463 research outputs found
A decorated tree approach to random permutations in substitution-closed classes
We establish a novel bijective encoding that represents permutations as
forests of decorated (or enriched) trees. This allows us to prove local
convergence of uniform random permutations from substitution-closed classes
satisfying a criticality constraint. It also enables us to reprove and
strengthen permuton limits for these classes in a new way, that uses a
semi-local version of Aldous' skeleton decomposition for size-constrained
Galton--Watson trees.Comment: New version including referee's corrections, accepted for publication
in Electronic Journal of Probabilit
Random non-crossing plane configurations: A conditioned Galton-Watson tree approach
We study various models of random non-crossing configurations consisting of
diagonals of convex polygons, and focus in particular on uniform dissections
and non-crossing trees. For both these models, we prove convergence in
distribution towards Aldous' Brownian triangulation of the disk. In the case of
dissections, we also refine the study of the maximal vertex degree and validate
a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of
an underlying Galton-Watson tree structure.Comment: 24 pages, 9 figure
Random real trees
We survey recent developments about random real trees, whose prototype is the
Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain
the formalism of real trees, which yields a neat presentation of the theory and
in particular of the relations between discrete Galton-Watson trees and
continuous random trees. We then discuss the particular class of self-similar
random real trees called stable trees, which generalize the CRT. We review
several important results concerning stable trees, including their branching
property, which is analogous to the well-known property of Galton-Watson trees,
and the calculation of their fractal dimension. We then consider spatial trees,
which combine the genealogical structure of a real tree with spatial
displacements, and we explain their connections with superprocesses. In the
last section, we deal with a particular conditioning problem for spatial trees,
which is closely related to asymptotics for random planar quadrangulations.Comment: 25 page
On scaling limits of multitype Galton-Watson trees with possibly infinite variance
In this work, we study asymptotics of multitype Galton-Watson trees with
finitely many types. We consider critical and irreducible offspring
distributions such that they belong to the domain of attraction of a stable
law, where the stability indices may differ. We show that after a proper
rescaling, their corresponding height process converges to the continuous-time
height process associated with a strictly stable spectrally positive L\'evy
process. This gives an analogue of a result obtained by Miermont in the case of
multitype Galton-Watson trees with finite covariance matrices of the offspring
distribution. Our approach relies on a remarkable decomposition for multitype
trees into monotype trees introduced by Miermont.Comment: 30 pages, 2 figure
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