Clustering in coagulation -fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws

We develop a unified approach to the problem of clustering in the three different fields of applications, as indicated in the title the paper. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fawler method in statistical physics. Our main result is the derivation of asymptotic formulae for probabilities of clusters (= groups) of certain sizes as the number of particles goes to infinity. Based on these formulae we prove the zero-one law for the distribution of the largest cluster and establish the threshold function in the phase transition from 0 to 1 in the above law.