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219
Coalescents With Multiple Collisions
 Ann. Probab
, 1999
"... For each finite measure on [0 ..."
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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 103 (8 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...
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 87 (21 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...
Smoluchowski’s coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent
 Ann. Appl. Probab
, 1999
"... 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 nonuniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of ..."
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Cited by 66 (3 self)
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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 nonuniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski’s equation. 1.
Approach to SelfSimilarity in Smoluchowski’s Coagulation Equations
, 2003
"... We consider the approach to selfsimilarity (or dynamical scaling) in Smoluchowski’s equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known selfsimilar solutions with exponential tails, there are oneparameter families of solutions with algebraic deca ..."
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Cited by 58 (8 self)
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We consider the approach to selfsimilarity (or dynamical scaling) in Smoluchowski’s equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known selfsimilar solutions with exponential tails, there are oneparameter families of solutions with algebraic decay, whose form is related to heavytailed distributions wellknown in probability theory. For K = 2 the size distribution is MittagLeffler, and for K = x + y and K = xy it is a powerlaw rescaling of a maximally skewed αstable Lévy distribution. We characterize completely the domains of attraction of all selfsimilar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.
TreeValued Markov Chains Derived From GaltonWatson Processes.
 Ann. Inst. Henri Poincar'e
, 1997
"... Let G be a GaltonWatson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the treevalued Markov process (G u ; 0 u 1) and an analogous process (G u ; 0 u 1) in which G 1 is a critical or subcritical GaltonWatson tree conditio ..."
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Cited by 56 (9 self)
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Let G be a GaltonWatson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the treevalued Markov process (G u ; 0 u 1) and an analogous process (G u ; 0 u 1) in which G 1 is a critical or subcritical GaltonWatson tree conditioned to be infinite. Results simplify and are further developed in the special case of Poisson() offspring distribution. Running head. Treevalued Markov chains. Key words. Borel distribution, branching process, conditioning, GaltonWatson process, generalized Poisson distribution, htransform, pruning, random tree, sizebiasing, spinal decomposition, thinning. AMS Subject classifications 05C80, 60C05, 60J27, 60J80 Research supported in part by N.S.F. Grants DMS9404345 and 9622859 1 Contents 1 Introduction 2 1.1 Related topics : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2 Background and technical setup 5 2.1 Notation and terminology for trees : : : : : : : : : : : : : : :...
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 ; : ..."
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Cited by 53 (14 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...
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 48 (16 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...
PoissonDirichlet and GEM invariant distributions for splitandmerge transformations of an interval partition
, 2001
"... This paper introduces a splitandmerge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov [10, 11] and another studied by Tsilevich [30, 29] and MayerWolf, Zeitouni and Zerner [20]. The invariance under this splitandmerge transformatio ..."
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Cited by 48 (0 self)
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This paper introduces a splitandmerge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov [10, 11] and another studied by Tsilevich [30, 29] and MayerWolf, Zeitouni and Zerner [20]. The invariance under this splitandmerge transformation of the interval partition generated by a suitable Poisson process yields a simple proof of the recent result of [20] that a PoissonDirichlet distribution is invariant for a closely related fragmentationcoagulation process. Uniqueness and convergence to the invariant measure are established for the splitandmerge transformation of interval partitions, but the corresponding problems for the fragmentationcoagulation process remain open.
On selfsimilarity and stationary problem for fragmentation and coagulation
"... We prove the existence of a stationary solution of any given mass to the coagulationfragmentation equation without assuming a detailed balance condition, but assuming instead that aggregation dominates fragmentation for small particles whisle fragmentation predominates for large particles. We also s ..."
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Cited by 47 (9 self)
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We prove the existence of a stationary solution of any given mass to the coagulationfragmentation equation without assuming a detailed balance condition, but assuming instead that aggregation dominates fragmentation for small particles whisle fragmentation predominates for large particles. We also show the existence of a self similar solution of any given mass to the coagulation equation and to the fragmentation equation for kernels satisfying a scaling property. These results are obtained by applying a simple abstract result of functional analysis based on the Tykhonov fixed point theorem, and which slightly generalizes a method introduced in [26]. Moreover, we show that the solutions to the fragmentation equation with initial data of a given mass behaves, as t → +∞, as the self similar solution of the same mass. 1 Introduction and