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Applications of Random Matrices in Physics pp 489–513Cite as

LARGE N ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS FROM INTEGRABILITY TO ALGEBRAIC GEOMETRY

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Part of the book series: NATO Science Series II: Mathematics, Physics and Chemistry ((NAII,volume 221))

Abstract

Random matrices play an important role in physics and mathematics [28, 19, 6, 14, 25, 34, 13]. It has been observed more and more in the recent years how deeply random matrices are related to integrability (τ -functions), and algebraic geometry.

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References

  1. M. Bertola, “Bilinear semi-classical moment functionals and their integral representation” J. App. Theory (at press), math.CA/0205160

    Google Scholar 

  2. M. Bertola, B. Eynard, J. Harnad, “Heuristic asymptotics of biorthogonal polynomials” Presentation by B.E. at AMS Northeastern regional meeting, Montreal May 2002.

    Google Scholar 

  3. M. Bertola, B. Eynard, J. Harnad, “Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions" SPHT T04/019. CRM-3169 (2004), xxx, nlin.SI/0410043.

    Google Scholar 

  4. Differential systems for bi-orthogonal polynomials appearing in two-matrix models and the associated Riemann-Hilbert problem. (M. Bertola, B.E., J. Harnad), 60 pages, SPHT 02/097, CRM-2852. Comm. Math. Phys. 243 no.2 (2003) 193–240, xxx, nlin.SI/0208002.

    Google Scholar 

  5. Partition functions for Matrix Models and Isomonodromic Tau Functions. (M. Bertola, B.E., J. Harnad), 17 pages, SPHT 02/050, CRM 2841, J. Phys. A Math. Gen. 36 No 12 (28 March 2003) 3067–3083. xxx, nlin.SI/0204054.

    Google Scholar 

  6. P.M. Bleher and A.R. Its, eds., “Random Matrix Models and Their Applications” MSRI Research Publications 40, Cambridge Univ. Press, (Cambridge, 2001).

    Google Scholar 

  7. P. Bleher, A. Its, “Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model" Ann. of Math.(2) 150, no. 1, 185–266 (1999).

    Google Scholar 

  8. P. Bleher, A. Its, “On asymptotic analysis of orthogonal polynomials via the Riemann- Hilbert method” Symmetries and integrability of difference equations (Canterbury, 1996), 165–177, London Math. Soc. Lecture Note Ser., 255, Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  9. S.S. Bonan, D.S. Clark, “Estimates of the Hermite and the Freud polynomials” J. Approx. Theory 63, 210–224 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Bonnet, F. David, and B. Eynard, “Breakdown of Universality in multi-cut matrix models” J. Phys A33, 6739 (2000), xxx, cond-mat/0003324.

    Google Scholar 

  11. P. Deift, T. Kriecherbauer, K. T. R. McLaughlin, S. Venakides, Z. Zhou, “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).

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Deift, T. Kriecherbauer, K. T. R. McLaughlin, S. Venakides, Z. Zhou, “Strong asymptotics of orthogonal polynomials with respect to exponential weights” Commun. Pure Appl. Math. 52, 1491–1552, (1999).

    Google Scholar 

  13. P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach, Courant (New York University Press, ., 1999).

    Google Scholar 

  14. P. Di Francesco, P. Ginsparg, J. Zinn-Justin, "2D Gravity and Random Matrices” Phys. Rep. 254, 1 (1995).

    Article  MathSciNet  ADS  Google Scholar 

  15. F.J. Dyson, “Correlations between the eigenvalues of a random matrix” Comm. Math. Phys. 19 (1970) 235–250.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Correlation functions of eigenvalues of multi-matrix models, and the limit of a time dependent matrix. (B.E.), 27 pages. SPHT 98/001 DTP 97–59, Journal of Physics A 40 (1998) 8081, xxx, cond-mat/9801075.

    Google Scholar 

  17. Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices. (B.E.), 31 pages. SPHT 97031. Nuc. Phys. B506,3 633–664 (1997). xxx, condmat/ 9707005.

    Google Scholar 

  18. B. Eynard, “Polynômes biorthogonaux, probleme de Riemann-Hilbert et geometrie algebrique” Habilitation “a diriger les recherches, universite Paris VII, (2005).

    Google Scholar 

  19. B. Eynard “An introduction to random matrices” lectures given at Saclay, October 2000, notes available at http://www-spht.cea.fr/articles/t01/014/.

    Google Scholar 

  20. B. Eynard, “Asymptotics of skew orthogonal polynomials” J. Phys A. 34(2001) 7591, cond-mat/0012046.

    Google Scholar 

  21. B. Eynard, “Master loop equations, free energy and correlations for the chain of matrices” JHEP11(2003)018, xxx, hep-th/0309036.

    Google Scholar 

  22. H.M. Farkas, I. Kra, “Riemann surfaces" 2nd edition, Springer Verlag, 1992.

    Google Scholar 

  23. J. Fay, “Theta Functions on Riemann Surfaces” Lectures Notes in Mathematics, Springer Verlag, 1973.

    Google Scholar 

  24. A. Fokas, A. Its, A. Kitaev, “The isomonodromy approach to matrix models in 2D quantum gravity” Commun. Math. Phys. 147, 395–430 (1992).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. T. Guhr, A. Mueller-Groeling, H.A. Weidenmuller, “Random matrix theories in quantum physics: Common concepts” Phys. Rep. 299, 189 (1998).

    Article  MathSciNet  ADS  Google Scholar 

  26. A.R. Its, A.V. Kitaev, and A.S. Fokas, “An isomonodromic Approach in the Theory of Two-Dimensional Quantum Gravity” Usp. Matem. Nauk, 45, 6 (276), 135–136 (1990), (Russian), translation in Russian Math. Surveys, 45, no. 6, 155–157 (1990).

    Google Scholar 

  27. M. Jimbo, T. Miwa and K. Ueno, “Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients I, II, III.” Physica 2D, 306–352 (1981), Physica 2D, 407–448 (1981); ibid., 4D, 26–46 (1981).

    Google Scholar 

  28. V.A. Kazakov, A. Marshakov, “Complex Curve of the Two Matrix Model and its Tau function” J.Phys. A36(2003) 3107–3136, hep-th/0211236.

    Google Scholar 

  29. I. Krichever, “The τ-Function of the Universal Whitham Hierarchy, Matrix Models and Topological Field Theories” Comm. Pure Appl. Math. 47(1994), no. 4, 437–475.

    Article  MathSciNet  Google Scholar 

  30. M.L. Mehta, Random Matrices, 2nd edition, (Academic Press, New York, 1991).

    MATH  Google Scholar 

  31. G. Moore, “Geometry of the string equations” Comm. Math. Phys. 133, no. 2, 261–304 (1990).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  32. G. Szego, “Orthogonal Polynomials” American Mathematical Society, Providence, 1967.

    Google Scholar 

  33. C.A. Tracy and H. Widom, “Fredholm determinants, differential equations and matrix models” Commun. Math. Phys. 161, 289–309 (1994).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. P. Van Moerbeke, Random Matrices and their applications, MSRI-publications 40, 4986 (2000).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Service de Physique Théorique de Saclay, Gif-sur-Yvette Cedex, F-91191, France

    B. Eynard

Authors
  1. B. Eynard
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Editor information

Editors and Affiliations

  1. Laboratoire de Physique Théorique, Ecole Normale Supérieure, Paris, France

    Édouard Brézin

  2. Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, Université Paris-VI, Paris, France

    Vladimir Kazakov

  3. Service de Physique Théorique, CEA Saclay, Gif-sur-Yvette Cedex, France

    Didina Serban

  4. James Frank Institute, University of Chicago, Chicago, IL, U.S.A.

    Paul Wiegmann

  5. Institute of Biochemical Physics, Moscow, Russia

    Anton Zabrodin

  6. ITEP, Moscow, Russia

    Anton Zabrodin

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© 2006 Springer

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Eynard, B. (2006). LARGE N ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS FROM INTEGRABILITY TO ALGEBRAIC GEOMETRY. In: Brézin, É., Kazakov, V., Serban, D., Wiegmann, P., Zabrodin, A. (eds) Applications of Random Matrices in Physics. NATO Science Series II: Mathematics, Physics and Chemistry, vol 221. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4531-X_13

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