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On the characteristic polynomial of a random unitary matrix. (English) Zbl 0987.60039
Summary: We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial $$Z$$ of a random $$N\times N$$ unitary matrix, as $$N\to\infty$$. First we show that $$\ln Z\Bigl/\sqrt{{1\over 2}\ln N}$$, 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 $$\ln Z$$ in a Sobolev space of generalized 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 $$\sqrt{\ln N}\ll A\ll\ln N$$. 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.

##### MSC:
 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 11Z05 Miscellaneous applications of number theory 33C90 Applications of hypergeometric functions 60F10 Large deviations 11M50 Relations with random matrices
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