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On the Characteristic Polynomial of a Random Unitary Matrix
Hughes, C.P.; Keating, J.P.; O'Connell, Neil
HPLBRIMS200017
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, 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 , evaluated at a finite set of distinct points, can be obtained for. For higherorder 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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