We consider a system of particles with arms that are activated randomly to grab other particles as a toy model for polymerization. We assume that the following two rules are fulfilled: Once a particle has been grabbed then it cannot be grabbed again,
and an arm cannot grab a particle that belongs to its own cluster. We are interested in the shape of a typical polymer in the situation when the initial number of monomers is large and the numbers of arms of monomers are given by i.i.d. random variables. Our main result is a limit theorem for the empirical distribution of polymers, where limit is expressed in terms of a Galton-Watson tree.
The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowskis coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concent
rations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowskis coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors.
