Stochastic coalescence revisited
Abstract: Consider N particles merging into clusters with the following random rule:
a cluster of size x and a cluster of size y merge at rate K(x,y).
The function K is called the rate kernel.
Such a process is called stochastic coalescent. For special rate functions,
K(x,y)=xy and K(x,y)=x+y, discussed in the talk,
the random processes giving the size of the clusters
according to the time correspond
The asymptotic of a picture of a large coalescence system taken at some special time t
has some description in terms of some Brownian processes. The size of the components correspond
to the sizes of the excursions of this process.
- to the size of the connected components in the Erdös-Rényi graph for K(x,y)=xy (Aldous)
- to the size of the blocks of consecutive occupied places in a circular parking (Chassaing-Louchard).
In the above paragraph we used the word 'picture' figures the fact that the process indexed by the time,
"the video of the coalescence process"
is not known to converge to some continuous coalescent built with the Brownian processes evoked above,
even if such continuous processes are well defined and exist (Aldous, Pitman, Bertoin, Chassaing-Louchard).
The talk is devoted to this question. We provide a new description of the coalesence processes which makes it
possible to prove this convergence.
Joint work with Nicolas Broutin.