294 research outputs found
Self-similar fragmentations derived from the stable tree II: splitting at nodes
We study a natural fragmentation process of the so-called stable tree
introduced by Duquesne and Le Gall, which consists in removing the nodes of the
tree according to a certain procedure that makes the fragmentation self-similar
with positive index. Explicit formulas for the semigroup are given, and we
provide asymptotic results. We also give an alternative construction of this
fragmentation, using paths of Levy processes, hence echoing the two alternative
constructions of the standard additive coalescent by fragmenting the Brownian
continuum random tree or using Brownian paths, respectively due to
Aldous-Pitman and Bertoin.Comment: 32 page
Self-similar scaling limits of non-increasing Markov chains
We study scaling limits of non-increasing Markov chains with values in the
set of non-negative integers, under the assumption that the large jump events
are rare and happen at rates that behave like a negative power of the current
state. We show that the chain starting from and appropriately rescaled,
converges in distribution, as , to a non-increasing
self-similar Markov process. This convergence holds jointly with that of the
rescaled absorption time to the time at which the self-similar Markov process
reaches first 0. We discuss various applications to the study of random walks
with a barrier, of the number of collisions in -coalescents that do
not descend from infinity and of non-consistent regenerative compositions.
Further applications to the scaling limits of Markov branching trees are
developed in our paper, Scaling limits of Markov branching trees, with
applications to Galton--Watson and random unordered trees (2010).Comment: Published in at http://dx.doi.org/10.3150/10-BEJ312 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees
We consider a family of random trees satisfying a Markov branching property.
Roughly, this property says that the subtrees above some given height are
independent with a law that depends 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 so-called
self-similar fragmentation trees as scaling limits in the
Gromov-Hausdorff-Prokhorov topology. The main application of these results is
that the scaling limit of random uniform unordered trees is the Brownian
continuum random tree. This extends a result by Marckert-Miermont and fully
proves a conjecture by Aldous. We also recover, and occasionally extend,
results on scaling limits of consistent Markov branching models and known
convergence results of Galton-Watson trees toward the Brownian and stable
continuum random trees.Comment: Published in at http://dx.doi.org/10.1214/11-AOP686 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Self-similar fragmentations derived from the stable tree I: splitting at heights
The basic object we consider is a certain model of continuum random tree,
called the stable tree. We construct a fragmentation process
out of this tree by removing the vertices located under height . Thanks to a
self-similarity property of the stable tree, we show that the fragmentation
process is also self-similar. The semigroup and other features of the
fragmentation are given explicitly. Asymptotic results are given, as well as a
couple of related results on continuous-state branching processes.Comment: 30 page
Radius and profile of random planar maps with faces of arbitrary degrees
We prove some asymptotic results for the radius and the profile of large
random rooted planar maps with faces of arbitrary degrees. Using a bijection
due to Bouttier, Di Francesco and Guitter between rooted planar maps and
certain four-type trees with positive labels, we derive our results from a
conditional limit theorem for four-type spatial Galton-Watson trees.Comment: 25 pages, 2 figure
The genealogy of self-similar fragmentations with negative index as a continuum random tree
We encode a certain class of stochastic fragmentation processes, namely
self-similar fragmentation processes with a negative index of self-similarity,
into a metric family tree which belongs to the family of Continuum Random Trees
of Aldous. When the splitting times of the fragmentation are dense near 0, the
tree can in turn be encoded into a continuous height function, just as the
Brownian Continuum Random Tree is encoded in a normalized Brownian excursion.
Under mild hypotheses, we then compute the Hausdorff dimensions of these trees,
and the maximal H\"older exponents of the height functions
Compact Brownian surfaces I. Brownian disks
We show that, under certain natural assumptions, large random plane bipartite
maps with a boundary converge after rescaling to a one-parameter family
(, ) of random metric spaces homeomorphic to the
closed unit disk of , the space being called the
Brownian disk of perimeter and unit area. These results can be seen as an
extension of the convergence of uniform plane quadrangulations to the Brownian
map, which intuitively corresponds to the limit case where . Similar
results are obtained for maps following a Boltzmann distribution, in which the
perimeter is fixed but the area is random
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