V. Kitaev: The isomonodromy approach to matrix models in $2d$ quantum gravity, Comm. Math. Phys., 147, 395-430 (1992). | MR | Zbl

[16] P. J. Forrester: The spectrum edge of random matrix ensembles, Nucl. Phys. B, (1993). | MR | Zbl

[17] I. M. Gessel: Symmetric functions and $P$-recursiveness, J. of Comb. Theory, Ser A, 53, 257-285 (1990). | MR | Zbl

[18] J. M. Hammersley: A few seedlings of research, Proc. Sixth. Berkeley Symp. Math. Statist. and Probability, Vol. 1, 345-394, University of California Press (1972). | MR | Zbl

[19] M. Hisakado: Unitary matrix models and Painlevé III, Mod. Phys. Letters, A 11 3001-3010 (1996). | MR | Zbl

[20] Cl. Itzykson, J.-B. Zuber: The planar approximation, J. Math. Phys. 21, 411-421 (1980). | MR | Zbl

[21] M. Jimbo, T. Miwa, Y. Mori and M. Sato: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica 1D, 80-158 (1980). | MR

[22] K. Johansson: The Longest Increasing Subsequence in a Random Permutation and a Unitary Random Matrix Model, Math. Res. Lett., 5, no. 1-2, 63-82 (1998). | MR | Zbl

[23] B. F. Logan and L. A. Shepp: A variational problem for random Young tableaux, Advances in Math., 26, 206-222 (1977). | MR | Zbl

[24] R. D. Kamien, H. D. Politzer, M. B. Wise: Universality of random-matrix predictions for the statistics of energy levels Phys. rev. letters 60,1995-1998 (1988). | MR

[25] M. Kontsevich: Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147, 1-23 (1992). | MR | Zbl

[26] D. Knuth: "The Art of Computer programming, Vol III: Searching and Sorting", Addison-Wesley, Reading, MA, 1973. | MR | Zbl

[27] G. Mahoux, M. L. Mehta: A method of integration over matrix variables: IV, J. Phys. I (France) 1, 1093-1108 (1991). | MR

[28] M. L. Mehta: Random matrices, 2nd ed. Boston: Acad. Press, 1991. | MR | Zbl

[29] T. Nagao, M. Wadati: Correlation functions of random matrix ensembles related to classical orthogonal polynomials, J. Phys. Soc of Japan, 60 3298-3322 (1991). | MR

[30] A. M. Odlyzko: On the distribution of spacings between the zeros of the zeta function 48, 273-308 (1987). | MR | Zbl

[31] A. Okounkov: Random matrices and random permutations (1999), math.CO/ 9903176 | MR

[32] L. A. Pastur: On the universality of the level spacing distribution for some ensembles of random matrices, Letters Math. Phys., 25 259-265 (1992). | MR | Zbl

[33] E. M. Rains: Topics in Probability on compact Lie groups, Harvard University doctoral dissertation, (1995). 5B

[34] E. M. Rains: Increasing subsequences and the classical groups, Elect. J. of Combinatorics, 5, R12 (1998). | MR | Zbl

[35] C. A. Tracy and H. Widom: Level-Spacings distribution and the Airy kernel, Commun. Math. Phys., 159, 151-174 (1994). | MR | Zbl

[36] C. A. Tracy and H. Widom: Random unitary matrices, permutations and Painlevé (1999), math.CO/9811154 | MR

[37] C. A. Tracy and H. Widom: On the distribution of the lengths of the longest monotone subsequences in random words (1999), math.CO/9904042

[38] S. M. Ulam: Monte Carlo calculations in problems of mathematical physics, in Modern Mathematics for the Engineers, E.F. Beckenbach ed., McGraw-Hill, 261- 281 (1961). | MR

[39] P. Van Moerbeke: Integrable lattices: random matrices and permutations, MSRI-volume on Random matrices and exactly solvable models, Eds.: P. Bleher, A. Its, Oxford University press, 2000 (math.CO/00-10). | Zbl

[40] A. M. Vershik and S. V. Kerov: Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux, Soviet Math. Dokl., 18, 527-531 (1977). | Zbl