42 research outputs found
A phase transition in block-weighted random maps
We consider the model of random planar maps of size biased by a weight
per -connected block, and the closely related model of random planar
quadrangulations of size biased by a weight per simple component. We
exhibit a phase transition at the critical value . If , a
condensation phenomenon occurs: the largest block is of size .
Moreover, for quadrangulations we show that the diameter is of order ,
and the scaling limit is the Brownian sphere. When , the largest block
is of size , the scaling order for distances is , and
the scaling limit is the Brownian tree. Finally, for , the largest block
is of size , the scaling order for distances is , and
the scaling limit is the stable tree of parameter .Comment: 59 pages, 14 figure
Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005
This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop
Two Results in Drawing Graphs on Surfaces
In this work we present results on crossing-critical graphs drawn on non-planar surfaces and results on edge-hamiltonicity of graphs on the Klein bottle. We first give an infinite family of graphs that are 2-crossing-critical on the projective plane. Using this result, we construct 2-crossing-critical graphs for each non-orientable surface. Next, we use 2-amalgamations to construct 2-crossing-critical graphs for each orientable surface other than the sphere. Finally, we contribute to the pursuit of characterizing 4-connected graphs that embed on the Klein bottle and fail to be edge-hamiltonian. We show that known 4-connected counterexamples to edge-hamiltonicity on the Klein bottle are hamiltonian and their structure allows restoration of edge-hamiltonicity with only a small change
Results on Select Combinatorial Problems With an Extremal Nature
This dissertation is split into three sections, each containing new results on a particular combinatorial problem. In the first section, we consider the set of 3-connected quadrangulations on n vertices and the set of 5-connected triangulations on n vertices. In each case, we find the minimum Wiener index of any graph in the given class, and identify graphs that obtain this minimum value. Moreover, we prove that these graphs are unique up to isomorphism.
In the second section, we work with structures emerging from the biological sciences called tanglegrams. In particular, our work pertains to an invariant of tanglegrams called the tangle crossing number, an invariant which is NP-hard to compute. Czabarka, Székely, and Wagner found a finite characterization of tanglegrams with tangle crossing number equal to 0, which motivated the work here. In particular, our aim was to find a similar finite (and minimal) characterization of tanglegrams with tangle crossing number at least k, for any fixed k ≥ 2. We set out to prove this using an elegant order-theoretic argument, but came to another surprising result instead; we proved that the set of tanglegrams with the induced subtanglegram relation is not a well partial order.
In the final section, we work on the problem of finding an upper bound on the diameter of graphs with particular properties. It was proven independently by several groups that for fixed minimum degree , every connected graph of order satisfies diam as . Erd\H{o}s, Pach, Pollack, and Tuza noticed that the graphs which achieve the aforementioned bound all contain complete subgraphs whose order increases with , and conjectured that if we disallowed complete subgraphs of a given fixed size, then we could improve the bound. Czabarka, Singgih, and Sz\\u27ekely recently found a counterexample to part of the conjecture of Erd\H{o}s \emph{et al.} and formulated a new conjecture. Under a stronger assumption, we verify two cases of this new conjecture using a novel unified duality approach
Distance Related Graph Invariants in Triangulations and Quadrangulations of the Sphere
The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. I provide asymptotic upper bounds and sharp lower bounds for the Wiener index of simple triangulations and quadrangulations with given connectivity. Additionally, I make conjectures for the extremal triangulations and quadrangulations which maximize the Wiener index based on computational evidence. If σ(v) denotes the arithmetic mean of the distances from v to all other vertices of G, then the remoteness and proximity of G are defined as the largest and smallest value of σ(v) over all vertices v of G, respectively. I give sharp upper bounds on the remoteness and asymptotic upper bounds on the proximity of simple triangulations and quadrangulations of given order and connectivity
Quantum Loewner Evolution
What is the scaling limit of diffusion limited aggregation (DLA) in the
plane? This is an old and famously difficult question. One can generalize the
question in two ways: first, one may consider the {\em dielectric breakdown
model} -DBM, a generalization of DLA in which particle locations are
sampled from the -th power of harmonic measure, instead of harmonic
measure itself. Second, instead of restricting attention to deterministic
lattices, one may consider -DBM on random graphs known or believed to
converge in law to a Liouville quantum gravity (LQG) surface with parameter
.
In this generality, we propose a scaling limit candidate called quantum
Loewner evolution, QLE. QLE is defined in terms of the radial
Loewner equation like radial SLE, except that it is driven by a measure valued
diffusion derived from LQG rather than a multiple of a standard
Brownian motion. We formalize the dynamics of using an SPDE. For each
, there are two or three special values of for which
we establish the existence of a solution to these dynamics and explicitly
describe the stationary law of .
We also explain discrete versions of our construction that relate DLA to
loop-erased random walk and the Eden model to percolation. A certain
"reshuffling" trick (in which concentric annular regions are rotated randomly,
like slot machine reels) facilitates explicit calculation.
We propose QLE as a scaling limit for DLA on a random
spanning-tree-decorated planar map, and QLE as a scaling limit for the
Eden model on a random triangulation. We propose using QLE to endow
pure LQG with a distance function, by interpreting the region explored by a
branching variant of QLE, up to a fixed time, as a metric ball in a
random metric space.Comment: 132 pages, approximately 100 figures and computer simulation
Liouville Quantum Gravity and KPZ
Consider a bounded planar domain D, an instance h of the Gaussian free field
on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma <
2. The Liouville quantum gravity measure on D is the weak limit as epsilon
tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz,
where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h
on the circle of radius epsilon centered at z. Given a random (or
deterministic) subset X of D one can define the scaling dimension of X using
either Lebesgue measure or this random measure. We derive a general quadratic
relation between these two dimensions, which we view as a probabilistic
formulation of the KPZ relation from conformal field theory. We also present a
boundary analog of KPZ (for subsets of the boundary of D). We discuss the
connection between discrete and continuum quantum gravity and provide a
framework for understanding Euclidean scaling exponents via quantum gravity.Comment: 56 pages. Revised version contains more details. To appear in
Inventione