Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality in a class of fragmentation-coalescence processes

A. E. Kyprianou, S. W. Pagett, and T. Rogers

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Abstract

We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non-random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit a universal cluster size distribution regardless of the details of the rates, following a power law with exponent $3/2$.

Résumé

Nous introduisons et étudions une classe de processus de fragmentation-coalescence définis sur des systèmes finis de particules organisées en amas. Lors d’un événement de coalescence, de multiples amas fusionnent simultanément en un amas plus gros, et dans le cas d’un événement de fragmentation, un amas est détruit en une collection de singletons. Sous des hypothèses faibles sur les taux de coalescence, nous montrons que la loi des tailles des amas devient déterministe dans la limite thermodynamique. De plus, nous montrons que dans la limite des petits taux de fragmentation, ces processus admettent une loi universelle pour la les tailles des amas, indépendante de la valeur précise des taux, et qui suit une loi de puissance avec un exposant $3/2$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 1134-1151.

Dates
Received: 23 December 2015
Revised: 30 March 2017
Accepted: 30 March 2017
First available in Project Euclid: 25 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1524643242

Digital Object Identifier
doi:10.1214/17-AIHP834

Mathematical Reviews number (MathSciNet)
MR3795079

Zentralblatt MATH identifier
06897981

Subjects
Primary: 60G99: None of the above, but in this section

Keywords
Fragmentation-coalescence Cluster distribution Thermodynamic limit

Citation

Kyprianou, A. E.; Pagett, S. W.; Rogers, T. Universality in a class of fragmentation-coalescence processes. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 1134--1151. doi:10.1214/17-AIHP834. https://projecteuclid.org/euclid.aihp/1524643242


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