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

## 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 |
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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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