User menu

Accès à distance ? S'identifier sur le proxy UCLouvain

Universality of the double scaling limit in random matrix models

Primary tabs

Claeys, Tom [UCL] Kuijlaars, ABJ []

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painleve equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann-Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann-Hilbert problem is done with the Deift-Zhou steepest-descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. (c) 2005 Wiley Periodicals, Inc.

Document type Article de périodique (Journal article)
Access type Accès restreint
Publication date 2006
Language Anglais
Journal information "Communications on pure and applied mathematics" - Vol. 59, no. 11, p. 1573-1603 (2006)
Peer reviewed yes
issn 0010-3640
Publication status Publié
Affiliation UCL - SC/MATH - Département de mathématique
Links
  1. Akemann, Nuclear Phys B, 519, 682 (1998)
  2. Aptekarev, Comm Math Phys, 259, 367 (2005)
  3. Baik, J Amer Math Soc, 12, 1119 (1999)
  4. Baik, Geom Funct Anal, 10, 702 (2000)
  5. Baik, Comm Math Phys, 223, 627 (2001)
  6. Baik, J Statist Phys, 100, 523 (2000)
  7. Baik, Duke Math J, 109, 205 (2001)
  8. Bleher, J Phys A, 36, 3085 (2003)
  9. Bleher, Ann of Math (2), 150, 185 (1999)
  10. Bleher, Comm Pure Appl Math, 56, 433 (2003)
  11. M. Bleher Pavel, Its Alexander, Asymptotics of the partition function of a random matrix model, 10.5802/aif.2147
  12. Bleher, Comm Math Phys, 252, 43 (2004)
  13. Bleher, Int Math Res Not, 2004, 109 (2004)
  14. Buyarov, Sb Math, 190, 791 (1999)
  15. ; ; Multi-critical unitary random matrix ensembles and the general Painlevé II equation. arXiv: math-ph/0508062, 2005.
  16. Damelin, Trans Amer Math Soc, 351, 4561 (1999)
  17. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. American Mathematical Society, Providence, R.I., 1999.
  18. Deift, J Approx Theory, 95, 388 (1998)
  19. Deift, Comm Pure Appl Math, 52, 1491 (1999)
  20. Deift, Comm Pure Appl Math, 52, 1335 (1999)
  21. Deift, Internat Math Res Notices, 1997, 286 (1997)
  22. Deift, Ann of Math (2), 137, 295 (1993)
  23. Deift, Comm Pure Appl Math, 48, 277 (1995)
  24. Flaschka, Comm Math Phys, 76, 65 (1980)
  25. Fokas, Comm Math Phys, 142, 313 (1991)
  26. Fokas, Comm Math Phys, 147, 395 (1992)
  27. Boundary value problems. Pergamon Press, Oxford-New York-Paris; Addison-Wesley, Reading, Mass.-London, 1966.
  28. Hastings, Arch Rational Mech Anal, 73, 31 (1980)
  29. Its, Nonlinearity, 16, 363 (2003)
  30. Its Alexander R., Novokshenov Victor Yu., The Isomonodromic Deformation Method in the Theory of Painlevé Equations, ISBN:9783540164838, 10.1007/bfb0076661
  31. Riemann-Hilbert analysis for orthogonal polynomials. Orthogonal polynomials and special functions (Leuven, 2002), 167–210. Lecture Notes in Mathematics, 1817. Springer, Berlin, 2003.
  32. Kuijlaars, Comm Pure Appl Math, 53, 736 (2000)
  33. Kuijlaars, Comm Math Phys, 243, 163 (2003)
  34. Random matrices. 2nd edition. Academic Press, Boston, 1991.
  35. Ransford Thomas, Potential Theory in the Complex Plane, ISBN:9780511623776, 10.1017/cbo9780511623776
  36. Saff Edward B., Totik Vilmos, Logarithmic Potentials with External Fields, ISBN:9783642081736, 10.1007/978-3-662-03329-6
  37. Stahl Herbert, Totik Vilmos, General Orthogonal Polynomials : Encyclopedia of Mathematics and its Applications, ISBN:9780511759420, 10.1017/cbo9780511759420
  38. Totik Vilmos, Weighted Approximation with Varying Weight, ISBN:9783540577058, 10.1007/bfb0076133
  39. Tracy, Comm Math Phys, 161, 289 (1994)
Bibliographic reference Claeys, Tom ; Kuijlaars, ABJ. Universality of the double scaling limit in random matrix models . In: Communications on pure and applied mathematics, Vol. 59, no. 11, p. 1573-1603 (2006)
Permanent URL http://hdl.handle.net/2078.1/73337