Results 1  10
of
17
Clustering in coagulationfragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and ZeroOne law
"... The equilibrium distribution of a reversible coagulationfragmentation process (CFP) and the joint distribution of components of a random combinatorial structure are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (=compon ..."
Abstract

Cited by 19 (11 self)
 Add to MetaCart
(Show Context)
The equilibrium distribution of a reversible coagulationfragmentation process (CFP) and the joint distribution of components of a random combinatorial structure 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) = kp−1, k ≥ 1, 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 random combinatorial structures (RCS’s) and for RCS’s, corresponding to the case p < 0. 1 Summary. Our main result is a central limit theorem (Theorem 2.4) for the number of groups at steady state for a class of reversible CFP’s and for the corresponding class of RCS’s. In Section 2, we provide a definition of a reversible kCFP admitting interactions of up to k groups, as a generalization of the standard 2CFP. The steady state of the processes considered is fully defined by a parameter function a ≥ 0 on the set of integers. It was observed by Kelly ([11], p. 183) that for all 2 ≤ k ≤ N the kCFP’s have the same invariant measure on the set of partitions of a given
Asymptotic enumeration and logical limit laws for expansive multisets
 J. London Math. Soc
, 2006
"... A multiset is an unordered sample from a set of object types in which the number of items is variable, but the total weight of the objects equals a parameter n. The number of types of objects of weight j is aj. Let cn be the number of multiset representatives of total weight n. Then, for Tn = ∑n j=1 ..."
Abstract

Cited by 10 (9 self)
 Add to MetaCart
(Show Context)
A multiset is an unordered sample from a set of object types in which the number of items is variable, but the total weight of the objects equals a parameter n. The number of types of objects of weight j is aj. Let cn be the number of multiset representatives of total weight n. Then, for Tn = ∑n j=1 jZj where Zj are independent but not identically distributed negative binomial random variables with appropriate j=1 1 − e−σj) −aj P(Tn = n), where σ is an arparameters, cn = enσ ∏n bitrary parameter partially determining the distribution of the Zj. When aj ≍ jr−1yj for some r> 0, y ≥ 1, then we say that the multiset is expansive. For expansive multisets we prove a local limit lemma for Tn under the condition that σ is chosen so that E(Tn) = n. Moreover, we prove that cn/cn+1 → 1 and that cn/cn+1 < 1 for large enough n. This allows us to prove Monadic Second Order Limit Laws for expansive multisets. The above results are extended to a class of expansive multisets with oscillation. If the condition aj = Kjr−1y j + O(yνj) is imposed, where K> 0, r> 0, y> 1, ν ∈ (0,1), we are then able to find an explicit asymptotic formula for cn.
Limit shapes of Gibbs distributions on the set of integer partitions: The expansive case,
 Ann. Inst. H. Poincar Probab. Statist.
, 2008
"... ..."
(Show Context)
Asymptotics of counts of small components in random structures and models of coagulation –fragmentation.
, 2008
"... ..."
Limit shapes of multiplicative measures associated with coagulationfragmentation processes and random combinatorial structures
, 2008
"... ..."
Compton's Method for Proving Logical Limit Laws
 CONTEMPORARY MATHEMATICS
, 2003
"... Developments in the study of logical limit laws for both labelled and unlabelled structures, based on the methods of Compton (1987/1989), are surveyed, and a sandwich theorem is proved for multiplicative systems. ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Developments in the study of logical limit laws for both labelled and unlabelled structures, based on the methods of Compton (1987/1989), are surveyed, and a sandwich theorem is proved for multiplicative systems.
Partition Identities I  Sandwich Theorems and Logical 01 Laws
, 2004
"... The Sandwich Theorems proved in this paper give a new method to show that the partition function a(n) of a partition identity A(x):= ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The Sandwich Theorems proved in this paper give a new method to show that the partition function a(n) of a partition identity A(x):=
unknown title
, 2008
"... Asymptotics of counts of small components in random combinatorial structures and models of coagulationfragmentation. ..."
Abstract
 Add to MetaCart
(Show Context)
Asymptotics of counts of small components in random combinatorial structures and models of coagulationfragmentation.