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. 429451. ISSN 14320916
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, lowerorder 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 higherorder 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.
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Authors/Creators: 


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 