Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight

Nonlinearity

Volumn 19, Issue 10, 2006, Pages 2211-2245

  Duits, M ^a   Kuijlaars, A B J ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33748874925     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/19/10/001     Document Type: Article

References (28)

1. Baik J, Deift P, McLaughlin K, Miller P and Zhou X 2001 Optimal tail estimates for directed last passage site percolation with geometric random variables Adv. Theor. Math. Phys. 5 1207-50
2. Bleher P and Its A 1999 Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the random matrix model Ann. Math. 150 185-266
3. Bleher P and Its A 2003 Double scaling limit in the random matrix model: the Riemann-Hilbert approach Commun. Pure Appl. Math. 56 433-516
4. Bleher P M and Kuijlaars A B J 2006 Large n limit of Gaussian random matrices with external source, part III: double scaling limit Commun. Math. Phys. at press (Preprint math-ph/0602064)
5. Claeys T and Kuijlaars A B J 2006 Universality of the double scaling limit in random matrix models Commun. Pure Appl. Math. at press (Preprint math-ph/0501074)
6. Claeys T, Kuijlaars A B J and Vanlessen M 2005 Multi-critical unitary random matrix ensembles and the general Painlevé II equation Preprint math-ph/0508062
7. Claeys T and Vanlessen M 2006 Universality of a double scaling limit near singular edge points in random matrix models Preprint math-ph/0607043
8. Costin O 1999 Correlation between pole location and asymptotic behavior for Painlevé I solutions Commun. Pure Appl. Math. 52 461-78
9. Deift P A 1999 Orthogonal polynomials and random matrices: a Riemann-Hilbert approach Courant Lecture Notes in Mathematics vol 3 (New York: New York University, Courant Institute of Mathematical Sciences)
10. Deift P, Kriecherbauer T and McLaughlin K T-R 1998 New results on the equilibrium measure for logarithmic potentials in the presence of an external field J. Approx. Theory 95 388-475
11. Deift P, Kriecherbauer T, McLaughlin K T-R, Venakides S and Zhou X 1999 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 1335-425
12. Deift P, Kriecherbauer T, McLaughlin K T-R, Venakides S and Zhou X 1999 Strong asymptotics of orthogonal polynomials with respect to exponential weights Commun. Pure Appl. Math. 52 1491-552
13. Deift P, Venakides S and Zhou X 1997 New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems Int. Math. Res. Not. 1997 286-99
14. Deift P and Zhou X 1993 steepest A descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation Ann. Math. 137 295-368
15. Di Francesco P 1999 2-D quantum and topological gravities, matrix models, and integrable differential systems. The Painlevé property (CRM Ser. Math. Phys.) (New York: Springer) pp 229-85
16. Di Francesco P, Ginsparg P and Zinn-Justin J 1995 2D gravity and random matrices Phys. Rep. 254 133
17. Fokas A S, Its A R and Kitaev A V 1991 Discrete Painlevé equations and their appearance in quantum gravity Commun. Math. Phys. 142 313-44
18. Fokas A S, Its A R and Kitaev A V 1992 The isomonodromy approach to matrix models in 2D quantum gravity Commun. Math. Phys. 147 395-430
19. Jimbo M and Miwa T 1981 Monodromy preserving deformation of linear ordinary differential equations with rational coefficients II Physica D 2 407-48
20. Joshi N and Kitaev A V 2001 On Boutroux's tritronquée solutions of the first Painlevé equation Stud. Appl. Math. 107 253-91
21. Kapaev A A 2004 Quasi-linear Stokes phenomenon for the Painlevé first equation J. Phys. A: Math. Gen. 37 11149-67
22. Kapaev A A and Kitaev A V 1993 Connection formulae for the first Painlevé transcendent in the complex domain Lett. Math. Phys. 27 243-52
23. Kuijlaars A B J and Martínez-Finkelshtein A 2004 Strong asymptotics for Jacobi polynomials with varying nonstandard parameters J. Anal. Math. 94 195-234
24. Kuijlaars A B J and McLaughlin K T-R 2001 Riemann-Hilbert analysis for Laguerre polynomials with large negative parameter Comput. Methods Funct. Theory 1 205-33
25. Kuijlaars A B J, McLaughlin K T-R, Van Assche W and Vanlessen M 2004 The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1, 1] Adv. Math. 188 337-98
26. Magnus A P 1999 Freud's equations for orthogonal polynomials as discrete Painlevé equations Symmetries and Integrability of Difference Equations (Canterbury, 1996) (London Math. Soc. Lecture Note Ser. vol 255) (Cambridge: Cambridge University Press) pp 228-43
27. Mahoux G 1999 Introduction to the theory of isomonodromic deformations of linear ordinary differential equations with rational coefficients The Painlevé Property (CRM Ser. Math. Phys.) (New York: Springer) pp 35-76
28. Saff E B and Totik V 1997 Logarithmic potentials with external fields Grundlehren der Mathematischen Wissenschaften vol 316 (Berlin: Springer)