8 research outputs found
Uniform Infinite Planar Triangulations
The existence of the weak limit as n --> infinity of the uniform measure on
rooted triangulations of the sphere with n vertices is proved. Some properties
of the limit are studied. In particular, the limit is a probability measure on
random triangulations of the plane.Comment: 36 pages, 4 figures; Journal revised versio
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
Logical limit laws for minor-closed classes of graphs
Let be an addable, minor-closed class of graphs. We prove that
the zero-one law holds in monadic second-order logic (MSO) for the random graph
drawn uniformly at random from all {\em connected} graphs in on
vertices, and the convergence law in MSO holds if we draw uniformly at
random from all graphs in on vertices. We also prove analogues
of these results for the class of graphs embeddable on a fixed surface,
provided we restrict attention to first order logic (FO). Moreover, the
limiting probability that a given FO sentence is satisfied is independent of
the surface . We also prove that the closure of the set of limiting
probabilities is always the finite union of at least two disjoint intervals,
and that it is the same for FO and MSO. For the classes of forests and planar
graphs we are able to determine the closure of the set of limiting
probabilities precisely. For planar graphs it consists of exactly 108
intervals, each of length . Finally, we analyse
examples of non-addable classes where the behaviour is quite different. For
instance, the zero-one law does not hold for the random caterpillar on
vertices, even in FO.Comment: minor changes; accepted for publication by JCT
Limit of normalized quadrangulations: The Brownian map
Consider a random pointed quadrangulation chosen equally likely among
the pointed quadrangulations with faces. In this paper we show that, when
goes to , suitably normalized converges weakly in a certain
sense to a random limit object, which is continuous and compact, and that we
name the Brownian map. The same result is shown for a model of rooted
quadrangulations and for some models of rooted quadrangulations with random
edge lengths. A metric space of rooted (resp. pointed) abstract maps that
contains the model of discrete rooted (resp. pointed) quadrangulations and the
model of the Brownian map is defined. The weak convergences hold in these
metric spaces.Comment: Published at http://dx.doi.org/10.1214/009117906000000557 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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0-1 Laws for Maps
A class of finite structures has a 0--1 law with respect to a logic if every property expressible in the logic has a probability approaching a limit of 0 or 1 as the structure size grows. To formulate 0--1 laws for maps (i.e., embeddings of graphs in a surface), it is necessary to represent maps as logical structures. Three such representations are given, the most general being the full cross representation based on Tutte's theory of combinatorial maps. The main result says that if a class of maps has two properties, richness and large representativity, then the corresponding class of full cross representations has a 0--1 law with respect to first-order logic. As a corollary the following classes of maps on a surface of fixed type have a first-order 0--1 law: all maps, smooth maps, 2-connected maps, 3-connected maps, triangular maps, 2-connected triangular maps, and 3-connected triangular maps. c fl ??? John Wiley & Sons, Inc. Keywords: 0--1 law, maps 1. INTRODUCTION. In probability..