Brownian excursions, critical random graphs and the multiplicative coalescent

Let (Bᵗ (s), 0 s <) be reflecting inhomogeneous Brownian motion with drift t - s at time s, started with Bᵗ (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 Bᵗ (Corollary 2). The dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors x of nonnegative real cluster sizes (xᵢ), and clusters with sizes xᵢ and xⱼ merge at rate xᵢ xⱼ. The multiplicative coalescent is shown to be a Feller process on l₂. The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time -; the existence of such a process is not obvious.