2005/11/15 by Boris L. Granovsky, Granovsky, Boris L.
Computer Science · Mathematics · #60C05 #60K35 #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.CO #math.PR #msc:60C05 #msc:60K35
paper · pdf · doi:10.48550/arxiv.math/0511381
This is a final version that will be pubilshed in the ESAIM:Probability and Statistics
openalex publication_date 2005/11/15 · arxiv created 2012/02/27 · arxiv updated 2012/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.