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Universality of a double scaling limit near singular edge points in random matrix models

Bibliographic reference Claeys, Tom ; Vanlessen, M. Universality of a double scaling limit near singular edge points in random matrix models. In: Communications in mathematical physics, Vol. 273, no. 2, p. 499-532 (2007)
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  1. Bleher P. and Its A. (1999). Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem and universality in the matrix model. Ann. Math. 150: 185–266
  2. Bleher P. and Its A. (2003). Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Comm. Pure Appl. Math. 56: 433–516
  3. Bowick M.J. and Brézin E. (1991). Universal scaling of the tail of the density of eigenvalues in random matrix models, Phys. Lett. B 268(1): 21–28
  4. Brézin E., Marinari E. and Parisi G. (1990). A non-perturbative ambiguity free solution of a string model. Phys. Lett. B 242(1): 35–38
  5. Claeys T. and Kuijlaars A.B.J. (2006). Universality of the double scaling limit in random matrix models. Comm. Pure Appl. Math. 59: 1573–1603
  6. Claeys, T., Kuijlaars, A.B.J., Vanlessen, M.: Multi- critical unitary random matrix ensembles and the general Painlevé II equation., to appear in Ann. Math.
  7. Claeys T. and Vanlessen M. (2007). The existence of a real pole- free solution of the fourth order analogue of the Painlevé I equation. Nonlinearity 20: 1163–1184
  8. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann- Hilbert Approach. Courant Lecture Notes 3, New York: New York University, 1999
  9. 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
  10. 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. Comm. Pure Appl. Math. 52: 1335–1425
  11. Deift P., Kriecherbauer T., McLaughlin K.T-R., Venakides S. and Zhou X. (1999). Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52: 1491–1552
  12. Deift, P., McLaughlin, K.T-R.: A continuum limit of the Toda lattice. Vol. 131, Memoirs of the Amer. Math. Soc. 624, Providence, RI: Amer. Math. Soc., 1998
  13. Deift P. and Zhou X. (1993). A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137: 295–368
  14. Dubrovin B., Liu S.-Q. and Zhang Y. (2006). On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasi-triviality of bi-Hamiltonian perturbations. Comm. Pure Appl. Math. 59(4): 559–615
  15. Dubrovin B. (2006). On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour. Commun. Math. Phys. 267: 117–139
  16. Duits M. and Kuijlaars A.B.J. (2006). Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight. Nonlinearity 19: 2211–2245
  17. 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
  18. Hastings S.P. and McLeod J.B. (1980). A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rat. Mech. Anal. 73: 31–51
  19. Kapaev A.A. (1995). Weakly nonlinear solutions of equation P I 2 . J. Math. Sc. 73(4): 468–481
  20. Kawai T., Koike T., Nishikawa Y. and Takei Y. (2004). On the Stokes geometry of higher order Painlevé equations. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. II. Astérisque 297: 117–166
  21. Kuijlaars A.B.J. and McLaughlin K.T-R. (2000). Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields. Comm. Pure Appl. Math. 53: 736–785
  22. Kuijlaars A.B.J., McLaughlin K.T-R., Vanlessen M. and Assche W. (2004). The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials. Adv. Math. 188(2): 337–398
  23. Kudryashov N.A. and Soukharev M.B. (1998). Uniformization and transcendence of solutions for the first and second Painlevé hierarchies, Phys. Lett. A 237(4–5): 206–216
  24. Kuijlaars A.B.J. and Vanlessen M. (2003). Universality for eigenvalue correlations at the origin of the spectrum. Commun. Math. Phys. 243: 163–191
  25. Mehta M.L. (1991). Random Matrices. Academic Press, Boston
  26. Moore G. (1990). Geometry of the string equations. Commun. Math. Phys. 133(2): 261–304
  27. Pastur L. and Shcherbina M. (1997). Universality of the local eigennvalue statistics for a class of unitary invariant random matrix ensembles. J. Stat. Phys. 86(1–2): 109–147
  28. Saff Edward B., Totik Vilmos, Logarithmic Potentials with External Fields, ISBN:9783642081736, 10.1007/978-3-662-03329-6
  29. Shcherbina, M.: Double scaling limit for matrix models with non analytic potentials. http:// arxiv./org/list/cond-math/0511161, 2005
  30. Szegõ, G.: “Orthogonal polynomials”. 3 rd ed., Providence, RI: Amer. Math. Soc. 1974
  31. Vanlessen M. (2003). Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight. J. Approx. Theory 125: 198–237
  32. Vanlessen, M.: Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory., 2005, to appear in Constr. Approx