A classical result by Erdos-Renyi provides a law of large numbers for the size
of the giant component of a sparse random graph. In this talk, I will discuss
some related asymptotics which concern the sizes of connected components, their
numbers of excess edges, and the number of components.
In particular, conditional laws of large numbers will be presented which
establish the existence of additional giant components given that the graph has
a number of giant components of certain size. These asymptotics are derived as
consequences of the limit behaviour of a certain stochastic process associated
with the random graph.