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Transactions of the American Mathematical Society
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
Published electronically: October 15, 2009
MathSciNet review: 2563725
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-09-04542-5
PII: S 0002-9947(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.