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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 ..."
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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...
Enumerations Of Trees And Forests Related To Branching Processes And Random Walks
 Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci
, 1997
"... In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p ..."
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Cited by 38 (15 self)
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In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn n\Gammak\Gamma1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies an...
Approach to SelfSimilarity in Smoluchowski’s Coagulation Equations, preprint
, 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 34 (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.
SPINAL PARTITIONS AND INVARIANCE UNDER REROOTING OF CONTINUUM RANDOM TREES
"... We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees ..."
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Cited by 20 (12 self)
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We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform rerooting. 1. Introduction. Starting from a rooted combinatorial tree T[n] with n leaves labeled by [n] ={1,...,n}, we call the path from the root to the leaf labeled 1 the spine of T[n]. Deleting each edge along the spine of T[n] defines a graph whose connected components we call bushes. If, as well as cutting each edge on the spine, we cut each edge connected to a spinal vertex, each bush is further decomposed
The Percolation Process on a Tree Where Infinite Clusters are Frozen
 Math. Proc. Cambridge Philos. Soc
, 1999
"... Modify the usual percolation process on the infinite binary tree by forbidding infinite clusters to grow further. The ultimate configuration will consist of both infinite and finite clusters. We give a rigorous construction of a version of this process and show that one can do explicit calculations ..."
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Cited by 17 (2 self)
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Modify the usual percolation process on the infinite binary tree by forbidding infinite clusters to grow further. The ultimate configuration will consist of both infinite and finite clusters. We give a rigorous construction of a version of this process and show that one can do explicit calculations of various quantities, for instance the law of the time (if any) that the cluster containing a fixed edge becomes infinite. Surprisingly, the distribution of the shape of a cluster which becomes infinite at time t ? 1=2 does not depend on t; it is always distributed as the incipient infinite percolation cluster on the tree. Similarly, a typical finite cluster at each time t ? 1=2 has the distribution of a critical percolation cluster. This elaborates an observation of Stockmayer (1942). AMS 1991 subject classification: 60K35, 05C80 Research supported by N.S.F. Grant DMS9622859 1 Introduction Let T = (V; E) be the infinite binary tree, wherein each vertex has degree three: V is the ver...
The structure of the allelic partition of the total population for GaltonWatson processes with neutral mutations
"... We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and ..."
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Cited by 15 (3 self)
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We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and the number of mutantchildren of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given. 1. Introduction. We consider a Galton–Watson process, that is, a population model with asexual reproduction such that at every generation, each individual gives birth to a random number of children according to a fixed distribution and independently of the other individuals in the population. We are interested in the situation where a child can be either a clone, that
Growth of Lévy trees
 Probab. Theory Related Fields 139 313–371. MR2322700
, 2007
"... We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuousstate branching processes. More precisely, we define a growing family of discrete GaltonWatson trees with i.i.d. exponential branch lengths that is consistent under Bernoulli ..."
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Cited by 14 (4 self)
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We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuousstate branching processes. More precisely, we define a growing family of discrete GaltonWatson trees with i.i.d. exponential branch lengths that is consistent under Bernoulli percolation on leaves; we define the Lévy tree as the limit of this growing family with respect to the GromovHausdorff topology on metric spaces. This elementary approach notably includes supercritical trees and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuousstate branching processes. We construct the mass measure of Lévy trees and we give a decomposition along the ancestral subtree of a Poisson sampling directed by the mass measure.
Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times
 ELECTRON. J. PROBAB
, 1999
"... For a random process X consider the random vector defined by the values of X at times 0
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Cited by 12 (3 self)
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For a random process X consider the random vector defined by the values of X at times 0 <U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n,i are the order statistics of n independent uniform (0, 1) variables, independent of X . The joint law of this random vector is explicitly described when X is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at n independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae a...
Treevalued Markov chains and PoissonGaltonWatson distributions
 Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci
, 1998
"... . The PoissonGaltonWatson distribution on finite trees, and the related PGW 1 (1) distribution on infinite trees with one end, arise in several contexts, in particular as n !1 weak limits within various sizen combinatorial models. We review this topic, introducing slick notation for describing ..."
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Cited by 10 (3 self)
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. The PoissonGaltonWatson distribution on finite trees, and the related PGW 1 (1) distribution on infinite trees with one end, arise in several contexts, in particular as n !1 weak limits within various sizen combinatorial models. We review this topic, introducing slick notation for describing such distributions. We then describe a family of continuoustime Markov chains whose marginal distributions are of PoissonGaltonWatson type. Different chains in this family have connections with different parts of the extensive literature on tree and forestvalued chains (the random graph process, stochastic coalescence, spanning tree chains) and also illustrate the classical statespacecompactification theory of continuoustime countablestate chains. 1. Introduction Tree and forestvalued Markov chains have been studied in several areas. ffl Evolution of treebased data structures (Knuth [13], Mahmoud [16]). ffl The Markov chain tree theorem (Pemantle [18], LyonsPeres [15]), which t...
A continuumtreevalued Markov process
"... Abstract. We present a construction of a Lévy continuum random tree (CRT) associated with a supercritical continuous state branching process using the socalled exploration process and a Girsanov’s theorem. We also extend the pruning procedure to this supercritical case. Let ψ be a critical branchi ..."
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Cited by 10 (9 self)
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Abstract. We present a construction of a Lévy continuum random tree (CRT) associated with a supercritical continuous state branching process using the socalled exploration process and a Girsanov’s theorem. We also extend the pruning procedure to this supercritical case. Let ψ be a critical branching mechanism. We set ψθ(·) = ψ( · + θ) − ψ(θ). Let Θ = (θ∞,+∞) or Θ = [θ∞,+∞) be the set of values of θ for which ψθ is a branching mechanism. The pruning procedure allows to construct a decreasing LévyCRTvalued Markov process (Tθ, θ ∈ Θ), such that Tθ has branching mechanism ψθ. It is subcritical if θ> 0 and supercritical if θ < 0. We then consider the explosion time A of the CRT: the smaller (negative) time θ for which Tθ has finite mass. We describe the law of A as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to A. We also study the evolution of the CRTvalued process after the explosion time. This extends results from Aldous and Pitman on GaltonWatson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous ’ CRT. 1.