Results 1  10
of
89
Random trees and applications
, 2005
"... We discuss several connections between discrete and continuous ..."
Abstract

Cited by 78 (14 self)
 Add to MetaCart
(Show Context)
We discuss several connections between discrete and continuous
The topological structure of scaling limits of large planar maps
 Invent. Math
"... We discuss scaling limits of large bipartite planar maps. If p ≥ 2 is a fixed integer, we consider, for every integer n ≥ 2, a random planar map Mn which is uniformly distributed over the set of all rooted 2pangulations with n faces. Then, at least along a suitable subsequence, the metric space con ..."
Abstract

Cited by 71 (13 self)
 Add to MetaCart
(Show Context)
We discuss scaling limits of large bipartite planar maps. If p ≥ 2 is a fixed integer, we consider, for every integer n ≥ 2, a random planar map Mn which is uniformly distributed over the set of all rooted 2pangulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of Mn, equipped with the graph distance rescaled by the factor n −1/4, converges in distribution as n → ∞ towards a limiting random compact metric space, in the sense of the GromovHausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4. 1
Tessellations of random maps of arbitrary genus
, 2009
"... We investigate Voronoilike tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the ..."
Abstract

Cited by 43 (5 self)
 Add to MetaCart
We investigate Voronoilike tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost
Betacoalescents and continuous stable random trees
, 2006
"... Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case t ..."
Abstract

Cited by 42 (13 self)
 Add to MetaCart
(Show Context)
Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Λ is the Beta(2 − α, α) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here we use a recent result of Birkner et al. to prove that Betacoalescents can be embedded in continuous stable random trees, about which much is known due to recent progress of Duquesne and Le Gall. Our proof is based on a construction of the DonnellyKurtz lookdown process using continuous random trees which is of independent interest. This produces a number of results concerning the smalltime behavior of Betacoalescents. Most notably, we recover an almost sure limit theorem of the authors for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and allele frequency spectrum associated with mutations in the context of population genetics.
Scaling limits of bipartite planar maps are homeomorphic to the 2sphere
 Geom. Funct. Anal
"... ..."
Smalltime behavior of beta coalescents
, 2008
"... For a finite measure Λ on [0, 1], the Λcoalescent is a coalescent process such that, whenever there are b clusters, each ktuple of clusters merges into one at rate ∫ 1 0 xk−2 (1 − x) b−k Λ(dx). It has recently been shown that if 1 < α < 2, the Λcoalescent in which Λ is the Beta(2−α, α) dist ..."
Abstract

Cited by 32 (12 self)
 Add to MetaCart
For a finite measure Λ on [0, 1], the Λcoalescent is a coalescent process such that, whenever there are b clusters, each ktuple of clusters merges into one at rate ∫ 1 0 xk−2 (1 − x) b−k Λ(dx). It has recently been shown that if 1 < α < 2, the Λcoalescent in which Λ is the Beta(2−α, α) distribution can be used to describe the genealogy of a continuousstate branching process (CSBP) with an αstable branching mechanism. Here we use facts about CSBPs to establish new results about the smalltime asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λcoalescents for which Λ has the same asymptotic behavior near zero as the Beta(2 − α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study smalltime properties of Λcoalescents.
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 ..."
Abstract

Cited by 29 (12 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 26 (7 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 24 (4 self)
 Add to MetaCart
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