  We present the results of systematic numerical computations relating to the
extreme value statistics of the characteristic polynomials of random unitary
matrices drawn from the Circular Unitary Ensemble (CUE) of Random Matrix
Theory. In particular, we investigate a range of recent conjectures and
theoretical results inspired by analogies with the theory of
logarithmically-correlated Gaussian random fields. These include phenomena
related to the conjectured freezing transition. Our numerical results are
consistent with, and therefore support, the previous conjectures and theory. We
also go beyond previous investigations in several directions: we provide the
first quantitative evidence in support of a correlation between extreme values
of the characteristic polynomials and large gaps in the spectrum, we
investigate the rate of convergence to the limiting formulae previously
considered, and we extend the previous analysis of the CUE to the C$\beta$E
which corresponds to allowing the degree of the eigenvalue repulsion to become
a parameter.
