Brownian Excursions, Critical Random Graphs and the Multiplicative Coalescent

Let (B t (s); 0 s < 1) be reeecting inhomogeneous Brownian motion with drift t ? s at time s, started with B t (0) = 0. Consider the random graph G(n; n ?1 +tn ?4=3), whose largest components have size of order n 2=3. Normalizing by n ?2=3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (Corollary 2). The dynamics of merging of components as t increases are abstracted to deene the multiplicative coalescent process. The states of this process are vectors x of nonnegative real cluster sizes (x i), and clusters with sizes x i and x j merge at rate x i x j. The multiplicative coalescent is shown to be a Feller process on l 2. The random graph limit speciies the standard multiplicative coalescent, which starts from innnitesimally small clusters at time ?1: the existence of such a process is not obvious.