The size of the giant component of a random graph with a given degree sequence

Given a sequence of nonnegative real numbers lambda(0),lambda(1),.. that sum to 1, we consider a random graph having approximately lambda(i)n vertices of degree i. In [12] the authors essentially show that if Sigma i(i - 2)lambda(i) > 0 then the graph a.s. has a giant component, while if Sigma i(i - 2)lambda(i) < 0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine epsilon,lambda(0)',lambda(1)'... such that a.s. the giant component, C, has epsilon n + o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n' = n - \C\ vertices, and with lambda(i)'n' of them of degree i.