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87
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 ..."
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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...
Brownian Excursions, Critical Random Graphs and the Multiplicative Coalescent
, 1996
"... 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 d ..."
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Cited by 86 (10 self)
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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...
Random trees, Lévy processes and spatial branching processes
 Astérisque
"... 0.1 Discrete trees................................ 5 0.2 GaltonWatson trees............................ 7 0.3 The continuous height process....................... 9 0.4 From discrete to continuous trees..................... 12 ..."
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Cited by 74 (3 self)
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0.1 Discrete trees................................ 5 0.2 GaltonWatson trees............................ 7 0.3 The continuous height process....................... 9 0.4 From discrete to continuous trees..................... 12
The Standard Additive Coalescent
, 1997
"... Regard an element of the set \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; X i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman (1997) is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g mer ..."
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Cited by 63 (22 self)
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Regard an element of the set \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; X i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman (1997) is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i +x j at rate x i +x j . They showed that a version (X 1 (t); \Gamma1 ! t ! 1) of this process arises as a n !1 weak limit of the process started at time \Gamma 1 2 log n with n clusters of mass 1=n. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. We describe the distribution of X 1 (t) on \Delta at a fixed time t. We show that the size of the cluster containing a given atom, as a process in t, has a simple representation in terms of the stable subordinator of index 1=2. As t ! \Gamma1, we establish a Gaussian limit for (centered and norm...
Probabilistic and fractal aspects of Lévy trees
 Probab. Th. Rel. Fields
, 2005
"... We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete GaltonWatson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted Rtrees, which i ..."
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Cited by 49 (14 self)
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We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete GaltonWatson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted Rtrees, which is equipped with the GromovHausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the wellknown property for GaltonWatson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching
Probability Distributions on Cladograms
 In Random Discrete Structures
, 1996
"... By analogy with the theory surrounding the Ewens sampling formula in neutral population genetics, we ask whether there exists a natural oneparameter family of probability distributions on cladograms ("evolutionary trees") which plays a central role in neutral evolutionary theory. Unfortunately the a ..."
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Cited by 48 (2 self)
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By analogy with the theory surrounding the Ewens sampling formula in neutral population genetics, we ask whether there exists a natural oneparameter family of probability distributions on cladograms ("evolutionary trees") which plays a central role in neutral evolutionary theory. Unfortunately the answer seems to be "no"  see Conjecture 2. But we can embed the two most popular models into an interesting family which we call "betasplitting" models. We briefly describe some mathematical results about this family, which exhibits qualitatively different behavior for different ranges of the parameter fi. 1 Probability distributions on partitions and neutral population genetics The first few sections give some conceptual background. The reader wishing to "get right to the point" should skim these and proceed to section 3. For each n there is a finite set of partitions of f1; 2; : : : ; ng into unordered families fA 1 ; A 2 ; : : : ; A k g of subsets. A oneparameter family (P (n) ` ) o...
Rayleigh processes, real trees, and root growth with regrafting
, 2004
"... Abstract. The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous’s Brownian continuum random tree, the random treelike object naturally associated with ..."
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Cited by 48 (11 self)
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Abstract. The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous’s Brownian continuum random tree, the random treelike object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N → ∞ of both a critical GaltonWatson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N → ∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–
Construction Of Markovian Coalescents
 Ann. Inst. Henri Poincar'e
, 1997
"... Partitionvalued and measurevalued 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 ma ..."
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Cited by 44 (20 self)
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Partitionvalued and measurevalued 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 nonnegative, 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 Fellerlike processes. A number of further results are obtained for the additive coalescent with collision kernel (x; y) = x + y. This process, which arises fro...
TreeBased Models for Random Distribution of Mass
 J. Stat. Phys
, 1993
"... A mathematical model for distribution of mass in ddimensional space, based upon randomly embedding random trees into space, is introduced and studied. The model is a variant of the superBrownian motion process studied by mathematicians. We present calculations relating to (i) distribution of posit ..."
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Cited by 39 (0 self)
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A mathematical model for distribution of mass in ddimensional space, based upon randomly embedding random trees into space, is introduced and studied. The model is a variant of the superBrownian motion process studied by mathematicians. We present calculations relating to (i) distribution of position of typical mass element (ii) moments of the center of mass (iii) large deviation behavior (iv) a recursive selfsimilarity property. Key words. Spatial distribution, random tree, superBrownian process, large deviations, recursive selfsimilarity. Journal of Statistical Physics 73 (1993) 625641. Research supported by N.S.F. Grants DMS9001710 and DMS9224857 and by the Miller Institute for Basic Research in Science 1 Introduction To start with an analogy, it has long been accepted that the mathematically fundamental model for a quantity varying randomly but continuously with time is (mathematical) Brownian motion (alternatively called the Wiener process or integrated white noise). ...
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...