## The Annals of Probability

- Ann. Probab.
- Volume 18, Number 1 (1990), 319-341.

### Random Tree-Type Partitions as a Model for Acyclic Polymerization: Holtsmark (3/2-Stable) Distribution of the Supercritical Gel

B. Pittel, W. A. Woyczynski, and J. A. Mann

#### Abstract

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 three papers studies the asymptotic behavior (in the thermodynamic limit as $n \rightarrow \infty$) 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 a classical Flory-Stockmayer polymerization theory with the spirit of more recent developments in the Erdos-Renyi 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.

#### Article information

**Source**

Ann. Probab. Volume 18, Number 1 (1990), 319-341.

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176990951

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176990951

**Mathematical Reviews number (MathSciNet)**

MR1043950

**Subjects**

Primary: 60J25: Continuous-time Markov processes on general state spaces

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82A51

**Keywords**

Random trees polymerization Gaussian and 3/2-stable limit behavior near critical supercritical phase transition Markov process stationary distribution

#### Citation

Pittel, B.; Woyczynski, W. A.; Mann, J. A. Random Tree-Type Partitions as a Model for Acyclic Polymerization: Holtsmark (3/2-Stable) Distribution of the Supercritical Gel. The Annals of Probability 18 (1990), no. 1, 319--341. doi:10.1214/aop/1176990951. http://projecteuclid.org/euclid.aop/1176990951.

#### See also

#### Corrections

- See Correction: B. Pittel, W. A. Woyczynski, J. A. Mann. Correction: Random Tree-Type Partitions as a Model for Acyclic Polymerization: Holtsmark (3/2-Stable) Distribution of the Supercritical Gel. Ann. Probab., Volume 20, Number 2 (1992), 1105--1106.Project Euclid: euclid.aop/1176989821