  We introduce and study a discrete random model for Smoluchowski's equation
with limited aggregations. The latter is a model of coagulation introduced by
Bertoin which may exhibit gelation.
  In our model, a large number of particles are initially given a prescribed
number of arms. These arms are activated independently after exponential times
and successively linked together. However, when the size of a cluster goes
above a fixed threshold, it falls instantaneously into the gel, meaning that it
no longer interacts with the other particles in solution.
  The concentrations in this model asymptotically obey Smoluchowski's equation
with limited aggregations. In this article, we study the discrete features of
this model. We are able to argue that it remains closely related with a
configuration model. We specifically obtain explicit expressions for the
parameters of this configuration model, which show that it is subcritical
before gelation, but remains critical afterwards. As a consequence, the
asymptotic distribution of a typical cluster in solution is that of a
subcritical Galton-Watson tree before gelation, while it is that of a critical
Galton-Watson tree after gelation. Our model therefore exhibits self-organized
criticality.
  Our study relies heavily on the study of the configuration model in the
critical window. One of the main results of the paper is an extension of a
result of Janson and Luczak on the size of the largest components in this
critical window.
  Finally, as a consequence of our explicit expressions, we provide an
explanation of analytic formulas established by Normand and Zambotti regarding
the limiting concentrations in Smoluchowski's equation.
