An old approach to the giant component problem

In 1998, Molloy and Reed showed that, under suitable conditions, if a sequence dndn of degree sequences converges to a probability distribution D  , then the proportion of vertices in the largest component of the random graph associated to dndn is asymptotically ρ(D)ρ(D), where ρ(D)ρ(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.