Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations

Journal of Physics A: Mathematical and General

Volumn 36, Issue 12 SPEC. ISS., 2003, Pages 3085-3105

  Bleher, Pavel ^a   Eynard, Bertrand ^b,c  

^a INDIANA UNIVERSITY   (United States)

^b CEA SACLAY   (France)

^c UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037471286     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/36/12/314     Document Type: Article

References (34)

1. [BDE] Bonnet G, David F and Eynard B 2000 Breakdown of universality in multi-cut matrix models J. Phys. A: Math. Gen. 33 6739-68
2. [BDJ] Baik J, Deift P and Johansson K 1999 On the distribution of the length of the longest increasing subsequence of random permutations J. Am. Math. Soc. 12 1119-78
3. [BDJT] Brower R C, Deo N, Jain S and Tan C-I 1993 Symmetry breaking in the double-well Hermitian matrix models Nucl. Phys. B 405 166-87
4. [BI1] Bleher P M and Its A R 1999 Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model Ann. Math. 150 185-266
5. [BI2] Bleher P and Its A 2002 Double scaling limit in the matrix model: the Riemann-Hilbert approach Preprint math-ph/0201003
6. [BIPZ] Brézin E, Itzykson C, Parisi G and Zuber J B 1978 Planar diagrams Commun. Math. Phys. 59 35-51
7. [BIZ] Bessis D, Itzykson C and Zuber J-B 1980 Quantum field theory techniques in graphical enumeration Adv. Appl. Math. 1 109-57
8. [BKa] Brézin E and Kazakov V A 1990 Exactly solvable field theories of closed strings Phys. Lett. B 236 144-50
9. [BPS] Boutet de Monvel A, Pastur L and Shcherbina M 1995 On the statistical mechanics approach in the random matrix theory: integrated density of states J. Stat. Phys. 79 585-611
10. [BZ] Brézin E and Zee A 1993 Universality of the correlations between eigenvalues of large random matrices Nucl. Phys. B 402 613-27
11. [CDM] Crnkovic C, Douglas M and Moore G 1992 Loop equations and the topological phase of multi-cut matrix models Int. J. Mod. Phys. A 7 7693-711
12. [CG] Chen Y and Grava T 2002 Eigenvalue correlations on hyperelliptic Riemann surfaces J Phys. A: Math. Gen. 35 L45-9
13. [CI1] Chen Y and Ismail M 1997 Ladder operators and differential equations for orthogonal polynomials J. Phys. A: Math. Gen. 30 7817-29
14. [CI2] Chen Y and Ismail M 1997 Thermodynamic relations of the Hermitian matrix ensembles J. Phys. A: Math. Gen. 30 6633-54
15. [DDJT] Demeterfi K, Deo N, Jain S and Tan C-I 1990 Multiband structure and critical behavior of matrix models Phys.Rev. D 42 4105-22
16. [DGZ] Di Francesco Ph, Ginsparg P and Zinn-Justin J 1995 2D gravity and random matrices Phys. Rep. 254 133
17. [DKMVZ] 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
18. [DS] Douglas M and Shenker S 1990 Strings in less than one dimension Nucl. Phys. B 335 635-54
19. [DSS] Douglas M, Seiberg N and Shenker S 1990 Flow and instability in quantum gravity Phys. Lett. B 244 381-6
20. [Eyn1] Eynard B 1997 Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices Nucl. Phys. B 506 633-64
21. [Eyn2] Eynard B 2001 A concise expression for the ODE's of orthogonal polynomials Preprint math-ph/0109018
22. [FIK] 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
23. [GM] Gross D and Migdal A 1990 Nonperturbative two-dimensional quantum gravity Phys. Rev. Lett. 64 127-30 Gross D and Migdal A 1990 A nonperturbative treatment of two-dimensional quantum gravity Nucl. Phys. B 340 333-65
24. [GM] Gross D and Migdal A 1990 Nonperturbative two-dimensional quantum gravity Phys. Rev. Lett. 64 127-30 Gross D and Migdal A 1990 A nonperturbative treatment of two-dimensional quantum gravity Nucl. Phys. B 340 333-65
25. [GW] Gross D and Witten E 1980 Possible third-order phase transition in the large-N lattice gauge theory Phys. Rev. D 21 446-53
26. [HMPN] Hollowood T, Miramontes L, Pasquinucci A and Nappi C 1992 Hermitian versus anti-Hermitian one-matrix models and their hierarchies Nucl. Phys. B 373 247-80
27. [IN] Its A and Novokshenov V 1986 The isomonodromic deformation method in the theory of Painlevé equations Lecture Notes in Mathematics vol 1191 (New York: Springer)
28. [Joh] Johansson K 1988 On fluctuations of eigenvalues of random Hermitian matrices Duke Math. J. 91 151-204
29. [Kit] Kitaev A 1991 Turning points of linear systems and double asymptotics of Painlevé transcendents Diff. Geom. Lie Groups Mech. 12 (LOMI vol 187) 53-74
30. [Meh] Mehta M L 1991 Random Matrices, 2nd edn (New York: Academic)
31. [Moo] Moore G 1990 Geometry of the string equations Commun. Math. Phys. 133 261-304
32. [PeS] Periwal V and Shevitz D 1990 Exactly solvable unitary matrix models: multicritical potentials and correlations Nucl. Phys. B 333 731-46
33. [Sze] Szegö G 1967 Orthogonal Polynomials 3rd edn (Providence, RI: American Mathematical Society)
34. [TW] Tracy C and Widom H 1993 Introduction to random matrices Geometric and Quantum Aspects of Integrable Systems (Scheveningen, 1992) (Lecture Notes in Physics vol 424) (Berlin: Springer) pp 103-30