10,452 research outputs found
Random Planar Lattices and Integrated SuperBrownian Excursion
In this paper, a surprising connection is described between a specific brand
of random lattices, namely planar quadrangulations, and Aldous' Integrated
SuperBrownian Excursion (ISE). As a consequence, the radius r_n of a random
quadrangulation with n faces is shown to converge, up to scaling, to the width
r=R-L of the support of the one-dimensional ISE. More generally the
distribution of distances to a random vertex in a random quadrangulation is
described in its scaled limit by the random measure ISE shifted to set the
minimum of its support in zero.
The first combinatorial ingredient is an encoding of quadrangulations by
trees embedded in the positive half-line, reminiscent of Cori and Vauquelin's
well labelled trees. The second step relates these trees to embedded (discrete)
trees in the sense of Aldous, via the conjugation of tree principle, an
analogue for trees of Vervaat's construction of the Brownian excursion from the
bridge.
From probability theory, we need a new result of independent interest: the
weak convergence of the encoding of a random embedded plane tree by two contour
walks to the Brownian snake description of ISE.
Our results suggest the existence of a Continuum Random Map describing in
term of ISE the scaled limit of the dynamical triangulations considered in
two-dimensional pure quantum gravity.Comment: 44 pages, 22 figures. Slides and extended abstract version are
available at http://www.loria.fr/~schaeffe/Pub/Diameter/ and
http://www.iecn.u-nancy.fr/~chassain
The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
A unicellular map is a map which has only one face. We give a bijection
between a dominant subset of rooted unicellular maps of fixed genus and a set
of rooted plane trees with distinguished vertices. The bijection applies as
well to the case of labelled unicellular maps, which are related to all rooted
maps by Marcus and Schaeffer's bijection.
This gives an immediate derivation of the asymptotic number of unicellular
maps of given genus, and a simple bijective proof of a formula of Lehman and
Walsh on the number of triangulations with one vertex. From the labelled case,
we deduce an expression of the asymptotic number of maps of genus g with n
edges involving the ISE random measure, and an explicit characterization of the
limiting profile and radius of random bipartite quadrangulations of genus g in
terms of the ISE.Comment: 27pages, 6 figures, to appear in PTRF. Version 2 includes corrections
from referee report in sections 6-
The vertical profile of embedded trees
Consider a rooted binary tree with n nodes. Assign with the root the abscissa
0, and with the left (resp. right) child of a node of abscissa i the abscissa
i-1 (resp. i+1). We prove that the number of binary trees of size n having
exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is with n_{l-1}=n_{r+1}=0. The
sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the
tree. The vertical profile of a uniform random tree of size n is known to
converge, in a certain sense and after normalization, to a random mesure called
the integrated superbrownian excursion, which motivates our interest in the
profile. We prove similar looking formulas for other families of trees whose
nodes are embedded in Z. We also refine these formulas by taking into account
the number of nodes at abscissa j whose parent lies at abscissa i, and/or the
number of vertices at abscissa i having a prescribed number of children at
abscissa j, for all i and j. Our proofs are bijective.Comment: 47 page
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
The lineage process in Galton--Watson trees and globally centered discrete snakes
We consider branching random walks built on Galton--Watson trees with
offspring distribution having a bounded support, conditioned to have nodes,
and their rescaled convergences to the Brownian snake. We exhibit a notion of
``globally centered discrete snake'' that extends the usual settings in which
the displacements are supposed centered. We show that under some additional
moment conditions, when goes to , ``globally centered discrete
snakes'' converge to the Brownian snake. The proof relies on a precise study of
the lineage of the nodes in a Galton--Watson tree conditioned by the size, and
their links with a multinomial process [the lineage of a node is the vector
indexed by giving the number of ancestors of having children
and for which is a descendant of the th one]. Some consequences
concerning Galton--Watson trees conditioned by the size are also derived.Comment: Published in at http://dx.doi.org/10.1214/07-AAP450 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
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