Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle

International Mathematics Research Notices

Volumn , Issue 25, 2004, Pages 1249-1272

  Krasovsky, I V ^a  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

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EID: 2442427555     PISSN: 10737928     EISSN: None     Source Type: Journal    
DOI: 10.1155/s1073792804140221     Document Type: Article

References (31)

1. M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, DC, 1965.
2. N. I. Akhieser, On polynomials orthogonal on a circular arc, Soviet Math. Dokl. 1 (1960), 31-34.
3. J. Baik, P. A. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119-1178.
4. D. Barrios Rolanía and G. López Lagomasino, Ratio asymptotics for polynomials orthogonal on arcs of the unit circle, Constr. Approx. 15 (1999), no. 1, 1-31.
5. M. Bello Hernández and G. López Lagomasino, Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle, J. Approx. Theory 92 (1998), no. 2, 216-244.
6. P. Bleher and A. Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2) 150 (1999), no. 1, 185-266.
7. P. A. Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics, vol. 3, New York University Courant Institute of Mathematical Sciences, New York, 1999.
8. -, Spectral theory and inverse spectral theory for Jacobi operators, AMS-IMS-SIAM Summer Research Conference, AMS-IMS-SIAM, Utah, June 2003.
9. P. A. Deift, A. R. Its, and X. Zhou, 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), no. 1, 149-235.
10. P. A. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), no. 12, 1491-1552.
11. -, 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.
12. P. A. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295-368.
13. J. des Cloizeaux and M. L. Mehta, Asymptotic behavior of spacing distributions for the eigenvalues of random matrices, J. Mathematical Phys. 14 (1973), 1648-1650.
14. F. J. Dyson, Fredholm determinants and inverse scattering problems, Comm. Math. Phys. 47 (1976), no. 2, 171-183.
15. A. S. Fokas, A. R. Its, and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2, 395-430.
16. L. Golinskii, Akhiezer's orthogonal polynomials and Bernstein-Szegö method for a circular arc, J. Approx. Theory 95 (1998), no. 2, 229-263.
17. L. Golinskii, P. Nevai, and W. Van Assche, Perturbation of orthogonal polynomials on an arc of the unit circle, J. Approx. Theory 83 (1995), no. 3, 392-422.
18. I. I. Hirschman Jr., On a formula of Kac and Achiezer, J. Math. Mech. 16 (1967), 167-196.
19. M. Jimbo, T. Miwa, Y. Môri, and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D 1 (1980), no. 1, 80-158.
20. I. V. Krasovsky, Asymptotics for Toeplitz determinants on a circular arc, preprint, 2003, http://www.arxiv.org/abs/math.FA/0401256.
21. -, Some computable Wiener-Hopf determinants and polynomials orthogonal on an arc of the unit circle, preprint, 2003, http://www.arxiv.org/abs/math.FA/0310172.
22. A. B. J. Kuijlaars, K. T.-R. McLaughlin, W. Van Assche, and M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1], to appear in Adv. Math., http://www.arxiv.org/abs/math.CA/0111252.
23. A.B. J. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Int. Math. Res. Not. 2002 (2002), no. 30, 1575-1600.
24. M. L. Mehta, Random Matrices, Academic Press, New York, 1990.
25. G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, Rhode Island, 1959.
26. C. A. Tracy and H. Widom, Introduction to random matrices, Geometric and Quantum Aspects of Integrable Systems (Scheveningen, 1992) (G. F. Helminck, ed.), Lecture Notes in Phys., vol. 424, Springer, Berlin, 1993, pp. 103-130.
27. -, Fredholm determinants, differential equations and matrix models, Comm. Math. Phys. 163 (1994), no. 1, 33-72.
28. H. Widom, The strong Szegö limit theorem for circular arcs, Indiana Univ. Math. J. 21 (1971), 277-283.
29. -, Toeplitz determinants with singular generating functions, Amer. J. Math. 95 (1973), 333-383.
30. -, The asymptotics of a continuous analogue of orthogonal polynomials, J. Approx. Theory 77 (1994), no. 1, 51-64.
31. A. Zhedanov, On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval, J. Approx. Theory 94 (1998), no. 1, 73-106.