93 research outputs found
The topological structure of scaling limits of large planar maps
We discuss scaling limits of large bipartite planar maps. If p is a fixed
integer strictly greater than 1, we consider a random planar map M(n) which is
uniformly distributed over the set of all 2p-angulations with n faces. Then, at
least along a suitable subsequence, the metric space M(n) equipped with the
graph distance rescaled by the factor n to the power -1/4 converges in
distribution as n tends to infinity towards a limiting random compact metric
space, in the sense of the Gromov-Hausdorff distance. We prove that the
topology of the limiting space is uniquely determined independently of p, and
that this space can be obtained as the quotient of the Continuum Random Tree
for an equivalence relation which is defined from Brownian labels attached to
the vertices. We also verify that the Hausdorff dimension of the limit is
almost surely equal to 4.Comment: 45 pages Second version with minor modification
Breadth first search coding of multitype forests with application to Lamperti representation
We obtain a bijection between some set of multidimensional sequences and this
of -type plane forests which is based on the breadth first search algorithm.
This coding sequence is related to the sequence of population sizes indexed by
the generations, through a Lamperti type transformation. The same
transformation in then obtained in continuous time for multitype branching
processes with discrete values. We show that any such process can be obtained
from a dimensional compound Poisson process time changed by some integral
functional. Our proof bears on the discretisation of branching forests with
edge lengths
Positive solutions of Schr\"odinger equations and fine regularity of boundary points
Given a Lipschitz domain in and a nonnegative
potential in such that is bounded
in we study the fine regularity of boundary points with respect to
the Schr\"odinger operator in . Using potential
theoretic methods, several conditions equivalent to the fine regularity of are established. The main result is a simple (explicit if
is smooth) necessary and sufficient condition involving the size of
for to be finely regular. An essential intermediate result consists in
a majorization of for
positive harmonic in and . Conditions for
almost everywhere regularity in a subset of are also
given as well as an extension of the main results to a notion of fine
-regularity, if , being two potentials, with and a second order elliptic operator.Comment: version 1. 23 pages version 3. 28 pages. Mainly a typo in Theorem 1.1
is correcte
Examining the Frontal Subcortical Brain Vulnerability Hypothesis in Children With Neurofibromatosis Type 1: Are T2-Weighted Hyperintensities Related to Executive Dysfunction?
Objective: It was hypothesized that neuropsychological impairments in children with neurofibromatosis type I (NF1) are associated with brain areas of increased T2-weighted signal intensity on MRI. Systematic and extensive examination of this hypothesis remains however scarce, particularly regarding executive dysfunction whereas hyperintensities are located preferentially in frontal-sub-cortical networks. In this study, we compared the executive functioning profile with characteristics of brain hyperintensities in children with NF1. Method: A sample of 36 school-age children with NF1 (7-12 years) underwent a detailed examination of executive function, including performance-based tests and child\u27s behavior rating in daily life. Executive function measures were compared with the characteristics of the T2-weighted hyperintensities on parallel MRI scans. The presence, number, and size of hyperintensities in the whole brain were considered as well as their main cerebral locations. Results: Executive dysfunction including traditional cognitive and ecological measures in children with NF1 is not significantly influenced by T2-weighted hyperintensities, in terms of presence or not, number, size, and location, whether in the whole brain or according to involved specific brain areas. Conclusion: T2-weighted hyperintensities, as they are currently measured, cannot be used as a strong indicator of executive dysfunction in children with NF1. Based on the available NF1 cognitive impairment pathogenesis models, a critical discussion on anatomical-functional relationships between hyperintensities and neuropsychological profile is proposed, especially the executive dysfunction. (PsycINFO Database Record (c) 2014 APA, all rights reserved)
Self-intersection local time of planar Brownian motion based on a strong approximation by random walks
The main purpose of this work is to define planar self-intersection local
time by an alternative approach which is based on an almost sure pathwise
approximation of planar Brownian motion by simple, symmetric random walks. As a
result, Brownian self-intersection local time is obtained as an almost sure
limit of local averages of simple random walk self-intersection local times. An
important tool is a discrete version of the Tanaka--Rosen--Yor formula; the
continuous version of the formula is obtained as an almost sure limit of the
discrete version. The author hopes that this approach to self-intersection
local time is more transparent and elementary than other existing ones.Comment: 36 pages. A new part on renormalized self-intersection local time has
been added and several inaccuracies have been corrected. To appear in Journal
of Theoretical Probabilit
Random walks and polymers in the presence of quenched disorder
After a general introduction to the field, we describe some recent results
concerning disorder effects on both `random walk models', where the random walk
is a dynamical process generated by local transition rules, and on `polymer
models', where each random walk trajectory representing the configuration of a
polymer chain is associated to a global Boltzmann weight. For random walk
models, we explain, on the specific examples of the Sinai model and of the trap
model, how disorder induces anomalous diffusion, aging behaviours and Golosov
localization, and how these properties can be understood via a strong disorder
renormalization approach. For polymer models, we discuss the critical
properties of various delocalization transitions involving random polymers. We
first summarize some recent progresses in the general theory of random critical
points : thermodynamic observables are not self-averaging at criticality
whenever disorder is relevant, and this lack of self-averaging is directly
related to the probability distribution of pseudo-critical temperatures
over the ensemble of samples of size . We describe the
results of this analysis for the bidimensional wetting and for the
Poland-Scheraga model of DNA denaturation.Comment: 17 pages, Conference Proceedings "Mathematics and Physics", I.H.E.S.,
France, November 200
Random Convex Hulls and Extreme Value Statistics
In this paper we study the statistical properties of convex hulls of
random points in a plane chosen according to a given distribution. The points
may be chosen independently or they may be correlated. After a non-exhaustive
survey of the somewhat sporadic literature and diverse methods used in the
random convex hull problem, we present a unifying approach, based on the notion
of support function of a closed curve and the associated Cauchy's formulae,
that allows us to compute exactly the mean perimeter and the mean area enclosed
by the convex polygon both in case of independent as well as correlated points.
Our method demonstrates a beautiful link between the random convex hull problem
and the subject of extreme value statistics. As an example of correlated
points, we study here in detail the case when the points represent the vertices
of independent random walks. In the continuum time limit this reduces to
independent planar Brownian trajectories for which we compute exactly, for
all , the mean perimeter and the mean area of their global convex hull. Our
results have relevant applications in ecology in estimating the home range of a
herd of animals. Some of these results were announced recently in a short
communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special
issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting
Asymptotic behavior of solutions to the -Yamabe equation near isolated singularities
-Yamabe equations are conformally invariant equations generalizing
the classical Yamabe equation. In an earlier work YanYan Li proved that an
admissible solution with an isolated singularity at to the
-Yamabe equation is asymptotically radially symmetric. In this work
we prove that an admissible solution with an isolated singularity at to the -Yamabe equation is asymptotic to a radial
solution to the same equation on . These results
generalize earlier pioneering work in this direction on the classical Yamabe
equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli
et al, we formulate and prove a general asymptotic approximation result for
solutions to certain ODEs which include the case for scalar curvature and
curvature cases. An alternative proof is also provided using
analysis of the linearized operators at the radial solutions, along the lines
of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.Comment: 55 page
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