On the Characteristic Polynomial of a Random Unitary Matrix

TLDR: In this paper, a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix are presented.

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 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 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