21828 articles – 15613 Notices  [english version]

hal-00339779, version 1

## The structure of typical clusters in large sparse random configurations

Jean Bertoin () 12, Vladas Sidoravicius 34

Résumé : The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors.

• 1 :  Département de Mathématiques et Applications (DMA)
• CNRS : UMR8553 – Ecole normale supérieure de Paris - ENS Paris
• 2 :  Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
• CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
• 3 :  UMI CNRS-IMPA (UCI)
• Institut National de Mathématiques Pures – CNRS : UMI2924
• 4 :  Center for Mathematics and Computer Science (CWI)
• Netherlands Organisation for Scientific Research
• Domaine : Mathématiques/Probabilités
• Mots-clés : configuration model – Galton-Watson tree

• hal-00339779, version 1
• oai:hal.archives-ouvertes.fr:hal-00339779
• Contributeur :
• Soumis le : Mardi 18 Novembre 2008, 20:20:37
• Dernière modification le : Mardi 18 Novembre 2008, 21:33:03