Reversible coagulation–fragmentation processes and random combinatorial structures: Asymptotics for the number of groups

Abstract

The equilibrium distribution of a reversible coagulation‐fragmentation process (CFP) and the joint distribution of components of a random combinatorial structure (RCS) are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (= components) in the case a(k) = qkp−1, k ≥ 1, q, p > 0, where a(k), k ≥ 1, is the parameter function that induces the invariant measure. The result obtained is compared with the ones for logarithmic RCS's and for RCS's, corresponding to the case p < 0. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004

Meinardus' theorem on weighted partitions: Extensions and a probabilistic proof

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“…Proposition 1. Let F be defined by (7) and condition (8) holds. If ρ 1 < 1 then F is analytic in the disk |z| < ρ 1 and has a singularity at ρ = ρ 1 .…”

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