Asymptotics of counts of small components in random structures and models of coagulation-fragmentation

We establish necessary and sufficient conditions for convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. We show that this convergence is equivalent to asymptotic independence of finite sizes of components. The multiplicative measures depict component spectra of random structures, the equilibrium of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We then apply Schur's tauberian lemma and some results from additive number theory and enumerative combinatorics, in order to verify the conditions derived in important special cases. Our results demostrate that the common belief that interacting groups in mean field models become independent as the number of particles goes to infinity, is not true in general.