1,513 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
Counting Triangulations and other Crossing-Free Structures Approximately
We consider the problem of counting straight-edge triangulations of a given
set of points in the plane. Until very recently it was not known
whether the exact number of triangulations of can be computed
asymptotically faster than by enumerating all triangulations. We now know that
the number of triangulations of can be computed in time,
which is less than the lower bound of on the number of
triangulations of any point set. In this paper we address the question of
whether one can approximately count triangulations in sub-exponential time. We
present an algorithm with sub-exponential running time and sub-exponential
approximation ratio, that is, denoting by the output of our
algorithm, and by the exact number of triangulations of , for some
positive constant , we prove that . This is the first algorithm that in sub-exponential time computes a
-approximation of the base of the number of triangulations, more
precisely, . Our algorithm can be
adapted to approximately count other crossing-free structures on , keeping
the quality of approximation and running time intact. In this paper we show how
to do this for matchings and spanning trees.Comment: 19 pages, 2 figures. A preliminary version appeared at CCCG 201
Liftings and stresses for planar periodic frameworks
We formulate and prove a periodic analog of Maxwell's theorem relating
stressed planar frameworks and their liftings to polyhedral surfaces with
spherical topology. We use our lifting theorem to prove deformation and
rigidity-theoretic properties for planar periodic pseudo-triangulations,
generalizing features known for their finite counterparts. These properties are
then applied to questions originating in mathematical crystallography and
materials science, concerning planar periodic auxetic structures and ultrarigid
periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual
Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201
Planar maps, circle patterns and 2d gravity
Via circle pattern techniques, random planar triangulations (with angle
variables) are mapped onto Delaunay triangulations in the complex plane. The
uniform measure on triangulations is mapped onto a conformally invariant
spatial point process. We show that this measure can be expressed as: (1) a sum
over 3-spanning-trees partitions of the edges of the Delaunay triangulations;
(2) the volume form of a K\"ahler metric over the space of Delaunay
triangulations, whose prepotential has a simple formulation in term of ideal
tessellations of the 3d hyperbolic space; (3) a discretized version (involving
finite difference complex derivative operators) of Polyakov's conformal
Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes,
thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17
figure
Growth and isoperimetric profile of planar graphs
Let G be a planar graph such that the volume function of G satisfies V(2n)<
CV(n) for some constant C > 0. Then for every vertex v of G and integer n,
there is a domain \Omega such that B(v,n) \subset \Omega, \Omega \subset B(v,
6n) and the size of the boundary of \Omega is at most order n.Comment: 8 page
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