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The structure of typical clusters in large sparse random configurations


Bertoin, Jean (2009). The structure of typical clusters in large sparse random configurations. Journal of Statistical Physics, 135(1):87-105.

Abstract

The initial purpose of this work is to provide a probabilistic explanation of recent results 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. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors.

Abstract

The initial purpose of this work is to provide a probabilistic explanation of recent results 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. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2009
Deposited On:15 May 2013 15:23
Last Modified:05 Apr 2016 16:47
Publisher:Springer
ISSN:0022-4715
Publisher DOI:https://doi.org/10.1007/s10955-009-9728-y
Official URL:http://link.springer.com/article/10.1007/s10955-009-9728-y
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1168.82028
http://arxiv.org/abs/0811.2988

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