Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case

Michael M. Erlihson; Boris L. Granovsky

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 5, page 915-945
  • ISSN: 0246-0203

Abstract

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We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak∼Ckp−1, k→∞, p>0, where C is a positive constant. The measures considered are associated with the generalized Maxwell–Boltzmann models in statistical mechanics, reversible coagulation–fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence (given their masses). Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation.

How to cite

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Erlihson, Michael M., and Granovsky, Boris L.. "Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case." Annales de l'I.H.P. Probabilités et statistiques 44.5 (2008): 915-945. <http://eudml.org/doc/77997>.

@article{Erlihson2008,
abstract = {We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak∼Ckp−1, k→∞, p&gt;0, where C is a positive constant. The measures considered are associated with the generalized Maxwell–Boltzmann models in statistical mechanics, reversible coagulation–fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence (given their masses). Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation.},
author = {Erlihson, Michael M., Granovsky, Boris L.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Gibbs distributions on the set of integer partitions; limit shapes; random combinatorial structures and coagulation–fragmentation processes; local and integral central limit theorems; combinatorial probability; random integer partitions; multiplicative probability measures; limit theorems; Young diagrams; central limit theorem; Maxwell-Boltzmann models; gelation; coagulation-fragmentation processes},
language = {eng},
number = {5},
pages = {915-945},
publisher = {Gauthier-Villars},
title = {Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case},
url = {http://eudml.org/doc/77997},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Erlihson, Michael M.
AU - Granovsky, Boris L.
TI - Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 5
SP - 915
EP - 945
AB - We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak∼Ckp−1, k→∞, p&gt;0, where C is a positive constant. The measures considered are associated with the generalized Maxwell–Boltzmann models in statistical mechanics, reversible coagulation–fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence (given their masses). Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation.
LA - eng
KW - Gibbs distributions on the set of integer partitions; limit shapes; random combinatorial structures and coagulation–fragmentation processes; local and integral central limit theorems; combinatorial probability; random integer partitions; multiplicative probability measures; limit theorems; Young diagrams; central limit theorem; Maxwell-Boltzmann models; gelation; coagulation-fragmentation processes
UR - http://eudml.org/doc/77997
ER -

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