The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the "birth of a cut" transition

International Mathematics Research Notices

Volumn 2008, Issue 1, 2008, Pages

  Mo, Man Yue ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67649130853     PISSN: 10737928     EISSN: 16870247     Source Type: Journal    
DOI: 10.1093/imrn/rnn042     Document Type: Article

References (32)

1. Belokolos, E. D., A., I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev. Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer Series in Nonlinear Dynamics. Berlin: Springer, 1995.
2. Bleher, P., and B. Eynard. "Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations." Journal of Physics A 36, no, 12 (2003): 3085-3105.
3. Bleher, P., and A. Its. "Semiclassical asymptotics of orthogonal polynomials, RiemannHilbert problem, and universality in the matrix model." Annals of Mathematics (2) 150, no. 1 (1999): 185-266.
4. Bleher, P., and A. Its. "Double scaling limit in the random matrix model: The Riemann-Hilbert approach." Communications on Pure and Applied Mathematics 56, no. 4 (2003): 433-516.
5. Bleher, P., and A. B. J. Kuijlaars. "Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit." Communications in Mathematical Physics 270, no. 2 (2007): 481-517.
6. Buyarov, V. S., and E. A. Rakhmanov. "On families of measures that are balanced in the external field on the real axis." Matematicheski Sbornik 190, no. 6 (1999): 11-22.
7. Claeys, T. "The birth of a cut in unitary random matrix ensembles." International Mathematics Research Notices (2008) Article ID rnml66,40 pages.
8. Claeys, T., and A. B. J. Kuijlaars. "Universality of the double scaling limit in random matrix models." Communications on Pure and Applied Mathematics 59, no. 11 (2006): 1573-1603.
9. Claeys, T., and A. B. J. Kuijlaars, "Universality in unitary random matrix ensembles when the soft edge meets the hard edge." (2007): preprint arXiv:math-ph/0701003.
10. Claeys, T., A. B. J. Kuijlaars, and M. Vanlessen. "Multi-critical unitary random matrix ensembles and the general Painlevé II equation." (2005): preprint arXiv:math-ph/0508062.
11. Claeys, T., and M. Vanlessen. "Universality of a double scaling limit near singular edge points in random matrix models." Communications in Mathematical Physics 273, no. 2 (2007): 499-532.
12. Deift, P. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes, 3. New York: New York University, 1999.
13. Deift, P., T. Kriecherbauer, and K. T. R. McLaughlin. "New results on the equilibrium measure for logarithmic potentials in the presence of an external field." Journal of Approximation Theory 95, no. 3 (1998): 388-375.
14. Deift, P., T. Kriecherbauer, K. T. R. McLaughlin, and S. Venakides. "Strong asymptotics of orthogonal polynomials with respect to exponential weights." Communications on Pure and Applied Mathematics 52, no. 12 (1999): 1491-1552.
15. Deift, P., T. Kriecherbauer, K. T. R. McLaughlin, and S. Venakides. "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 52, no. 11 (1999): 1335-1425.
16. Duits, M., and A. B. J. Kuijlaars. "Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight." Nonlinearity 19, no. 10 (2006): 2211-45.
17. Dyson, F. J. "Correlation between the eigenvalues of a random matrix." Communications in Mathematical Physics 19 (1970): 235-50.
18. Eynard, B. "Universal distribution of random matrix eigenvalues near the "birth of a cut" transition." Journal of Statistical Mechanics 1 (2006): P07005.
19. Fokas, A. S., A. R. Its, and A. V. Kitaev. "The isomonodromy approach to matrix models in 2D quantum gravity." Communications in Mathematical Physics 147, no. 2 (1992): 395-430.
20. Its, A. R., A. B. J. Kuijlaars, and J. Ostensson. "Critical edge behavior in unitary random matrix ensembles and the thirty-fourth Painleve transcendent." International Mathematics Research Notices (2008). Article ID rnn017, 67 pages.
21. Its, A. R., F. Mezzadri, and M. Y. Mo. "Entanglement entropy in quantum spin chains with finite range interaction." (2008): preprint arXiv:0708.0161.
22. Johansson, K. "On fluctuations of eigenvalues of random Hermitian matrices." Duke Mathematical Journal 91, no. 1 (1998): 151-204.
23. Kuijlaars, A. B. J., and K. T. R. McLaughlin. "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 53, no. 6 (2000): 736-85.
24. Kuijlaars, A. B. J., and M. Vanlessen. "Universality for eigenvalue correlations at the origin of the spectrum." Communications in Mathematical Physics 243, no. 1 (2003): 163-91.
25. Mehta, M. L. Random Matrices. Amsterdam, The Netherlands: Elsevier, 2004.
26. Muskhelishvili, N. I. Singular Integral Equations. Boundary Problems of Function Theory and Their Application to Mathematical Physics. Translation by J. R. M. Radok. Groningen, The Netherlands: Noordhof, 1953.
27. Porter, C. E. ed, Statistical Theories of Spectra: Fluctuations, a Collection of Reprints, Original Papers, with an Introductory Review. New York: Academic Press, 1965.
28. Saff, E. B., and V. Totik. Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften, 316. Berlin, Germany: Springer, 1997.
29. Shcherbina, M. "Double scaling limit for matrix models with non analytic potentials." (2007): preprint arXiv:math-ph/0508062.
30. Szego, G. Orthogonal Polynomials. American Mathematical Society, Colloquium Publications, 23. Providence, RI: American Mathematical Society, 1939.
31. Totik, V. Weighted Approximation with Varying Weight. Lecture Notes in Mathematics, 1569. Berlin: Springer, 1994.
32. Vanlessen, M. "Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight." Journal of Approximation Theory 125, no. 2 (2003): 198-237.