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

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.