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43
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
A note on GromovHausdorffProkhorov distance between (locally) compact measure spaces
, 2012
"... Abstract. We present an extension of the GromovHausdorff metric on the set of compact metric spaces: the GromovHausdorffProkhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the noncompact case by describing a metric on the set of rooted complet ..."
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Cited by 22 (4 self)
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Abstract. We present an extension of the GromovHausdorff metric on the set of compact metric spaces: the GromovHausdorffProkhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the noncompact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the extended GromovHausdorffProkhorov metric is a Polish space. This generalization is needed to define Lévy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure. hal00673921, version 1 24 Feb 2012 1.
Geodesics in large planar maps and in the brownian map
, 2009
"... We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the ..."
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Cited by 21 (6 self)
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We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a noncompact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.
The threepoint function of planar quadrangulations
, 2008
"... We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete threepoint function converges to a simple universal scaling function, which is the continuous threepoint f ..."
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Cited by 20 (5 self)
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We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete threepoint function converges to a simple universal scaling function, which is the continuous threepoint function of pure 2D quantum gravity. We give explicit expressions for this universal threepoint function both in the grandcanonical and canonical ensembles. Various limiting regimes are studied when some of the distances become large or small. By considering the case where the marked vertices are aligned, we also obtain the probability law for the number of geodesic points, namely vertices that lie on a geodesic path between two given vertices, and at prescribed distances from these vertices. 1.
Scaling limits of random planar maps with large faces
, 2009
"... We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index α ∈ (1,2). When the number n of vertices of the map tends to infinity, the asymptotic behavior of distance ..."
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Cited by 17 (2 self)
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We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index α ∈ (1,2). When the number n of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative jumps and index α. In particular, the profile of distances in the map, rescaled by the factor n −1/2α, converges to a random measure defined in terms of the distance process. With the same rescaling of distances, the vertex set viewed as a metric space converges in distribution as n → ∞, at least along suitable subsequences, towards a limiting random compact metric space whose Hausdorff dimension is equal to 2α.
Exit times for an increasing Lévy treevalued process
, 2012
"... We give an explicit construction of the increasing treevalued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure ..."
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Cited by 16 (9 self)
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We give an explicit construction of the increasing treevalued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure of the jumps of the increasing treevalued process to describe its behavior at the first time the tree grows higher than a given height. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the treevalued process.
Distance statistics in quadrangulations with a boundary, or with a selfavoiding loop
, 2009
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The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
, 2010
"... A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all ro ..."
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Cited by 14 (5 self)
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A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer’s bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE.
Asymptotic enumeration of constellations and related families of maps on orientable surfaces
, 2009
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