BROWNIAN EXCURSIONS, CRITICAL RANDOM GRAPHS AND THE MULTIPLICATIVE COALESCENT

David Aldous

We introduce the {\em multiplicative coalescent process}, defined as follows.
The states are vectors ${\bf x}$ of nonnegative real cluster sizes
$(x_i)$, and clusters with sizes $x_i$ and $x_j$ merge at rate $x_ix_j$.
It turns out that this gives a Feller process on the space $l_2$.
There is a connection with the classical theory of random graphs.
Consider the random graph $\GG(n,n^{-1} + tn^{-4/3})$, whose
largest components have size of order $n^{2/3}$.
Normalizing by $n^{-2/3}$, there is an asymptotic joint distribution
$X(t) = (X_1(t),X_2(t),\ldots)$ of component sizes, and this must
evolve as the multiplicative coalescent, defined for all $- \infty < t < \infty$.
Informally, we can start the multiplicative coalescent at time $- \infty$
with infinitesimally small clusters.
A remarkable fact is that the distribution of $X(t)$ is the same as the
joint distribution of excursion lengths of a Brownian-type process
$(B^t(s),0 \leq s < \infty)$
defined to be reflecting inhomogeneous Brownian motion with drift $t-s$ at time $s$,
started with $B^t(0) = 0$.