70,291 research outputs found
Random enriched trees with applications to random graphs
We establish limit theorems that describe the asymptotic local and global
geometric behaviour of random enriched trees considered up to symmetry. We
apply these general results to random unlabelled weighted rooted graphs and
uniform random unlabelled -trees that are rooted at a -clique of
distinguishable vertices. For both models we establish a Gromov--Hausdorff
scaling limit, a Benjamini--Schramm limit, and a local weak limit that
describes the asymptotic shape near the fixed root
Multicritical continuous random trees
We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps
Scaling limit of critical random trees in random environment
We consider Bienaym\'e-Galton-Watson trees in random environment, where each
generation is attributed a random offspring distribution , and
is a sequence of independent and identically distributed
random probability measures. We work in the "strictly critical" regime where,
for all , the average of is assumed to be equal to almost
surely, and the variance of has finite expectation. We prove that, for
almost all realisations of the environment (more precisely, under some
deterministic conditions that the random environment satisfies almost surely),
the scaling limit of the tree in that environment, conditioned to be large, is
the Brownian continuum random tree. Standard techniques used for standard
Bienaym\'e-Galton-Watson trees do not apply to this case, and our proof
therefore provides an alternative approach for showing scaling limits of random
trees. In particular, we make a (to our knowledge) novel connection between the
Lukasiewicz path and the height process of the tree, by combining a discrete
version of the L\'evy snake introduced by Le Gall and the spine decomposition
Scaling limits of slim and fat trees
We consider Galton--Watson trees conditioned on both the total number of
vertices and the number of leaves . The focus is on the case in which
both and grow to infinity and , with . Assuming the exponential decay of the offspring distribution, we show
that the rescaled random tree converges in distribution to Aldous' Continuum
Random Tree with respect to the Gromov--Hausdorff topology. The scaling depends
on a parameter which we calculate explicitly. Additionally, we
compute the limit for the degree sequences of these random trees.Comment: 37 pages, 2 figure
The statistical geometry of scale-free random trees
The properties of scale-free random trees are investigated using both
preconditioning on non-extinction and fixed size averages, in order to study
the thermodynamic limit. The scaling form of volume probability is found, the
connectivity dimensions are determined and compared with other exponents which
describe the growth. The (local) spectral dimension is also determined, through
the study of the massless limit of the Gaussian model on such trees.Comment: 21 pages, 2 figures, revtex4, minor changes (published version
Scaling limit of the invasion percolation cluster on a regular tree
We prove existence of the scaling limit of the invasion percolation cluster
(IPC) on a regular tree. The limit is a random real tree with a single end. The
contour and height functions of the limit are described as certain diffusive
stochastic processes. This convergence allows us to recover and make precise
certain asymptotic results for the IPC. In particular, we relate the limit of
the rescaled level sets of the IPC to the local time of the scaled height
function.Comment: Published in at http://dx.doi.org/10.1214/11-AOP731 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
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