The asymptotic behavior in a reversible random coagulation-fragmentation polymerization process with sub-exponential decay

This paper studies the subcritical, near-critical and supercritical asymptotic behavior of a reversible random coagulation-fragmentation polymerization process as N→∞, with the number of distinct ways to form a k-clusters from k units satisfying fk=1+o1cr−ke−kαk−β, where 0<α<1 and β>0. When the cluster size is small, its distribution is proved to converge to the Gaussian distribution. For the medium clusters, its distribution will converge to Poisson distribution in supercritical stage, and no large clusters exist in this stage. Furthermore, the largest length of polymers of size N is of order lnN in the subcritical stage under α⩽1∕2.