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

Boris L. Granovsky — 2013

ESAIM: Probability and Statistics

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...

Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case

Michael M. Erlihson; Boris L. Granovsky — 2008

Annales de l'I.H.P. Probabilités et statistiques

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ∼ , →∞, >0, where is a positive constant. The measures considered are associated with the generalized Maxwell–Boltzmann models in statistical mechanics, reversible coagulation–fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of...

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