Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann-Hilbert Problem

Communications in Mathematical Physics

Volumn 243, Issue 2, 2003, Pages 193-240

  Bertola, M ^a,b   Eynard, B ^b,c   Harnad, J ^a,b  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

^b Gina Cody School of Engineering and Computer Science   (Canada)

^c CEA SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0346995411     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-003-0934-1     Document Type: Article

References (28)

1. Bertola, M., Eynard, B., Harnad, J.; An ansatz for the solution of the Riemann-Hilbert problem for biorthogonal polynomials. In preparation
2. Bertola, M., Eynard, B., Harnad, J.: Duality: Biorthogonal Polynomials and Multi-Matrix Models. Commun. Math. Phys. 229(1), 73-120 (2002)
3. Bertola, M., Eynard, B., Harnad, J.: Duality of spectral curves arising in two-matrix models. Theor. Math. Phys 134(1), (2003)
4. Bertola, M., Eynard, B., Harnad, J.: Genus zero large n asymptotics of bi-orthogonal polynomials involved in the random 2-matrix model. Presentation by B.E. at AMS Northeastern Regional Meeting, Montreal, May 3-5, 2002
5. Bertola, M.: Bilinear semi-classical moment functionals and their integral representation. J. App. Theory 121, 71-99 (2003)
6. Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. (2) 150(1), 185-266 (1999)
7. Bonnet, G., David, F., Eynard, B.: Breakdown of universality in multi-cut matrix models. J. Phys. A 33, 6739-6768 (2000)
8. Chihara, T.S.: An introduction to orthogonal polynomials. Mathematics and its Applications. Vol. 13 New York-London-Paris: Gordon and Breach Science Publishers, 1978
9. Daul, J.M., Kazakov, V., Kostov, I.K.: Rational Theories of 2D Gravity from the Two-Matrix Model. Nucl. Phys B409, 311-338 (1993), hep-th/9303093.
10. Deift, P., Kriecherbauer, T., McLaughlin, K.T.R., Venakides, S., Zhou, Z.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math 52, 1335-1425 (1999)
11. Deift, P., Kriecherbauer, T., McLaughlin, K.T.R., Venakides, S., Zhou, Z.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math 52, 1491-1552 (1999)
12. Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D Gravity and Random Matrices. Phys. Rep. 254, 1 (1995)
13. Ercolani, N.M., McLaughlin, K.T.-R.: Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model. Physica D 152-153, 232-268 (2001)
14. Eynard, B.: Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices. Nucl. Phys. B 506, 633 (1997), cond-mat/9707005
15. Eynard, B.: Correlation functions of eigenvalues of multi-matrix models, and the limit of a time dependent matrix. J. Phys. A: Math. Gen 31, 8081 (1998), cond-mat/9801075
16. Eynard, B., Mehta, M.L.: Matrices coupled in a chain: eigenvalue correlations. J. Phys. A: Math. Gen 31, 4449 (1998), cond-mat/9710230
17. Fokas, A., Its, A., Kitaev, A.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys 147, 395-430 (1992)
18. Guionnet, A., Zeitouni A.: Large deviations asymptotics for spherical integrals. J. F. A. 188, 461-515 (2002)
19. Its, A.R., Kitaev, A.V., Fokas, A.S.: Matrix models of two-dimensional quantum gravity and isomonodromy solutions of discrete Painleve equations. Zap. Nauch. Sem. LOMI 187, 3-30 (1991) (Russian), translation in J. Math. Sci. 73(4), 415-429 (1995)
20. Its, A.R., Kitaev, A.V., Fokas, A.S.: Matrix models of two-dimensional quantum gravity and isomonodromy solutions of discrete Painleve equations. Zap. Nauch. Sem. LOMI 187, 3-30 (1991) (Russian), translation in J. Math. Sci. 73(4), 415-429 (1995)
21. Its, A.R., Kitaev, A.V., Fokas, A.S.: An isomonodromic Approach in the Theory of Two-Dimensional Quantum Gravity. Usp. Matem. Nauk, 45(6), 135-136, 276 (1990) (Russian), translation in Russ Math. Surveys 45(6), 155-157 (1990)
22. Its, A.R., Kitaev, A.V., Fokas, A.S.: An isomonodromic Approach in the Theory of Two-Dimensional Quantum Gravity. Usp. Matem. Nauk, 45(6), 135-136, 276 (1990) (Russian), translation in Russ Math. Surveys 45(6), 155-157 (1990)
23. Kapaev, A.A.: The Riemann-Hilbert problem for the bi-orthogonal polynomials. nlin.SI/0207036
24. Kazakov, V.A.: Ising model on a dynamical planar random lattice: exact solution. Phys Lett. A119, 140-144 (1986)
25. Matytsin, A.: On the large N limit of the Itzykson Zuber Integral. Nuc. Phys B411, 805 (1994), hep-th/9306077
26. Mehta, M.L.: Random Matrices, Second edition. New York: Academic Press, 1991
27. Szegö, G.: Orthogonal Polynomials. Providence, Rhode Island: AMS, 1939
28. Ueno, K., Takasaki, K.: Toda Lattice Hierarchy. Adv. Studies Pure Math. 4, 1-95 (1984)