Summary: Random tree-type partitions for finite sets are used as a model of a chemical polymerization process when ring formation is forbidden. Technically, our series of papers [see authors, Graph theory and topology in chemistry, Collect. Pap. Int. Conf., Athens/GA 1987, Stud. Phys. Theor. Chem. 51, 362-370 (1987; Zbl 0662.05063); Random graphs ’87, Proc. 3rd Int. Semin., Poznań/Poland 1987, 223-273 (1990; Zbl 0731.60101) and the two papers reviewed below] studies the asymptotic behavior (in the thermodynamic limit as \(n\to \infty\)) of a particular probability distribution on the set of all forests of trees on a set of \(n\) elements (monomers). The study rigorously establishes the existence of three stages of polymerization dependent upon the ratio of association and dissociation rates of monomers. The subcritical stage has been analyzed in the other two papers of this series. The present paper, second in the series, concentrates on the analysis of the near-critical and supercritical stages. In the supercritical stage we discover that the molecular weight of the largest connected component (gel) has the Holtsmark distribution.
Our study combines elements of classical Flory-Stockmayer polymerization theory with the spirit of more recent developments in the Erdős-Rényi theory of random graphs. Although this paper has a chemical motivation, conceptually similar mathematical models have been found useful in other disciplines, such as computer science and biology, etc.