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Published March 2010 | Version Published
Journal Article Open

Powers of large random unitary matrices and Toeplitz determinants

Creators

  • 1. ROR icon KU Leuven
  • 2. ROR icon California Institute of Technology
  • 3. ROR icon Royal Institute of Technology

Abstract

We study the limiting behavior of Tr U^(k(n)), where U is an n x 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 Є N, the random variables TrU^(k_j(n))√min(k_j(n),n), j = 1,...,m, converge to independent standard complex normals.

Additional Information

© 2009 American Mathematical Society. The copyright for this article reverts to public domain after 28 years from publication. Received by editor(s): July 11, 2006 . Received by editor(s) in revised form: April 24, 2007 Posted: October 15, 2009. 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). The presented work was developed while the first author was staying at the Royal Institute of Technology in Stockholm during the spring term of 2006. The authors wish to thank Jens Hoppe for inviting the first author and for his generous hospitality during this period. The authors also wish to thank Zeev Rudnick for drawing attention to the papers [9] and [10]

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Additional details

Identifiers

Eprint ID
17918
Resolver ID
CaltechAUTHORS:20100409-135537147

Funding

Marie Curie Training Network
European Science Foundation Program
FWO-Flanders project
G.0455.04
K.U. Leuven research grant
OT/04/21
Belgian Interuniversity Attraction Pole
P06/02
Göran Gustafsson Foundation

Dates

Created
2010-04-12
Created from EPrint's datestamp field
Updated
2021-11-08
Created from EPrint's last_modified field