Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

Communications on Pure and Applied Mathematics

Volumn 52, Issue 11, 1999, Pages 1335-1425

  Deift, P ^a   Kriecherbauer, T ^b   McLaughlin, K T R ^c   Venakides, S ^d   Zhou, X ^d  

^a NEW YORK UNIVERSITY   (United States)

^b UNIVERSITY OF MUNICH   (Germany)

^c The University of Arizona   (United States)

^d DUKE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033440723     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1     Document Type: Article

References (46)

1. Abramowitz, M.; Stegun, I. A. Handbook of mathematical functions, with formulas, graphs, and mathematical tables. Dover, New York, 1966.
2. Beals, R.; Coifman, R. R. Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math., 37 (1984), 39-90.
3. Beals, R.; Deift, P.; Tomei, C. Direct and inverse scattering on the line. Mathematical Surveys and Monographs, 28. American Mathematical Society, Providence, R.I., 1988.
4. Bleher, P.; Its, A. Asymptotics of orthogonal polynomials and universality in matrix models. Preprint, 1996.
5. Clancey, K. F.; Gohberg, I. Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, 3. Birkhäuser, Basel-Boston, 1981.
6. Deift, P. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. Courant Institute, New York, 1999.
7. Deift, P. A.; Its, A. R.; Zhou, Z. 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), 149-235.
8. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T-R. New results for the asymptotics of orthogonal polynomials and related problems via the Lax-Levermore method. Recent advances in partial differential equations, Venice 1996, 87-104. Proc. Sympos. Appl. Math., 54. Amer. Math. Soc., Providence, R.I., 1998.
9. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T-R. New results for the asymptotics of orthogonal polynomials and related problems via the Lax-Levermore method. Recent advances in partial differential equations, Venice 1996, 87-104. Proc. Sympos. Appl. Math., 54. Amer. Math. Soc., Providence, R.I., 1998.
10. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T-R; Venakides, S.; Zhou, X. Asymptotics for poly-nomials orthogonal with respect to varying exponential weights. Internat. Math. Res. Notices 1997, no. 16, 759-782.
11. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T-R; Venakides, S.; Zhou, X. Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (1999), to appear.
12. Deift, P.; Venakides, S.; Zhou, X. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices 1997, no. 6, 286-299
13. Deift, P.; Zhou, X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. of Math. (2) 137 (1993), no. 2, 295-368.
14. Deift, P. A.; Zhou, X. Asymptotics for the Painlevé II equation. Comm. Pure Appl. Math. 48 (1995), no. 3, 277-337.
15. Deift, P.; Zhou, X. Long time behavior of the non-focusing nonlinear pasturSchrödinger equation - a case study. NS: Lectures in Mathematical Sciences, 5. University of Tokyo, 1994.
16. Dubrovin, B. A. Theta-functions and nonlinear equations. (Russian) Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11-80. Russ. Math. Surv. 36:2 (1981), 11-92.
17. Dubrovin, B. A. Theta-functions and nonlinear equations. (Russian) Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11-80. Russ. Math. Surv. 36:2 (1981), 11-92.
18. Ercolani, N. M.; Levermore, C. D.; Zhang, T. The behavior of the Weyl function in the zero-dispersion KdV limit. Comm. Math. Phys. 183 (1997), no. 1, 119-143.
19. Ercolani, N. M.; Levermore, C. D.; Zhang, T. The KdV zero dispersion limit via Dirichlet spectra and Weyl functions. Preprint, 1997.
20. Farkas, H. M.; Kra, I. Riemann surfaces. Graduate Texts in Mathematics, 71. Springer-Verlag, New York-Berlin, 1992.
21. Fokas, A. S.; Its, A. R.; Kitaev, A. V. Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (1991), no. 2, 313-344.
22. Fokas, A. S.; Its, A. R.; Kitaev, A. V. The isomonodromy approach to matrix models in 2D quantum gravity. Comm. Math. Phys. 147 (1992), no. 2, 395-430.
23. Gaudin, M. Sur la loi limite de l'es pacement des valeurs propres d'une matrice aléatoire. Nucl. Phys. 25 (1961), 447-458.
24. Its, A. R.; Novokshenov, V. Yu. The isomonodromic deformation method in the theory of Painlevé equations. Lecture Notes in Mathematics, 1191. Springer-Verlag, Berlin-New York, 1986.
25. Johansson, K. On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 (1998), no. 1, 151-204.
26. Katz, N. M.; Sarnak, P. Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, R.I., 1999.
27. Lax, P. D.; Levermore, C. D. The small dispersion limit of the Korteweg-de Vries equation. I. II. III. Comm. Pure Appl. Math. 36 (1983), no. 3, 253-290; 36 (1983), no. 5, 571-593; 36 (1983), no. 6, 809-829.
28. Lax, P. D.; Levermore, C. D. The small dispersion limit of the Korteweg-de Vries equation. I. II. III. Comm. Pure Appl. Math. 36 (1983), no. 3, 253-290; 36 (1983), no. 5, 571-593; 36 (1983), no. 6, 809-829.
29. Lax, P. D.; Levermore, C. D. The small dispersion limit of the Korteweg-de Vries equation. I. II. III. Comm. Pure Appl. Math. 36 (1983), no. 3, 253-290; 36 (1983), no. 5, 571-593; 36 (1983), no. 6, 809-829.
30. Lubinsky, D. S. Strong asymptotics for extremal errors and polynomials associated with Erdös-type weights. Pitman Research Notes in Mathematics Series, 202. Longman, Harlow; copublished in the United States with John Wiley, New York, 1989.
31. Mehta, M. L. Random matrices. Second edition. Academic Press, Boston, 1991.
32. Mehta, M. L.; Gaudin, M. On the density of eigenvalues of a random matrix. Nuclear Phys. 18 (1960), 420-427.
33. 4). SIAM J. Math. Anal. 15 (1984), no. 6, 1177-1187,
34. Pastur, L. A. Spectral and probabilistic aspects of matrix models. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 207-242. Math. Phys. Stud., 19. Kluwer, Dordrecht, 1996.
35. Pastur, L. A. Spectral and probabilistic aspects of matrix models. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 207-242. Math. Phys. Stud., 19. Kluwer, Dordrecht, 1996.
36. Pastur, L.; Shcherbina, M. Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Statist. Phys. 86 (1997), no. 1-2, 109-147.
37. Rakhmanov, E. A. Strong asymptotics for orthogonal polynomials. Methods of approximation theory in complex analysis and mathematical physics (Leningrad, 1991), 71-97. Lecture Notes in Math., 1550, Springer, Berlin, 1993.
38. Rakhmanov, E. A. Strong asymptotics for orthogonal polynomials. Methods of approximation theory in complex analysis and mathematical physics (Leningrad, 1991), 71-97. Lecture Notes in Math., 1550, Springer, Berlin, 1993.
39. Reed, M.; Simon, B. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich], New York-London, 1978.
40. Sabat, A. B. One-dimensional perturbations of a differential operator, and the inverse scattering problem. (Russian) Problems in mechanics and mathematical physics (Russian), 279-296, 298. Izdat. "Nauka," Moscow, 1976.
41. Saff, E. B.; Totik, V. Logarithmic potentials with external fields. Appendix B by Thomas Bloom. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 316. Springer, Berlin, 1997.
42. 6/6). J. Approx. Theory 50 (1987), no. 3, 232-293.
43. Simon, B. Trace ideals and their applications. London Mathematical Society Lecture Note Series, 35. Cambridge University Press, Cambridge-New York, 1979.
44. Szegö, G. Orthogonal Polynomials. American Mathematical Society Colloquium Publications, 23. American Mathematical Society, Providence, R.I., 1939.
45. Zhou, X. The Riemann-Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20 (1989), no. 4, 966-986.
46. Zhou, X. Riemann-Hilbert problems and integrable systems. Lecture notes, preprint.