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48
Random recursive trees and the BolthausenSznitman coalescent
 Electron. J. Probab
, 2005
"... We describe a representation of the BolthausenSznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the ..."
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Cited by 34 (2 self)
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We describe a representation of the BolthausenSznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n → ∞, and obtain a scaling law for the sizes of these blocks. We also consider the discretetime Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the timereversal of this Markov chain have limits as n → ∞. These results can be interpreted as describing a “postgelation ” phase of the BolthausenSznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed. 1
Asymptotic results concerning the total branch length of the Bolthausen–Sznitman coalescent
, 2007
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Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation (Extended Abstract)
, 2012
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The cuttree of large GaltonWatson trees and the Brownian CRT. The Annals of Applied Probability
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Percolation on random triangulations and stable looptrees
 In preparation
"... We study site percolation on Angel & Schramm’s Uniform Infinite Planar Triangulation. We compute several critical and nearcritical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes (subcritical, critical and supercritical). We prove in partic ..."
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Cited by 9 (4 self)
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We study site percolation on Angel & Schramm’s Uniform Infinite Planar Triangulation. We compute several critical and nearcritical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes (subcritical, critical and supercritical). We prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3/2, which was introduced in [13]. We also give a conjecture linking looptrees of any index α ∈ (1, 2) with scaling limits of cluster boundaries in random triangulations decorated with O(N) models. Figure 1: A site percolated triangulation and the interfaces separating the clusters.
Conditioned GaltonWatson trees do not grow
 In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees Combinatorics and Probability
, 2006
"... An example is given which shows that, in general, conditioned Galton–Watson trees cannot be obtained by adding vertices one by one, while this can be done in some important but special cases, as shown by Luczak and Winkler. Keywords: conditioned Galton–Watson trees, random trees, profile 1 Monotonic ..."
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Cited by 8 (2 self)
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An example is given which shows that, in general, conditioned Galton–Watson trees cannot be obtained by adding vertices one by one, while this can be done in some important but special cases, as shown by Luczak and Winkler. Keywords: conditioned Galton–Watson trees, random trees, profile 1 Monotonicity of conditioned Galton–Watson trees? A conditioned Galton–Watson tree is a random rooted tree that is (or has the same distribution as) the family tree of a Galton–Watson process with some given offspring distribution, conditioned on the total number of vertices. We let ξ be a random variable with the given offspring distribution; i.e., the number of offspring of each individual in the Galton–Watson process is a copy of ξ. We let ξ be fixed throughout the paper, and let Tn denote the corresponding conditioned Galton–Watson tree with n vertices. For simplicity, we consider only ξ such that P(ξ = 0)> 0 and P(ξ = 1)> 0; then Tn exists for all n ≥ 1. Furthermore, we assume that E ξ = 1 (the Galton–Watson process is critical) and σ 2: = Var(ξ) < ∞. The importance of this construction lies in that many combinatorially interesting random trees are of this type, for example the following:
PRECISE LOGARITHMIC ASYMPTOTICS FOR THE RIGHT TAILS OF SOME LIMIT RANDOM VARIABLES FOR RANDOM TREES
, 2007
"... For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the righthand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (i ..."
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Cited by 8 (3 self)
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For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the righthand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton–Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown.
Distances between pairs of vertices and Vertical Profile in conditioned GaltonWatson trees
, 2009
"... We consider a conditioned Galton–Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysi ..."
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We consider a conditioned Galton–Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the latter proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet–Mélou and Janson [5], saying that the vertical profile of a randomly labelled conditioned Galton–Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion).