Communications in Mathematical Physics

, Volume 220, Issue 2, pp 429–451

On the Characteristic Polynomial¶ of a Random Unitary Matrix

Authors

  • C. P. Hughes
    • BRIMS, Hewlett-Packard Labs, Bristol, BS34 8QZ, UK
  • J. P. Keating
    • BRIMS, Hewlett-Packard Labs, Bristol, BS34 8QZ, UK
  • Neil O'Connell
    • BRIMS, Hewlett-Packard Labs, Bristol, BS34 8QZ, UK

DOI: 10.1007/s002200100453

Cite this article as:
Hughes, C., Keating, J. & O'Connell, N. Commun. Math. Phys. (2001) 220: 429. doi:10.1007/s002200100453

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 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 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001