This is a chatty discussion, intended to be understandable to a Ph. D. student in theoretical or applied probability. Numbers like [55] refer to the bibliography and are accompanied by Math Reviews links, and paper gives you the paper in compressed Postscript.

My survey paper Deterministic and Stochastic Models for Coalescence ...., written July 1997 (appeared early 1999) gave my view of the subject then. On this page I

- indicate what's in the survey
- describe my technical work
- update subsequent work.

To make a deterministic model, write $n(x,t)$ for the average
number of mass-$x$ clusters per unit volume at time $t$.
The idea above is then formalized by a set of differential
equations for the $n(x,t)$ called the
* Smoluchowski coagulation equation*.
These were the subject of much study in the scientific
literature, peaking during the 1960s.

Around 1980 there was a wave of statistical physics interest
in an explicitly stochastic model,
*the Marcus-Lushnikov process*.
Start with $N$ separate atoms.
As time increases, atoms merge into clusters according to the rule:
each pair of clusters (sizes $x$ and $y$, say)
merge at rate $K(x,y)/N$.
Although it wasn't realized at the time, two special cases
had been studied in the probability literature:
the case $K(x,y) = 1$ is *Kingman's coalescent*
and the case $K(x,y) = xy$ is the process of component sizes
in the Erdos-Renyi random graph process.

Evans and Pitman
MR 1625867
introduced the
* general stochastic coalescent*.
Here we scale total mass to be $1$,
and allow countably many clusters.
So the state-space is
$\{(x_i): x_i > 0, \sum_i x_i = 1\}$
for continuous cluster-masses $x_i$.
The evolution rule is:
each pair of clusters (masses $x$ and $y$, say)
merge at rate $K(x,y)$.
If the kernel has a scaling property
then the Marcus-Lushnikov process is a special case of
the stochastic coalescent, by scaling mass and time.

My general theme has been to show that in special cases the stochastic coalescent can be described in terms of known stochastic processes, as described below. There are many other types of open problem in this area, described in the survey paper.

As another special model, consider the
continuum random tree
and for $0< \lambda < \infty$ split the tree into components
at the points of a Poisson process of rate $\lambda$ along the
skeleton of the tree. This gives a vector
$Y(\lambda) = (Y_1(\lambda),Y_2(\lambda),......)$
of masses of the components, which as $\lambda$ increases specifies
a *fragmentation process*. In
paper [82]
(with Jim Pitman)
it is shown that reversing the direction of time
by setting $\lambda = e^{-t}$ gives the *standard additive coalescent*,
that is the case $K(x,y) = x+y$ of the general stochastic coalescent.
The same construction applied to a more general family of
*inhomogeneous* continuum random trees
yields the general version of the additive coalescent, i.e.
we can identify the entrance boundary: see
paper [87]
(with Jim Pitman).

In a different direction, there is some non-rigorous
statistical physics literature
(van Dongen
MR 90f:82016)
concerning cluster sizes in the
* gelation* phase transition for rate kernels $K(x,y)$ more
general than $K(x,y) = xy$. Almost nothing is known rigorously,
but paper [77]
MR 99g:60128
makes a start by studying a special one-parameter family of kernels.
Jeon
MR 99g:82056
also has results of this type.