34,449 research outputs found
Branching Processes, and Random-Cluster Measures on Trees
Random-cluster measures on infinite regular trees are studied in conjunction
with a general type of `boundary condition', namely an equivalence relation on
the set of infinite paths of the tree. The uniqueness and non-uniqueness of
random-cluster measures are explored for certain classes of equivalence
relations. In proving uniqueness, the following problem concerning branching
processes is encountered and answered. Consider bond percolation on the
family-tree of a branching process. What is the probability that every
infinite path of , beginning at its root, contains some vertex which is
itself the root of an infinite open sub-tree
Information Ranking and Power Laws on Trees
We study the situations when the solution to a weighted stochastic recursion
has a power law tail. To this end, we develop two complementary approaches, the
first one extends Goldie's (1991) implicit renewal theorem to cover recursions
on trees; and the second one is based on a direct sample path large deviations
analysis of weighted recursive random sums. We believe that these methods may
be of independent interest in the analysis of more general weighted branching
processes as well as in the analysis of algorithms
General branching processes in discrete time as random trees
The simple Galton--Watson process describes populations where individuals
live one season and are then replaced by a random number of children. It can
also be viewed as a way of generating random trees, each vertex being an
individual of the family tree. This viewpoint has led to new insights and a
revival of classical theory. We show how a similar reinterpretation can shed
new light on the more interesting forms of branching processes that allow
repeated bearings and, thus, overlapping generations. In particular, we use the
stable pedigree law to give a transparent description of a size-biased version
of general branching processes in discrete time. This allows us to analyze the
condition for exponential growth of supercritical general processes
as well as relation between simple Galton--Watson and more general branching
processes.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ138 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Continuum random trees and branching processes with immigration
We study a genealogical model for continuous-state branching processes with
immigration with a (sub)critical branching mechanism. This model allows the
immigrants to be on the same line of descent. The corresponding family tree is
an ordered rooted continuum random tree with a single infinite end defined
thanks to two continuous processes denoted by
and that code the parts at resp. the left and
the right hand of the infinite line of descent of the tree. These processes are
called the left and the right height processes. We define their local time
processes via an approximation procedure and we prove that they enjoy a
Ray-Knight property. We also discuss the important special case corresponding
to the size-biased Galton-Watson tree in the continuous setting. In the last
part of the paper we give a convergence result under general assumptions for
rescaled discrete left and right contour processes of sequences of
Galton-Watson trees with immigration. We also provide a strong invariance
principle for a sequence of rescaled Galton-Watson processes with immigration
that also holds in the supercritical case.Comment: 35 page
Linear-Time Model Checking Branching Processes
(Multi-type) branching processes are a natural and well-studied model for generating random infinite trees. Branching processes feature both nondeterministic and probabilistic branching, generalizing both transition systems and Markov chains (but not generally Markov decision processes). We study the complexity of model checking branching processes against linear-time omega-regular specifications: is it the case almost surely that every branch of a tree randomly generated by the branching process satisfies the omega-regular specification? The main result is that for LTL specifications this problem is in PSPACE, subsuming classical results for transition systems and Markov chains, respectively. The underlying general model-checking algorithm is based on the automata-theoretic approach, using unambiguous Büchi automata
Fringe trees, Crump-Mode-Jagers branching processes and -ary search trees
This survey studies asymptotics of random fringe trees and extended fringe
trees in random trees that can be constructed as family trees of a
Crump-Mode-Jagers branching process, stopped at a suitable time. This includes
random recursive trees, preferential attachment trees, fragmentation trees,
binary search trees and (more generally) -ary search trees, as well as some
other classes of random trees.
We begin with general results, mainly due to Aldous (1991) and Jagers and
Nerman (1984). The general results are applied to fringe trees and extended
fringe trees for several particular types of random trees, where the theory is
developed in detail. In particular, we consider fringe trees of -ary search
trees in detail; this seems to be new.
Various applications are given, including degree distribution, protected
nodes and maximal clades for various types of random trees. Again, we emphasise
results for -ary search trees, and give for example new results on protected
nodes in -ary search trees.
A separate section surveys results on height, saturation level, typical depth
and total path length, due to Devroye (1986), Biggins (1995, 1997) and others.
This survey contains well-known basic results together with some additional
general results as well as many new examples and applications for various
classes of random trees
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
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