3,050 research outputs found
The skeleton of the UIPT, seen from infinity
We prove that geodesic rays in the Uniform Infinite Planar Triangulation
(UIPT) coalesce in a strong sense using the skeleton decomposition of random
triangulations discovered by Krikun. This implies the existence of a unique
horofunction measuring distances from infinity in the UIPT. We then use this
horofunction to define the skeleton "seen from infinity" of the UIPT and relate
it to a simple Galton--Watson tree conditioned to survive, giving a new and
particularly simple construction of the UIPT. Scaling limits of perimeters and
volumes of horohulls within this new decomposition are also derived, as well as
a new proof of the -point function formula for random triangulations in the
scaling limit due to Ambj{\o}rn and Watabiki.Comment: 34 pages, 14 figure
Formal study of plane Delaunay triangulation
This article presents the formal proof of correctness for a plane Delaunay
triangulation algorithm. It consists in repeating a sequence of edge flippings
from an initial triangulation until the Delaunay property is achieved. To
describe triangulations, we rely on a combinatorial hypermap specification
framework we have been developing for years. We embed hypermaps in the plane by
attaching coordinates to elements in a consistent way. We then describe what
are legal and illegal Delaunay edges and a flipping operation which we show
preserves hypermap, triangulation, and embedding invariants. To prove the
termination of the algorithm, we use a generic approach expressing that any
non-cyclic relation is well-founded when working on a finite set
Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find
Drawing together techniques from combinatorics and computer science, we
improve the census algorithm for enumerating closed minimal P^2-irreducible
3-manifold triangulations. In particular, new constraints are proven for face
pairing graphs, and pruning techniques are improved using a modification of the
union-find algorithm. Using these results we catalogue all 136 closed
non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten
tetrahedra.Comment: 37 pages, 34 figure
The infinite cyclohedron and its automorphism group
Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes
that can be constructed from centrally symmetric triangulations of even-sided
polygons. In this article we introduce an infinite-dimensional analogue and
prove that the group of symmetries of our construction is a semidirect product
of a degree 2 central extension of Thompson's infinite finitely presented
simple group T with the cyclic group of order 2. These results are inspired by
a similar recent analysis by the first author of the automorphism group of an
infinite-dimensional associahedron.Comment: 18 pages, 8 figure
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
Flipping Cubical Meshes
We define and examine flip operations for quadrilateral and hexahedral
meshes, similar to the flipping transformations previously used in triangular
and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th
International Meshing Roundtable. This version removes some unwanted
paragraph breaks from the previous version; the text is unchange
Triangulated Surfaces in Twistor Space: A Kinematical Set up for Open/Closed String Duality
We exploit the properties of the three-dimensional hyperbolic space to
discuss a simplicial setting for open/closed string duality based on (random)
Regge triangulations decorated with null twistorial fields. We explicitly show
that the twistorial N-points function, describing Dirichlet correlations over
the moduli space of open N-bordered genus g surfaces, is naturally mapped into
the Witten-Kontsevich intersection theory over the moduli space of N-pointed
closed Riemann surfaces of the same genus. We also discuss various aspects of
the geometrical setting which connects this model to PSL(2,C) Chern-Simons
theory.Comment: 35 pages, references added, slightly revised introductio
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