  In 1998, Molloy and Reed showed that, under suitable conditions, if a
sequence of degree sequences converges to a probability distribution $D$, then
the size of the largest component in corresponding $n$-vertex random graph is
asymptotically $\rho(D)n$, where $\rho(D)$ is a constant defined by the
solution to certain equations that can be interpreted as the survival
probability of a branching process associated to $D$. There have been a number
of papers strengthening this result in various ways; here we prove a strong
form of the result (with exponential bounds on the probability of large
deviations) under minimal conditions.
