Results 1 
9 of
9
Rescaled bipartite planar maps converge to the Brownian map
 In preparation
, 2013
"... Abstract. For every integer n ≥ 1, we consider a random planar map Mn which is uniformly distributed over the class of all rooted bipartite planar maps with n edges. We prove that the vertex set ofMn equipped with the graph distance rescaled by the factor (2n)−1/4 converges in distribution, in the G ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Abstract. For every integer n ≥ 1, we consider a random planar map Mn which is uniformly distributed over the class of all rooted bipartite planar maps with n edges. We prove that the vertex set ofMn equipped with the graph distance rescaled by the factor (2n)−1/4 converges in distribution, in the GromovHausdorff sense, to the Brownian map. This complements several recent results giving the convergence of various classes of random planar maps to the Brownian map. 1.
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
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).
figure shows a large critical Galton–Watson tree with finite variance.
, 2013
"... We study a particular type of subcritical Galton–Watson trees, which are called nongeneric trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned nongeneric trees, meaning that with high probability th ..."
Abstract
 Add to MetaCart
(Show Context)
We study a particular type of subcritical Galton–Watson trees, which are called nongeneric trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned nongeneric trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. Using recent results concerning subexponential distributions, we investigate this phenomenon by studying scaling limits of such trees and show that the situation is completely different from the critical case. In particular, the height of such trees grows logarithmically in their size. We also study fluctuations around the condensation vertex.