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We introduce and study a discrete random model for Smoluchowski's equationwith limited aggregations. The latter is a model of coagulation introduced byBertoin which may exhibit gelation.  In our model, a large number of particles are initially given a prescribednumber of arms. These arms are activated independently after exponential timesand successively linked together. However, when the size of a cluster goesabove a fixed threshold, it falls instantaneously into the gel, meaning that itno longer interacts with the other particles in solution.  The concentrations in this model asymptotically obey Smoluchowski's equationwith limited aggregations. In this article, we study the discrete features ofthis model. We are able to argue that it remains closely related with aconfiguration model. We specifically obtain explicit expressions for theparameters of this configuration model, which show that it is subcriticalbefore gelation, but remains critical afterwards. As a consequence, theasymptotic distribution of a typical cluster in solution is that of asubcritical Galton-Watson tree before gelation, while it is that of a criticalGalton-Watson tree after gelation. Our model therefore exhibits self-organizedcriticality.  Our study relies heavily on the study of the configuration model in thecritical window. One of the main results of the paper is an extension of aresult of Janson and Luczak on the size of the largest components in thiscritical window.  Finally, as a consequence of our explicit expressions, we provide anexplanation of analytic formulas established by Normand and Zambotti regardingthe limiting concentrations in Smoluchowski's equation.

Self-organized criticality in a discrete model for Smoluchowski's equation with limited aggregations