TY - JOUR

T1 - Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems

T2 - Asymptotic formulae and limiting laws

AU - Freiman, Gregory A.

AU - Granovsky, Boris L.

PY - 2005/6

Y1 - 2005/6

N2 - 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 1/l+1 is the threshold for the limiting distribution of the largest cluster. As a byproduct of our study, we prove the independence of the numbers of groups of fixed sizes, as n → ∞. 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.

AB - 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 1/l+1 is the threshold for the limiting distribution of the largest cluster. As a byproduct of our study, we prove the independence of the numbers of groups of fixed sizes, as n → ∞. 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.

KW - Additive number systems

KW - Coagulation-fragmentation process

KW - Distributions on the set of partitions

KW - Local limit theorem

KW - Random combinatorial structures

UR - http://www.scopus.com/inward/record.url?scp=20144376995&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-04-03617-7

DO - 10.1090/S0002-9947-04-03617-7

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AN - SCOPUS:20144376995

SN - 0002-9947

VL - 357

SP - 2483

EP - 2507

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -