  In this paper, we investigate a stochastic model describing the time
evolution of a polymerization process. A polymer is a macro-molecule resulting
from the aggregation of several elementary sub-units called monomers. Polymers
can grow by addition of monomers or can be split into several polymers. The
initial state of the system consists mainly of monomers. We study the time
evolution of the mass of polymers, in particular the asymptotic distribution of
the first instant when the fraction of monomers used in polymers is above some
positive threshold $\delta$. A scaling approach is used by taking the mass $N$
as a scaling parameter. The mathematical model used in this paper includes {\em
a nucleation property}: If $n_c$ is defined as the size of the nucleus,
polymers with a size less than $n_c$ are quickly fragmented into smaller
polymers, at a rate proportional to $\Phi(N)$ for some non-decreasing and
unbounded function $\Phi$. For polymers of size greater than $n_c$,
fragmentation still occurs but at bounded rates. If $T^N$ is the instant of
creation of the first polymer whose size is $n_c$, it is shown that, under
appropriate conditions, the variable $T^N{/}[\Phi(N)^{n_c{-}2}{/}N]$ converges
in distribution, and that the first instant $L_\delta^N$ when a fraction
$\delta$ of monomers is polymerized has the same order of magnitude. An
original feature proved for this model is the significant variability of the
variable $T^N$. This is a well known phenomenon observed in biological
experiments but few mathematical models of the literature have this property.
The results are proved via a series of technical estimates for occupation
measures of some functionals of the corresponding Markov processes on fast time
scales and by using coupling techniques.
