4,819 research outputs found
Confluence of geodesic paths and separating loops in large planar quadrangulations
We consider planar quadrangulations with three marked vertices and discuss
the geometry of triangles made of three geodesic paths joining them. We also
study the geometry of minimal separating loops, i.e. paths of minimal length
among all closed paths passing by one of the three vertices and separating the
two others in the quadrangulation. We concentrate on the universal scaling
limit of large quadrangulations, also known as the Brownian map, where pairs of
geodesic paths or minimal separating loops have common parts of non-zero
macroscopic length. This is the phenomenon of confluence, which distinguishes
the geometry of random quadrangulations from that of smooth surfaces. We
characterize the universal probability distribution for the lengths of these
common parts.Comment: 48 pages, 33 color figures. Final version, with one concluding
paragraph and one reference added, and several other small correction
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
Random trees between two walls: Exact partition function
We derive the exact partition function for a discrete model of random trees
embedded in a one-dimensional space. These trees have vertices labeled by
integers representing their position in the target space, with the SOS
constraint that adjacent vertices have labels differing by +1 or -1. A
non-trivial partition function is obtained whenever the target space is bounded
by walls. We concentrate on the two cases where the target space is (i) the
half-line bounded by a wall at the origin or (ii) a segment bounded by two
walls at a finite distance. The general solution has a soliton-like structure
involving elliptic functions. We derive the corresponding continuum scaling
limit which takes the remarkable form of the Weierstrass p-function with
constrained periods. These results are used to analyze the probability for an
evolving population spreading in one dimension to attain the boundary of a
given domain with the geometry of the target (i) or (ii). They also translate,
via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main
modifications in Sect. 5-6 and conclusio
Oh, Behave! Behavior as an Interaction between Genes & the Environment
This lesson is designed to teach students that behavior is a trait shaped by both genes and the environment. Students will read a scientific paper, discuss and generate predictions based on the ideas and data therein, and model the relationships between genes, the environment, and behavior. The lesson is targeted to meet the educational goals of undergraduate introductory biology, evolution, and animal behavior courses, but it is also suitable for advanced high school biology students. This lesson meets the criteria for the Next Generation Science Standard HS-LS4, Biological Evolution: Unity and Diversity (NGSS Lead States, 2013)
Distance statistics in large toroidal maps
We compute a number of distance-dependent universal scaling functions
characterizing the distance statistics of large maps of genus one. In
particular, we obtain explicitly the probability distribution for the length of
the shortest non-contractible loop passing via a random point in the map, and
that for the distance between two random points. Our results are derived in the
context of bipartite toroidal quadrangulations, using their coding by
well-labeled 1-trees, which are maps of genus one with a single face and
appropriate integer vertex labels. Within this framework, the distributions
above are simply obtained as scaling limits of appropriate generating functions
for well-labeled 1-trees, all expressible in terms of a small number of basic
scaling functions for well-labeled plane trees.Comment: 24 pages, 9 figures, minor corrections, new added reference
Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model
We use the nested loop approach to investigate loop models on random planar
maps where the domains delimited by the loops are given two alternating colors,
which can be assigned different local weights, hence allowing for an explicit
Z_2 domain symmetry breaking. Each loop receives a non local weight n, as well
as a local bending energy which controls loop turns. By a standard cluster
construction that we review, the Q = n^2 Potts model on general random maps is
mapped to a particular instance of this problem with domain-non-symmetric
weights. We derive in full generality a set of coupled functional relations for
a pair of generating series which encode the enumeration of loop configurations
on maps with a boundary of a given color, and solve it by extending well-known
complex analytic techniques. In the case where loops are fully-packed, we
analyze in details the phase diagram of the model and derive exact equations
for the position of its non-generic critical points. In particular, we
underline that the critical Potts model on general random maps is not self-dual
whenever Q \neq 1. In a model with domain-symmetric weights, we also show the
possibility of a spontaneous domain symmetry breaking driven by the bending
energy.Comment: 44 pages, 13 figure
Packing and Hausdorff measures of stable trees
In this paper we discuss Hausdorff and packing measures of random continuous
trees called stable trees. Stable trees form a specific class of L\'evy trees
(introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum
random tree (1991) which corresponds to the Brownian case. We provide results
for the whole stable trees and for their level sets that are the sets of points
situated at a given distance from the root. We first show that there is no
exact packing measure for levels sets. We also prove that non-Brownian stable
trees and their level sets have no exact Hausdorff measure with regularly
varying gauge function, which continues previous results from a joint work with
J-F Le Gall (2006).Comment: 40 page
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