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20
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models
 Ann. Probab
, 2008
"... Given any regularly varying dislocation measure, we identify a natural selfsimilar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s betasplitting models and Ford’s alpha models for phyl ..."
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Cited by 37 (13 self)
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Given any regularly varying dislocation measure, we identify a natural selfsimilar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s betasplitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.
Homogeneous fragmentation processes
, 2000
"... The purpose of this work is to define and study homogeneous fragmentation processes in continuous time, which are meant to describe the evolution of an object that breaks down randomly into pieces as time passes. Roughly, we show that the dynamic of such a fragmentation process is determined by som ..."
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Cited by 29 (4 self)
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The purpose of this work is to define and study homogeneous fragmentation processes in continuous time, which are meant to describe the evolution of an object that breaks down randomly into pieces as time passes. Roughly, we show that the dynamic of such a fragmentation process is determined by some exchangeable measure on the set of partitions of N, and results from the combination of two different phenomena: a continuous erosion and sudden dislocations. In particular, we determine the class of fragmentation measures which can arise in this setting, and investigate the evolution of the size of the fragment that contains a point pick at random at the initial time.
Scaling limits of Markov branching trees, with applications to GaltonWatson and random unordered trees
 Ann. Probab
"... We consider a family of random trees satisfying a Markov branching property. Roughly, this propertysaysthatthesubtreesabovesomegivenheightareindependentwithalawthatdepends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences ..."
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Cited by 24 (4 self)
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We consider a family of random trees satisfying a Markov branching property. Roughly, this propertysaysthatthesubtreesabovesomegivenheightareindependentwithalawthatdepends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences of distributions on partitions of the integers, that determine how the size of a tree is distributed in its different subtrees. Under some natural assumption on these distributions, stipulating that “macroscopic ” splitting events are rare, we show that Markov branching trees admit the socalled selfsimilar fragmentation trees as scaling limits in the GromovHausdorffProkhorov topology. Applications include scaling limits of consistent Markov branching model, and convergence of GaltonWatson trees towards the Brownian and stable continuum random trees. We also obtain that random uniform unordered trees have the Brownian tree as a scaling limit, hence
Exchangeable fragmentationcoalescence processes and their equilibrium distribution
 Electr. J. Prob
, 2004
"... We define and study a family of Markov processes with state space the compact set of all partitions of N that we call exchangeable fragmentationcoalescence processes. They can be viewed as a combination of exchangeable fragmentation as defined by Bertoin and of homogenous coalescence as defined by ..."
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Cited by 13 (1 self)
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We define and study a family of Markov processes with state space the compact set of all partitions of N that we call exchangeable fragmentationcoalescence processes. They can be viewed as a combination of exchangeable fragmentation as defined by Bertoin and of homogenous coalescence as defined by Pitman and Schweinsberg or Möhle and Sagitov. We show that they admit a unique invariant probability measure and we study some properties of their paths and of their equilibrium measure. Key words. Fragmentation, coalescence, invariant distribution. A.M.S. Classification. 60 J 25, 60 G 09. 1
A representation of exchangeable hierarchies by sampling from random real trees
, 2011
"... A hierarchy on a set S, also called a total partition of S, is a collection H of subsets of S such that S ∈ H, each singleton subset of S belongs to H, and if A,B ∈ H then A ∩ B equals either A or B or ∅. Every exchangeable random hierarchy of positive integers has the same distribution as a random ..."
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Cited by 7 (1 self)
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A hierarchy on a set S, also called a total partition of S, is a collection H of subsets of S such that S ∈ H, each singleton subset of S belongs to H, and if A,B ∈ H then A ∩ B equals either A or B or ∅. Every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy H associated as follows with a random real tree T equipped with root element 0 and a random probability distribution p on the Borel subsets of T: given (T, p), let t1, t2,... be independent and identically distributed according to p, and let H comprise all singleton subsets of N, and every subset of the form {j: tj ∈ Fx} as x ranges over T, where Fx is the fringe subtree of T rooted at x. There is also the alternative characterization: every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy H derived as follows from a random hierarchyH on [0, 1] and a family (Uj) of IID uniform [0,1] random variables independent ofH: let H comprise all sets of the form {j: Uj ∈ B} as B ranges over the members ofH.
Fragmentation of ordered partitions and intervals
 Electron. J. Probab
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TRAVELING WAVES AND HOMOGENEOUS FRAGMENTATION
, 2011
"... We formulate the notion of the classical Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit ..."
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Cited by 4 (0 self)
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We formulate the notion of the classical Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit
Energy efficiency of consecutive fragmentation processes
 J. Appl. Probab
, 2010
"... We present a first study on the energy required to reduce a unit mass fragment by consecutively using several devices, as it happens in the mining industry. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the selfsimilar energy model introdu ..."
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Cited by 3 (0 self)
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We present a first study on the energy required to reduce a unit mass fragment by consecutively using several devices, as it happens in the mining industry. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the selfsimilar energy model introduced by Bertoin and Martínez [7], we compute the average energy required to attain a size η0 with this twodevice procedure. We then asymptotically compare, as η0 goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that for certain range of parameters of the fragmentation processes and of their energy costfunctions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or conversely.
Equilibrium for Fragmentation With Immigration
, 2005
"... This paper introduces stochastic processes that describe the evolution of systems of particles in which particles immigrate according to a Poisson measure and split according to a selfsimilar fragmentation. Criteria for existence and absence of stationary distributions are established and uniquenes ..."
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Cited by 2 (0 self)
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This paper introduces stochastic processes that describe the evolution of systems of particles in which particles immigrate according to a Poisson measure and split according to a selfsimilar fragmentation. Criteria for existence and absence of stationary distributions are established and uniqueness is proved. Also, convergence rates to the stationary distribution are given. Linear equations which are the deterministic counterparts of fragmentation with immigration processes are next considered. As in the stochastic case, existence and uniqueness of solutions, as well as existence and uniqueness of stationary solutions, are investigated.