Geodesic
Asymptotics of counts of small components in random structures and models of coagulation-fragmentation
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 531-549

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

DOI : 10.1051/ps/2012007
Classification : 60C05, 60K35, 05A16, 82B05, 11M45
Keywords: multiplicative measures on the set of partitions, random structures, coagulation-fragmentation processes, Schur's lemma, models of ideal gas 
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     title = {Asymptotics of counts of small components in random structures and models of coagulation-fragmentation},
     journal = {ESAIM: Probability and Statistics},
     pages = {531--549},
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Granovsky, Boris L. Asymptotics of counts of small components in random structures and models of coagulation-fragmentation. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 531-549. doi: 10.1051/ps/2012007

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