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42
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
Random trees and applications
, 2005
"... We discuss several connections between discrete and continuous ..."
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Cited by 29 (4 self)
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We discuss several connections between discrete and continuous
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 27 (11 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.
Subtree prune and regraft: A reversible realtree valued Markov chain
 Ann. Prob
"... Abstract. We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technic ..."
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Cited by 25 (5 self)
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Abstract. We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov– Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion. 1.
Convergence in distribution of random metric measure spaces (Λcoalescent measure trees)
, 2007
"... We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topol ..."
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Cited by 23 (4 self)
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We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows provided the sequence is tight from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultrametric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra)metric measure spaces given by the random genealogies of the Λcoalescents. We show that the Λcoalescent defines an infinite (random) metric measure space if and only if the socalled “dustfree”property holds.
Limits of normalized quadrangulations. The Brownian map
 Ann. Probab
, 2004
"... Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper, we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name t ..."
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Cited by 20 (0 self)
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Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper, we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of Brownian map is defined. The weak convergences hold in these metric spaces. 1
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
Regenerative tree growth: binary selfsimilar continuum random trees and PoissonDirichlet compositions
, 2008
"... We use a natural ordered extension of the Chinese Restaurant Process to grow a twoparameter family of binary selfsimilar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a dist ..."
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Cited by 14 (7 self)
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We use a natural ordered extension of the Chinese Restaurant Process to grow a twoparameter family of binary selfsimilar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford’s trees. In general, the Markov branching trees induced by the twoparameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urnmodel description of sampling from Dirichlet random distributions. 1. Introduction. We
Conditioned Brownian trees
"... We consider a Brownian tree consisting of a collection of onedimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus ..."
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Cited by 14 (6 self)
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We consider a Brownian tree consisting of a collection of onedimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus yields a convenient representation of the socalled Integrated SuperBrownian Excursion (ISE), which can be viewed as the uniform probability measure on the tree of paths. We discuss different approaches that lead to the definition of the Brownian tree conditioned to stay on the positive halfline. We also establish a Verwaatlike theorem showing that this conditioned Brownian tree can be obtained by rerooting the unconditioned one at the vertex corresponding to the minimal spatial position. In terms of ISE, this theorem yields the following fact: Conditioning ISE to put no mass on]−∞, −ε [ and letting ε go to 0 is equivalent to shifting the unconditioned ISE to the right so that the leftmost point of its support becomes the origin. We derive a number of explicit estimates and formulas for our conditioned Brownian trees. In particular, the probability that ISE puts no mass on] − ∞, −ε [ is shown to behave like 2ε 4 /21 when ε goes to 0. Finally, for the conditioned Brownian tree with a fixed height h, we obtain a decomposition involving a spine whose distribution is absolutely continuous with respect to that of a ninedimensional Bessel process on the time interval [0,h], and Poisson processes of subtrees originating from this spine. 1
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.