UC Berkeley

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 script G sign(n, n-1 + tn-4/3), whose largest components have size of order n2/3. 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 xixj. 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.