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On the characteristic polynomial of a random unitary matrix

Hughes, C.P., Keating, J.P. and O'Connell, N. (2001) On the characteristic polynomial of a random unitary matrix. Communications in Mathematical Physics, 220 (2). pp. 429-451. ISSN 1432-0916

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Abstract

We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for . For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A= ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.

Item Type: Article
Institution: The University of York
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 10 Aug 2009 10:36
Last Modified: 10 Aug 2009 10:36
Published Version: http://dx.doi.org/10.1007/s002200100453
Status: Published
Publisher: Springer Verlag (Germany)
Identification Number: 10.1007/s002200100453
URI: http://eprints.whiterose.ac.uk/id/eprint/5903

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