On the Characteristic Polynomial of a Random Unitary Matrix

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 lnZ/ √ 1 2 lnN , 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 lnZ 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 lnZ/A, evaluated at a finite set of distinct points, can be obtained for √ lnN A lnN . For higher-order scalings we obtain large deviations results for lnZ/A evaluated at a single point. There is a phase transition at A = lnN (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.