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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Gregory A. Freiman and Boris L. Granovsky
Journal: Trans. Amer. Math. Soc. 357 (2005), 2483-2507
MSC (2000): Primary 60K35, 05A15; Secondary 05A16, 05C80, 11M45
Published electronically: November 23, 2004
MathSciNet review: 2140447
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Abstract: We develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper, in the case when the parametric function of the models is regularly varying with positive exponent. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fowler method in statistical physics. Our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters (= components), as the total size of a structure (= number of particles) goes to infinity. We discover that $n^{\frac{1}{l+1}}$ is the threshold for the limiting distribution of the largest cluster. As a by-product of our study, we prove the independence of the numbers of groups of fixed sizes, as $n\to \infty.$ This is in accordance with the general principle of asymptotic independence of sites in mean-field models. The latter principle is commonly accepted in statistical physics, but not rigorously proved.


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Additional Information

Gregory A. Freiman
Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel
Email: grisha@math.tau.ac.il

Boris L. Granovsky
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel
Email: mar18aa@techunix.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03617-7
PII: S 0002-9947(04)03617-7
Keywords: Coagulation-fragmentation process, random combinatorial structures, local limit theorem, distributions on the set of partitions, additive number systems
Received by editor(s): July 18, 2002
Received by editor(s) in revised form: January 7, 2004
Published electronically: November 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society