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17
SCALING LIMITS OF RANDOM PLANAR MAPS WITH A UNIQUE LARGE FACE
, 2012
"... We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We proof that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown tha ..."
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We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We proof that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n −1/2 is described by a Brownian excursion. The planar maps, with the graph metric rescaled by n −1/2, are then shown to converge in distribution towards Aldous’ Brownian tree in the Gromov–Hausdorff topology. In the proofs we rely on the Bouttier–di Francesco–Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.
SUBGAUSSIAN TAIL BOUNDS FOR THE WIDTH AND HEIGHT OF CONDITIONED GALTON–WATSON TREES.
, 2010
"... We study the height and width of a Galton–Watson tree with offspring distribution ξ satisfying E ξ = 1, 0 < Var ξ < ∞, conditioned on having exactly n nodes. Under this conditioning, we derive subGaussian tail bounds for both the width (largest number of nodes in any level) and height (greate ..."
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We study the height and width of a Galton–Watson tree with offspring distribution ξ satisfying E ξ = 1, 0 < Var ξ < ∞, conditioned on having exactly n nodes. Under this conditioning, we derive subGaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 ≤ k ≤ n.
RANDOM STABLE LOOPTREES
"... We introduce a class of random compact metric spaces Lα indexed by α ∈ (1, 2) and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can be informally be viewed as dual graphs of αstable Lévy trees. We study their properties and ..."
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We introduce a class of random compact metric spaces Lα indexed by α ∈ (1, 2) and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can be informally be viewed as dual graphs of αstable Lévy trees. We study their properties and prove in particular that the Hausdorff dimension of Lα is almost surely equal to α. We also show that stable looptrees are universal scaling limits, for the Gromov–Hausdorff topology, of various combinatorial models. In a companion paper, we prove that the stable looptree of parameter 3 2 is the scaling limit of cluster boundaries in critical sitepercolation on large random triangulations. Figure 1: An α = 1.1 stable tree, and its associated looptree L1.1, embedded non isometrically in the plane (this embedding of L1.1 contains intersecting loops, even though they are disjoint in the metric space).
Scaling limits of random graphs from subcritical classes
, 2014
"... We study the uniform random graph Cn with n vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph Cn/ n converges to the Brownian Continuum Random Tree Te multiplied by a constant scaling factor that depends on the class under consideration. In add ..."
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We study the uniform random graph Cn with n vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph Cn/ n converges to the Brownian Continuum Random Tree Te multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter D(Cn) and height H(C n) of the rooted random graph C n. We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on Cn, where we show the convergence to Te under an appropriate rescaling.
Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees
, 2013
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A NOTE ON CONDITIONED GALTONWATSON TREES
"... Abstract. We give a necessary and sufficient condition for the convergence in distribution of a conditioned GaltonWatson tree to Kesten’s tree. This yields elementary proofs of Kesten’s result as well as other known results on local limit of conditioned GaltonWatson trees. We then apply this condi ..."
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Abstract. We give a necessary and sufficient condition for the convergence in distribution of a conditioned GaltonWatson tree to Kesten’s tree. This yields elementary proofs of Kesten’s result as well as other known results on local limit of conditioned GaltonWatson trees. We then apply this condition to get new results, in the critical and subcritical cases, on the limit in distribution of a GaltonWatson tree conditioned on having a large number of individuals with outdegree in a given set. hal00813145, version 1 15 Apr 2013 1.
LOCAL LIMITS OF CONDITIONED GALTONWATSON TREES I: THE INFINITE SPINE CASE
"... Abstract. We give a necessary and sufficient condition for the convergence in distribution of a conditioned GaltonWatson tree to Kesten’s tree. This yields elementary proofs of Kesten’s result as well as other known results on local limits of conditioned GaltonWatson trees. We then apply this cond ..."
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Abstract. We give a necessary and sufficient condition for the convergence in distribution of a conditioned GaltonWatson tree to Kesten’s tree. This yields elementary proofs of Kesten’s result as well as other known results on local limits of conditioned GaltonWatson trees. We then apply this condition to get new results in the critical case, with general offspring distribution, and in the subcritical cases, with generic offspring distribution, on the limit in distribution of a GaltonWatson tree conditioned on having a large number of individuals with outdegree in a given set. hal00813145, version 3 10 Sep 2013 1.