  We study the largest component of a random (multi)graph on n vertices with a
given degree sequence. We let n tend to infinity. Then, under some regularity
conditions on the degree sequences, we give conditions on the asymptotic shape
of the degree sequence that imply that with high probability all the components
are small, and other conditions that imply that with high probability there is
a giant component and the sizes of its vertex and edge sets satisfy a law of
large numbers; under suitable assumptions these are the only two possibilities.
In particular, we recover the results by Molloy and Reed on the size of the
largest component in a random graph with a given degree sequence.
  We further obtain a new sharp result for the giant component just above the
threshold, generalizing the case of G(n,p) with np=1+omega(n)n^{-1/3}, where
omega(n) tends to infinity arbitrarily slowly.
  Our method is based on the properties of empirical distributions of
independent random variables, and leads to simple proofs.
