Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at stochastic rate K(x,y)/N, where K is a specified rate kernel. This Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x,y)=1 and K(x,y)=xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously; so many interesting open problems appear.
Publié le : 1999-02-14
Classification:
branching process,
coalescence,
continuum tree,
density-dependent Markov process,
gelation,
random graph,
random tree,
Smoluchowski coagulation equation
@article{1173707093,
author = {Aldous, David J.},
title = {Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists},
journal = {Bernoulli},
volume = {5},
number = {6},
year = {1999},
pages = { 3-48},
language = {en},
url = {http://dml.mathdoc.fr/item/1173707093}
}
Aldous, David J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli, Tome 5 (1999) no. 6, pp. 3-48. http://gdmltest.u-ga.fr/item/1173707093/