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 wide-ranging 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 DMS96-22859 1 Introduction Models, implicitly or explicitly stochastic, of coalescence (= coagulati...

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Let (B t (s); 0 s ! 1) be reflecting inhomogeneous Brownian motion with drift t \Gamma s at time s, started with B t (0) = 0. Consider the random graph G(n; n \Gamma1 +tn \Gamma4=3 ), whose largest components have size of order n 2=3 . Normalizing by n \Gamma2=3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (Corollary 2). The dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors x of nonnegative real cluster sizes (x i ), and clusters with sizes x i and x j merge at rate x i x j . The multiplicative coalescent is shown to be a Feller process on l 2 . The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time \Gamma1: the existence of such a process is not obvious. AMS 1991 subject classifications. 60C05, 60J50, Key words and phras...

Partition-valued and measure-valued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m, and whose evolution is determined by the following intuitive prescription: each pair of masses of magnitudes x and y runs the risk of a binary collision to form a single mass of magnitude x+y at rate (x; y), for some non-negative, symmetric collision rate kernel (x; y). Such processes with finitely many masses have been used to model polymerization, coagulation, condensation, and the evolution of galactic clusters by gravitational attraction. With a suitable choice of state space, and under appropriate restrictions on and the initial distribution of mass, it is shown that such processes can be constructed as Feller or Feller-like processes. A number of further results are obtained for the additive coalescent with collision kernel (x; y) = x + y. This process, which arises fro...

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...

Abstract. Sufficient conditions are given for existence and uniqueness in Smoluchowski’s coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of non-uniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski’s equation. 1.

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Component sizes in the usual random graph process are a special case of the Marcus-Lushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the Marcus-Lushnikov 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 continuous-time finite-state Markov process which we shall call the Research supported by N.S.F. Grant DMS96-22859 Marcus-Lushnikov process. The model was introduced by Marcus [16], and further studied by Lush...

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. -- This paper studies a stochastic particle method for the numerical treatment of Smoluchowski equation governing the coagulation of particles in a host gas. Convergence in probability is established for the Monte Carlo estimators, when the number of particles tends to infinity. The deterministic limit is characterized as the solution of a discrete in time version of the Smoluchowski equation. Under some restrictions it is shown that this stochastic finite difference scheme is convergent to the solution of the original Smoluchowski equation. Extensions on a nonhomogeneous Smoluchowski equation are given, and in particular, a coagulation process in an isotropic fully developed turbulent flow is studied. 1 Introduction The coagulation processes of aerosol particles or clusters in a spatially homogeneous flow are governed by the Smoluchowski equation (e.g., see, [22], [23]): @n l @t = 1 2 X i+j=l K ij n i n j \Gamma n l 1 X i=1 K li n i + F l (t) (1:1) with the initial condit...