The Structure of Typical Clusters in Large Sparse Random Configurations |
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Authors: | Jean Bertoin and Vladas Sidoravicius |
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Institution: | (1) Laboratoire de Probabilités, UPMC, 175 rue du Chevaleret, 75013 Paris, France;(2) DMA, ENS Paris, Paris, France;(3) IMPA, Estr. Dona Castorina 110, Rio de Janeiro, Brazil;(4) CWI, Kruislaan 413, 1098 SJ, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands |
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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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Keywords: | Configuration model Galton-Watson tree Smoluchowski coagulation equations |
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