1,188 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
Reflecting a Langevin Process at an Absorbing Boundary
We consider a Langevin process with white noise random forcing. We suppose
that the energy of the particle is instantaneously absorbed when it hits some
fixed obstacle. We show that nonetheless, the particle can be instantaneously
reflected, and study some properties of this reflecting solution
Some aspects of additive coalescents
We present some aspects of the so-called additive coalescence, with a focus
on its connections with random trees, Brownian excursion, certain bridges with
exchangeable increments, L\'evy processes, and sticky particle systems
A two-time-scale phenomenon in a fragmentation-coagulation process
Consider two urns, and , where initially contains a large number
of balls and is empty. At each step, with equal probability, either we
pick a ball at random in and place it in , or vice-versa (provided of
course that , or , is not empty). The number of balls in after
steps is of order , and this number remains essentially the same after
further steps. Observe that each ball in the urn after steps
has a probability bounded away from and to be placed back in the urn
after further steps. So, even though the number of balls in
does not evolve significantly between and , the precise contain
of urn does. This elementary observation is the source of an interesting
two-time-scale phenomenon which we illustrate using a simple model of
fragmentation-coagulation. Inspired by Pitman's construction of coalescing
random forests, we consider for every a uniform random tree with
vertices, and at each step, depending on the outcome of an independent fair
coin tossing, either we remove one edge chosen uniformly at random amongst the
remaining edges, or we replace one edge chosen uniformly at random amongst the
edges which have been removed previously. The process that records the sizes of
the tree-components evolves by fragmentation and coagulation. It exhibits
subaging in the sense that when it is observed after steps in the regime
with fixed, it seems to reach a statistical
equilibrium as ; but different values of yield distinct
pseudo-stationary distributions
The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations
We consider a (sub) critical Galton-Watson process with neutral mutations
(infinite alleles model), and decompose the entire population into clusters of
individuals carrying the same allele. We specify the law of this allelic
partition in terms of the distribution of the number of clone-children and the
number of mutant-children of a typical individual. The approach combines an
extension of Harris representation of Galton-Watson processes and a version of
the ballot theorem. Some limit theorems related to the distribution of the
allelic partition are also given.Comment: This version corrects a significant mistake in the first on
Almost giant clusters for percolation on large trees with logarithmic heights
This text is based on a lecture for the Sheffield Probability Day; its main
purpose is to survey some recent asymptotic results about Bernoulli bond
percolation on certain large random trees with logarithmic height. We also
provide a general criterion for the existence of giant percolation clusters in
large trees, which answers a question raised by David Croydon
A second order SDE for the Langevin process reflected at a completely inelastic boundary
It was shown recently that a Langevin process can be reflected at an energy
absorbing boundary. Here, we establish that the law of this reflecting process
can be characterized as the unique weak solution to a certain second order
stochastic differential equation with constraints, which is in sharp contrast
with a deterministic analog
Fires on trees
We consider random dynamics on the edges of a uniform Cayley tree with
vertices, in which edges are either inflammable, fireproof, or burt. Every
inflammable edge is replaced by a fireproof edge at unit rate, while fires
start at smaller rate on each inflammable edge, then propagate
through the neighboring inflammable edges and are only stopped at fireproof
edges. A vertex is called fireproof when all its adjacent edges are fireproof.
We show that as , the density of fireproof vertices converges to
when , to when , and to some non-degenerate
random variable when . We further study the connectivity of the
fireproof forest, in particular the existence of a giant component
- …