  We consider global fluctuations of the spectrum of the GUE. Using results on
the linear statistics of such matrices as well as variance bounds on the
eigenvalues, we show that under a suitable scaling, global fluctuations of the
spectrum can be asymptotically described in terms of a logarithmically
correlated Gaussian field. We also discuss briefly connections between
different objects in RMT giving rise to log-correlated fields: the logarithm of
the absolute value of the characteristic polynomial, the eigenvalue counting
function, and the field of fluctuations of the eigenvalues around their
expected locations.
