THE ENTRANCE BOUNDARY OF THE MULTIPLICATIVE COALESCENT
David J. Aldous and Vlada Limic
The multiplicative coalescent $X(t)$ is a $l^2$-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 {\it standard} version of
this process starting with infinitesimally small clusters at
time $- \infty$.
In this paper, stochastic calculus techniques are used to
describe all versions $(X(t);- \infty < t < \infty)$ of the
multiplicative coalescent. Roughly, an extreme version is
specified by translation and scale parameters, and a vector
$c \in l^3$ of relative sizes of large clusters at time
$- \infty$. Such a version may be characterized in three ways:
via its $t \to - \infty$ behavior, via a representation of
the marginal distribution $X(t)$ in terms of excursion-lengths
of a L{\'e}vy-type process, or via a weak limit of processes
derived from the standard version via a ``coloring" construction.