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Deterministic and Stochastic Models for Coalescence (Aggregation, Coagulation): a Review of the MeanField Theory for Probabilists
 Bernoulli
, 1997
"... Consider N particles, which merge into clusters according to the 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 MarcusLushnikov model of stochastic coalescence, and the underlying deterministic approximation given by ..."
Abstract

Cited by 142 (13 self)
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Consider N particles, which merge into clusters according to the 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 MarcusLushnikov 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 wideranging survey. General kernels are only now starting to be studied rigorously, so many interesting open problems appear. Keywords. branching process, coalescence, continuum tree, densitydependent Markov process, gelation, random graph, random tree, Smoluchowski coagulation equation Research supported by N.S.F. Grant DMS9622859 1 Introduction Models, implicitly or explicitly stochastic, of coalescence (= coagulati...
Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
Emergence of the Giant Component in Special MarcusLushnikov Processes
 Random Structures and Algorithms
, 1997
"... Component sizes in the usual random graph process are a special case of the MarcusLushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the MarcusLushnikov process. Essentially no rigorous results ar ..."
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Cited by 13 (4 self)
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Component sizes in the usual random graph process are a special case of the MarcusLushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the MarcusLushnikov process. Essentially no rigorous results are known; we make a start by proving a weak result, but our main purpose is to draw this topic to the attention of random graph theorists. 1 Introduction 1.1 Background At time zero there are n separate "atoms"; as time increases, these atoms coalesce into clusters according to the rule for each pair of clusters, of sizes fx; yg say, they coalesce into a single cluster of size x + y at rate K(x; y)=n where K(x; y) = K(y; x) 0 is some specified rate kernel. This rule specifies a continuoustime finitestate Markov process which we shall call the Research supported by N.S.F. Grant DMS9622859 MarcusLushnikov process. The model was introduced by Marcus [16], and further studied by Lush...
Triangle percolation in mean field random graphs – with PDE
, 712
"... We apply a PDEbased method to deduce the critical time and the size of the giant component of the “triangle percolation ” on the ErdősRényi random graph process investigated by Palla, Derényi and Vicsek in [8], [9]. 1 ..."
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Cited by 2 (0 self)
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We apply a PDEbased method to deduce the critical time and the size of the giant component of the “triangle percolation ” on the ErdősRényi random graph process investigated by Palla, Derényi and Vicsek in [8], [9]. 1
Triangle percolation in mean field random graphs — with PDE
, 2008
"... We apply a PDEbased method to deduce the critical time and the size of the giant component of the “triangle percolation ” on the ErdősRényi random graph process investigated by Palla, Derényi and Vicsek in [4], [9]. 1 ..."
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We apply a PDEbased method to deduce the critical time and the size of the giant component of the “triangle percolation ” on the ErdősRényi random graph process investigated by Palla, Derényi and Vicsek in [4], [9]. 1