Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Powers of large random unitary matrices and Toeplitz determinants


Authors: Maurice Duits and Kurt Johansson
Journal: Trans. Amer. Math. Soc. 362 (2010), 1169-1187
MSC (2000): Primary 60B15; Secondary 47B35, 15A52, 60F05
DOI: https://doi.org/10.1090/S0002-9947-09-04542-5
Published electronically: October 15, 2009
MathSciNet review: 2563725
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the limiting behavior of $ \operatorname{Tr}U^{k(n)}$, where $ U$ is an $ n\times n$ random unitary matrix and $ k(n)$ is a natural number that may vary with $ n$ in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szegö limit theorem for Toeplitz determinants associated to symbols depending on $ n$ in a particular way. As a consequence of this result, we find that for each fixed $ m\in \mathbb{N}$, the random variables $ \operatorname{Tr}U^{k_j(n)}/\sqrt{\min(k_j(n),n)}$, $ j=1,\ldots,m$, converge to independent standard complex normals.


References [Enhancements On Off] (What's this?)

  • 1. Basor, E., Widom, H., On a Toeplitz determinant identity of Borodin and Okounkov, Integral Equations Operator Theory, 2000, 37(4), 397-401. MR 1780119 (2001g:47042b)
  • 2. Basor, E., Widom, H., Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols, J. Funct. Anal., 1983, 3, 387-413. MR 695420 (85d:47026)
  • 3. Borodin, A., Okounkov, A., A Fredholm determinant formula for Toeplitz determinants, Integral Equations Operator Theory, 2000, 37(4), 386-396. MR 1780118 (2001g:47042a)
  • 4. Böttcher, A., Silbermann, B., Introduction to large truncated Toeplitz matrices, Springer-Verlag, New York, 1999. MR 1724795 (2001b:47043)
  • 5. Diaconis, P., Patterns in eigenvalues: The 70th Josiah Willard Gibbs Lecture, Bull. Amer. Math. Soc., 2003, 40(2), 155-178. MR 1962294 (2004d:15017)
  • 6. Diaconis, P., Evans, S., Linear functionals of eigenvalues of random matrices, Trans. Amer. Math. Soc., 2001, 353, 2615-2633. MR 1828463 (2002d:60003)
  • 7. Diaconis, P., Shahshahani, M., On the eigenvalues of random matrices, In Studies in Applied Probability. J. Appl. Probab.: Special Vol. 31A, 1994, 49-62. MR 1274717 (95m:60011)
  • 8. Geronimo, J. S., Case, K. M., Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys., 1979, 20(2), 299-310. MR 519213 (80f:81100)
  • 9. Hughes, C. P., Rudnick, Z., Linear statistics of low-lying zeros of $ L$-functions, Quart. J. Math., 2003, 54(3), 309-333. MR 2013141 (2005a:11131)
  • 10. Hughes, C. P., Rudnick, Z., Mock Gaussian behavior for linear statistics of classical compact groups, J. Phys. A, 2003, 36(2), 2919-2932. MR 1986399 (2004e:60012)
  • 11. Rains, E., High powers of random elements of compact Lie groups, Probab. Theory Related Fields, 1997, 107, 219-241. MR 1431220 (98b:15026)
  • 12. Soshnikov, A., The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Prob., 2000, 28, 1353-1370. MR 1797877 (2002f:15035)
  • 13. Widom, H., Asymptotic behavior of block Toeplitz matrices and determinants, II, Adv. in Math., 1976, 21(1), 1-29. MR 0409512 (53:13266b)
  • 14. Wieand, K., Eigenvalue distributions of random unitary matrices, Probab. Theory Related Fields, 2002, 123(2), 202-224. MR 1900322 (2003b:60016)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60B15, 47B35, 15A52, 60F05

Retrieve articles in all journals with MSC (2000): 60B15, 47B35, 15A52, 60F05


Additional Information

Maurice Duits
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium
Address at time of publication: Department of Mathematics, California Institute of Technology, 1200 E. California Boulevard, Pasadena, California 91101
Email: maurice.duits@wis.kuleuven.be

Kurt Johansson
Affiliation: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Email: kurtj@kth.se

DOI: https://doi.org/10.1090/S0002-9947-09-04542-5
Received by editor(s): July 11, 2006
Received by editor(s) in revised form: April 24, 2007
Published electronically: October 15, 2009
Additional Notes: The first author is a research assistant of the Fund for Scientific Research–Flanders and was supported by the Marie Curie Training Network ENIGMA, European Science Foundation Program MISGAM, FWO-Flanders project G.0455.04, K.U. Leuven research grant OT/04/21 and Belgian Interuniversity Attraction Pole P06/02
The second author was supported by the Göran Gustafsson Foundation (KVA)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society