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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn017, 67 pages, doi:10.1093/imrn/rnn017 published on March 10, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Critical Edge Behavior in Unitary Random Matrix Ensembles and the Thirty-Fourth Painlevé Transcendent

Alexander R. Its1, Arno B. J. Kuijlaars2 and Jörgen Östensson3

1 Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202-3216, USA
2 Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium
3 Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden

Correspondence: Correspondence to be sent to: arno.kuijlaars{at}wis.kuleuven.be

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles Formula , with {alpha} > – 1/2, defined on n x n Hermitian matrices M. Assuming that the limiting mean eigenvalue density is regular and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as n, N -> {infty} such that n2/3(n/N – 1) = O(1). We use the Deift–Zhou steepest descent method for the Riemann–Hilbert problem for polynomials orthogonal with respect to the weight |x|2{alpha}e–NV(x). Our main attention is on the construction of a local parametrix near the origin by means of the {psi}-functions associated with a distinguished solution of the Painlevé XXXIV equation.



References

  1. Abramowitz M., Stegun I. Handbook of Mathematical Functions (1992) New York: Dover Publications. Reprint of the 1972 edition.
  2. Bleher P., Bolibruch A., Its A., Kapaev A. Linearization of the P34 equation of Painlevé–Gambier. Unpublished manuscript.
  3. Bleher P., Its A. Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and universality in the matrix model. Annals of Mathematics (1999) 150:185–266.[CrossRef][ISI]
  4. Bleher P., Its A. Double scaling limit in the random matrix model: The Riemann–Hilbert approach. Communications on Pure and Applied Mathematics (2003) 56:433–516.[CrossRef][ISI]
  5. Borodin A., Deift P. Fredholm determinants, Jimbo–Miwa–Ueno {tau}-functions, and representation theory. Communications on Pure and Applied Mathematics (2002) 55:1160–230.[CrossRef][ISI]
  6. Bowick M. J., Brézin E. Universal scaling of the tail of the density of eigenvalues in random matrix models. Physics Letters B (1991) 268:21–8.[CrossRef][ISI]
  7. Carlson F. Sur une classe de séries de Taylor. (1914) Dissertation, Uppsala, Sweden.
  8. Claeys T., Kuijlaars A. B. J. Universality of the double scaling limit in random matrix models. Communications on Pure and Applied Mathematics (2006) 59:1573–603.[CrossRef][ISI]
  9. Claeys T., Kuijlaars A. B. J., Vanlessen M. Multi-critical unitary random matrix ensembles and the general Painlevé 2 equation. Annals of Mathematics. (forthcoming).
  10. Claeys T., Vanlessen M. Universality of a double scaling limit near singular edge points in random matrix models. Communications in Mathematical Physics (2007) 273:499–532.[CrossRef][ISI]
  11. Deift P. Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. (1999) New York: New York University. Courant Lecture Notes 3.
  12. Deift P., Gioev D. Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Communications on Pure and Applied Mathematics (2007) 60:867–910.[CrossRef][ISI]
  13. Deift P., Kriecherbauer T., McLaughlin K.T-R. New results on the equilibrium measure for logarithmic potentials in the presence of an external field. Journal of Approximation Theory (1998) 95:388–475.[CrossRef][ISI]
  14. Deift P., Kriecherbauer T., McLaughlin K.T-R, Venakides S., Zhou X. 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 (1999) 52:1335–425.[CrossRef][ISI]
  15. Deift P., Kriecherbauer T., McLaughlin K.T-R, Venakides S., Zhou X. Strong asymptotics of orthogonal polynomials with respect to exponential weights. Communications on Pure and Applied Mathematics (1999) 52:1491–552.[CrossRef][ISI]
  16. Deift P., Zhou X. A steepest descent method for oscillatory Riemann–Hilbert problems: Asymptotics for the MKdV equation. Annals of Mathematics (1993) 137:295–368.[CrossRef][ISI]
  17. Deift P., Zhou X. Perturbation theory for infinite-dimensional integrable systems on line. A case study. Acta Mathematica (2002) 188:163–262.[CrossRef][ISI]
  18. Deift P., Zhou X. A priori Lp-estimates for solutions of Riemann–Hilbert problems. International Mathematics Research Notices (2002) 2002:2121–54.[Abstract/Free Full Text]
  19. Deift P., Zhou X. Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Communications on Pure and Applied Mathematics (2003) 56:1029–77.[CrossRef][ISI]
  20. Duits M., Kuijlaars A. B. J. Painlevé 1 asymptotics for orthogonal polynomials with respect to a varying quadratic weight. Nonlinearity (2006) 19:2211–45.[CrossRef][ISI]
  21. Dyson F. J. Correlation between the eigenvalues of a random matrix. Communications in Mathematical Physics (1970) 19:235–250.[CrossRef][ISI]
  22. Flaschka H., Newell A. C. Monodromy and spectrum-preserving deformations 1. Communications in Mathematical Physics (1980) 76:65–116.[CrossRef][ISI]
  23. Fokas A. S., Its A. R., Kitaev A. V. The isomonodromy approach to matrix models in 2D quantum gravity. In: Communications in Mathematical Physics (1992) 147:395–430.[CrossRef][ISI]
  24. Fokas A. S., Its A. R., Kapaev A. A., Novokshenov V. Yu. Painlevé Transcendents, the Riemann–Hilbert approach. (2006) Providence, RI: American Mathematical Society. Mathematical Surveys and Monographs 128.
  25. Fokas A. S., Zhou X. On the solvability of Painlevé 2 and 4. Communications in Mathematical Physics (1992) 144:601–22.[CrossRef][ISI]
  26. Forrester P. J. The spectrum edge of random matrix ensembles. Nuclear Physics B (1993) 402:709–28.[CrossRef][ISI]
  27. Harnad J., Its A. R. Integrable Fredholm operators and dual isomonodromic deformations. Communications in Mathematical Physics (2002) 226:497–530.[CrossRef][ISI]
  28. Hastings S. P., McLeod J. B. A boundary value problem associated with the second Painlevé transcendent and the Korteweg–de Vries equation. Archive for Rational Mechanics and Analysis (1980) 73:31–51.[CrossRef][ISI]
  29. Ince E. L. Ordinary Differential Equations (1956) New York: Dover.
  30. Its A. R., Kapaev A. A. The irreducibility of the second Painlevé equation and the isomonodromy method. In: Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear—Howls C. J., Kawai T., Takei Y., eds. (2000) Kyoto, Japan: Kyoto University Press. 209–22.
  31. Jimbo M., Miwa T., Ueno K. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. Physica D (1981) 2:306–52.[CrossRef]
  32. Kamvissis S., McLaughlin K. D. T-R, Miller P. D. Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (2003) Princeton, NJ: Princeton University Press. Annals of Mathematics Studies 154.
  33. Kapaev A. A. Global asymptotics of the second Painlevé transcendent. Physics Letters A (1992) 167:356–62.
  34. Kapaev A. A. Quasi-linear Stokes phenomenon for the Hastings–McLeod solution of the second Painlevé equation. (2004) preprint arXiv:nlin.SI/0410009.
  35. Kapaev A. A., Hubert E. A note on the Lax pairs for Painlevé equations. Journal of Physic Mathematical and General (1999) 32:8145–56.[CrossRef][ISI]
  36. Kuijlaars A. B. J., McLaughlin K. T-R. 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 (2000) 53:736–85.[CrossRef][ISI]
  37. Kuijlaars A. B. J., Vanlessen M. Universality for eigenvalue correlations at the origin of the spectrum. Communications in Mathematical Physics (2003) 243:163–91.[CrossRef][ISI]
  38. Mehta M. L. Random Matrices (1991) 2nd. ed. Boston, MA: Academic Press.
  39. Moore G. Matrix models of 2D gravity and isomonodromic deformations. Progress of Theoretical Physics Supplement (1990) 102:255–85.[CrossRef]
  40. Saff E. B., Totik V. Logarithmic Potentials with External Fields (1997) New York: Springer.
  41. Reed M., Simon B. Methods of Modern Mathematical Physics 4 (1978) New York, London: Academic Press.
  42. Tracy C., Widom H. Level spacing distributions and the Airy kernel. Communications in Mathematical Physics (1994) 159:151–74.[CrossRef][ISI]
  43. Tracy C., Widom H. Airy kernel and Painlevé 2. In: Isomonodromic Deformations and Applications in Physics—Harnad J., Its A., eds. (2002) Providence, RI: American Mathematical Society. 85–96. CRM Proceedings & Lecture Notes 31.
  44. Zhou X. The Riemann–Hilbert problem and inverse scattering. SIAM Journal on Mathematical Analysis (1989) 20:966–86.[CrossRef][ISI]

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This Article
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Right arrow Articles by Östensson, J.
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