Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields

Communications on Pure and Applied Mathematics

Volumn 53, Issue 6, 2000, Pages 736-785

  Kuijlaars, A B J ^a   Mclaughlin, K T R ^b  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b UNIVERSITY OF ARIZONA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034398875     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(200006)53:6<736::AID-CPA2>3.0.CO;2-5     Document Type: Article

References (37)

1. Bleher, P.; Its, A. Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1999), no. 1, 185-266.
2. Boulet de Monvel, A.; Pastur, L.; Shcherbina, M. On the statistical mechanics approach in the random matrix theory: integrated density of states. J. Statist. Phys. 79 (1995), no. 3-4, 585-611.
3. Buyarov, V. S.; Rakhmanov, E. A. Families of equilibrium measures in an external field on the real axis. Mat. Sb. 190 (1999), no. 6, 11-22. English translation in Sb. Math. 190 (1999), no. 6, 791-802.
4. Buyarov, V. S.; Rakhmanov, E. A. Families of equilibrium measures in an external field on the real axis. Mat. Sb. 190 (1999), no. 6, 11-22. English translation in Sb. Math. 190 (1999), no. 6, 791-802.
5. Damelin, S. B.; Kuijlaars, A. B. J. The support of the equilibrium measure in the presence of a monomial external field on [-1,1]. Trans. Amer. Math. Soc. 351 (1999), no. 11, 4561-4584.
6. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T-R. New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95 (1998), no. 3, 388-475.
7. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T-R; Venakides, S.; Zhou, X. Asymptotics for polynomials orthogonal with respect to varying exponential weights. Internat. Math. Res. Notices 1997, no. 16, 759-782.
8. 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), no. 12, 1491-1552.
9. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T-R; Venakides, S.; Zhou, X. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (1999), no. 11, 1335-1425.
10. Deift, P.; McLaughlin, K. T-R. A continuum limit of the Toda lattice. Mem. Amer. Math. Soc. 131 (1998), no. 624.
11. Flaschka, H.; Forest, M. G.; McLaughlin, D. W. Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation. Comm. Pure Appl. Math. 33 (1980), no. 6, 739-784.
12. Gonchar, A. A.; Rakhmanov, E. A. The equilibrium measure and distribution of zeros of extremal polynomials. (Russian) Mat. Sb. (N. S.) 125(167) (1984), no. 1(9), 117-127; translation in Math. USSR-Sb. 53 (1986), 119-130.
13. Gonchar, A. A.; Rakhmanov, E. A. The equilibrium measure and distribution of zeros of extremal polynomials. (Russian) Mat. Sb. (N. S.) 125(167) (1984), no. 1(9), 117-127; translation in Math. USSR-Sb. 53 (1986), 119-130.
14. Jin, S.; Levermore, C. D.; McLaughlin, D. W. The behavior of solutions of the NLS equation in the semiclassical limit. Singular limits of dispersive waves (Lyon, 1991), 235-255, NATO Adv. Sci. Inst. Ser. B Phys., 320. Plenum, New York, 1994.
15. Jin, S.; Levermore, C. D.; McLaughlin, D. W. The semiclassical limit of the defocusing NLS hierarchy. Comm. Pure Appl. Math. 52 (1999), no. 5, 613-654.
16. Johansson, K. On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 (1998), no. 1, 151-204.
17. Kiessling, M. K.-H.; Spohn, H. A note on the eigenvalue density of random matrices. Comm. Math. Phys. 199 (1999), no. 3, 683-695.
18. Kuijlaars, A. B. J. Extremal problems for logarithmic potentials. Approximation theory IX, 201-222. C. K. Chui and L. L. Schumaker, eds. Vanderbilt University Press, Nashville, 1998.
19. Kuijlaars, A. B. J.; Dragnev, P. D. Equilibrium problems associated with fast decreasing polynomials. Proc. Amer. Math. Soc. 127 (1999), no. 4, 1065-1074.
20. 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.
21. 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.
22. 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.
23. 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.
24. Lubinsky, D. S. An update on orthogonal polynomials and weighted approximation on the real line. Rational approximation theory. Acta Appl. Math. 33 (1993), no. 2-3, 121-164.
25. Lubinsky, D. S.; Saff, E. B. Strong asymptotics for extremal polynomials associated with weights on R. Lecture Notes in Mathematics, 1305. Springer, Berlin-New York, 1988.
26. Mehta, M. L. Random matrices. Second edition. Academic Press, Boston, 1991.
27. Mhaskar, H. N.; Saff, E. B. Extremal problems for polynomials with exponential weights. Trans. Amer. Math. Soc. 285 (1984), no. 1, 203-234.
28. 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.
29. Rakhmanov, E. A. Asymptotic properties of orthogonal polynomials on the real axis. (Russian) Mat. Sb. (N.S.) 119(161) (1982), no. 2, 163-203, 303. English translation in Math. Sb. (N. S.) 47 (1984), 155-193.
30. Rakhmanov, E. A. Asymptotic properties of orthogonal polynomials on the real axis. (Russian) Mat. Sb. (N.S.) 119(161) (1982), no. 2, 163-203, 303. English translation in Math. Sb. (N. S.) 47 (1984), 155-193.
31. 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 Mathematics, 1550. Springer, Berlin, 1993.
32. Ransford, T. Potential theory in the complex plane. London Mathematical Society Student Texts, 28. Cambridge University Press, Cambridge, 1995.
33. 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.
34. Stahl, H.; Totik, V. General orthogonal polynomials. Encyclopedia of Mathematics and Its Applications, 43. Cambridge University Press, Cambridge, 1992.
35. Totik, V. Weighted approximation with varying weight. Lecture Notes in Mathematics, 1569. Springer, Berlin, 1994.
36. Tracy, C. A.; Widom, H. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1994), no. 1, 151-174.
37. Venakides, S. The Korteweg-de Vries equation with small dispersion: higher order Lax-Levermore theory. Comm. Pure Appl. Math. 43 (1990), no. 3, 335-361.