A system of grabbing particles related to Galton-Watson trees

TL;DR

This paper models polymerization using a particle system with arms, deriving a limit theorem for the distribution of polymers based on Galton-Watson trees, under large initial monomer conditions.

Contribution

It introduces a novel particle system model for polymerization with random arms and establishes a limit theorem linking polymer shapes to Galton-Watson trees.

Findings

Limit theorem for empirical polymer distribution

Connection between polymer shapes and Galton-Watson trees

Asymptotic behavior for large initial monomers

Abstract

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.