50 research outputs found
Conditioned Galton-Watson trees do not grow
An example is given which shows that, in general, conditioned Galton-Watson
trees cannot be obtained by adding vertices one by one, as has been shown in a
special case by Luczak and Winkler.Comment: 5 pages, 2 figure
Sub-Gaussian tail bounds for the width and height of conditioned Galton--Watson trees
We study the height and width of a Galton--Watson tree with offspring
distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having
exactly n nodes. Under this conditioning, we derive sub-Gaussian 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.Comment: 15 page
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
Destruction of very simple trees
We consider the total cost of cutting down a random rooted tree chosen from a
family of so-called very simple trees (which include ordered trees, -ary
trees, and Cayley trees); these form a subfamily of simply generated trees. At
each stage of the process an edge is chose at random from the tree and cut,
separating the tree into two components. In the one-sided variant of the
process the component not containing the root is discarded, whereas in the
two-sided variant both components are kept. The process ends when no edges
remain for cutting. The cost of cutting an edge from a tree of size is
assumed to be . Using singularity analysis and the method of moments,
we derive the limiting distribution of the total cost accrued in both variants
of this process. A salient feature of the limiting distributions obtained
(after normalizing in a family-specific manner) is that they only depend on
.Comment: 20 pages; Version 2 corrects some minor error and fixes a few typo
A construction of a -coalescent via the pruning of Binary Trees
Considering a random binary tree with labelled leaves, we use a pruning
procedure on this tree in order to construct a -coalescent
process. We also use the continuous analogue of this construction, i.e. a
pruning procedure on Aldous's continuum random tree, to construct a continuous
state space process that has the same structure as the -coalescent
process up to some time change. These two constructions unable us to obtain
results on the coalescent process such as the asymptotics on the number of
coalescent events or the law of the blocks involved in the last coalescent
event
The forest associated with the record process on a L\'evy tree
We perform a pruning procedure on a L\'evy tree and instead of throwing away
the removed sub-tree, we regraft it on a given branch (not related to the
L\'evy tree). We prove that the tree constructed by regrafting is distributed
as the original L\'evy tree, generalizing a result where only Aldous's tree is
considered. As a consequence, we obtain that the quantity which represents in
some sense the number of cuts needed to isolate the root of the tree, is
distributed as the height of a leaf picked at random in the L\'evy tree
Random recursive trees and the Bolthausen-Sznitman coalescent
We describe a representation of the Bolthausen-Sznitman coalescent in terms
of the cutting of random recursive trees. Using this representation, we prove
results concerning the final collision of the coalescent restricted to [n]: we
show that the distribution of the number of blocks involved in the final
collision converges as n tends to infinity, and obtain a scaling law for the
sizes of these blocks. We also consider the discrete-time Markov chain giving
the number of blocks after each collision of the coalescent restricted to [n];
we show that the transition probabilities of the time-reversal of this Markov
chain have limits as n tends to infinity. These results can be interpreted as
describing a ``post-gelation'' phase of the Bolthausen-Sznitman coalescent, in
which a giant cluster containing almost all of the mass has already formed and
the remaining small blocks are being absorbed.Comment: 28 pages, 2 figures. Revised version with minor alterations. To
appear in Electron. J. Proba
The k-Cut Model in Conditioned Galton-Watson Trees
The k-cut number of rooted graphs was introduced by Cai et al. [Cai and Holmgren, 2019] as a generalization of the classical cutting model by Meir and Moon [Meir and Moon, 1970]. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson [Janson, 2006]
Record process on the Continuum Random Tree
By considering a continuous pruning procedure on Aldous's Brownian tree, we
construct a random variable which is distributed, conditionally given
the tree, according to the probability law introduced by Janson as the limit
distribution of the number of cuts needed to isolate the root in a critical
Galton-Watson tree. We also prove that this random variable can be obtained as
the a.s. limit of the number of cuts needed to cut down the subtree of the
continuum tree spanned by leaves