HP Labs Home
Job Openings
Technical Reports
Palo Alto, USA
Bristol, UK

HP Labs Technical Reports

Click here for full text: Postscript PDF

On the Characteristic Polynomial of a Random Unitary Matrix

Hughes, C.P.; Keating, J.P.; O'Connell, Neil


Keyword(s): eigenvalue counting function; large deviations; fluctuations

Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial of a random * unitary matrix, as N * *. First we show that , evaluated at a finite set of distinct points, is asymptotically a collection of iid 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. We also obtain a central limit theorem for 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 , evaluated at a finite set of distinct points, can be obtained for. For higher-order scalings we obtain large deviations results for evaluated at a single point. There is a phase transition at = (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy. Notes: C.P. Hughes and J.P. Keating, School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

31 Pages

Back to Index

HP Bottom Banner
Terms of Use Privacy Statement