  We study the $k$-core of a random (multi)graph on $n$ vertices with a given
degree sequence. In our previous paper [Random Structures Algorithms 30 (2007)
50--62] we used properties of empirical distributions of independent random
variables to give a simple proof of the fact that the size of the giant
$k$-core obeys a law of large numbers as ${{n\to \infty}}$. Here we develop the
method further and show that the fluctuations around the deterministic limit
converge to a Gaussian law above and near the threshold, and to a non-normal
law at the threshold. Further, we determine precisely the location of the phase
transition window for the emergence of a giant $k$-core. Hence, we deduce
corresponding results for the $k$-core in $G(n,p)$ and $G(n,m)$.
