The multiplicative coalescent is a Markov process taking values in ordered @math . It is a mean-field process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (1998) each extreme eternal version @math of the multiplicative coalescent was described in three different ways. One of these specifications matches the (marginal) law of @math to that of the ordered excursion lengths above past minima of @math , where @math is a certain Levy-type process which (modulo shift and scaling) has infinitesimal drift @math at time @math . Using a modification of the breadth-first-walk construction from Aldous (1997) and Aldous and Limic (1998), and some new insight from the thesis by Uribe (2007), this work settles an open problem (3) from Aldous (1997), in the more general context of Aldous and Limic (1998). Informally speaking, @math is entirely encoded by @math , and contrary to Aldous' original intuition, the evolution of time for @math does correspond to the linear increase in the constant part of the drift of @math . In the "standard multiplicative coalescent" context of Aldous (1997), this result was first announced by Armendariz in 2001, and obtained in a recent preprint by Broutin and Marckert, who simultaneously account for the process of excess edge counts (or marks). The novel argument presented here is based on a sequence of relatively elementary observations. Some of its components (for example, the new dynamic random graph construction via "simultaneous" breadth-first walks) are of independent interest, and may be useful for obtaining more sophisticated asymptotic results on near critical random graphs and related processes.
Let (B t (s), 0 ≤ s < ∞) be reflecting inhomogeneous Brownian motion with drift t - s at time s, started with B t (0) = 0. Consider the random graph script G sign(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 define 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 specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time -∞; the existence of such a process is not obvious. The multiplicative coalescent @math is a @math -valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From random graph asymptotics it is known (Aldous (1997)) that there exists a standard version of this process starting with infinitesimally small clusters at time @math . In this paper, stochastic calculus techniques are used to describe all versions @math of the multiplicative coalescent. Roughly, an extreme version is specified by translation and scale parameters, and a vector @math of relative sizes of large clusters at time @math . Such a version may be characterized in three ways: via its @math behavior, via a representation of the marginal distribution @math in terms of excursion-lengths of a Levy-type process, or via a weak limit of processes derived from the standard version via a "coloring" construction.