# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Random tree-type partitions as a model for acyclic polymerization: Holtsmark (3/2-stable) distribution of the supercritical gel. (English) Zbl 0743.60110

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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J25 Continuous-time Markov processes on general state spaces 82B26 Phase transitions (general)