HP Labs Technical Reports

On the Characteristic Polynomial of a Random Unitary Matrix

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

HPL-BRIMS-2000-17

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

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