2,145 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
Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus
We extend the notion of canonical ordering (initially developed for planar
triangulations and 3-connected planar maps) to cylindric (essentially simple)
triangulations and more generally to cylindric (essentially internally)
-connected maps. This allows us to extend the incremental straight-line
drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case)
and of Kant (in the -connected case) to this setting. Precisely, for any
cylindric essentially internally -connected map with vertices, we
can obtain in linear time a periodic (in ) straight-line drawing of that
is crossing-free and internally (weakly) convex, on a regular grid
, with and ,
where is the face-distance between the two boundaries. This also yields an
efficient periodic drawing algorithm for graphs on the torus. Precisely, for
any essentially -connected map on the torus (i.e., -connected in the
periodic representation) with vertices, we can compute in linear time a
periodic straight-line drawing of that is crossing-free and (weakly)
convex, on a periodic regular grid
, with and
, where is the face-width of . Since ,
the grid area is .Comment: 37 page
Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense
Recent works have shown that random triangulations decorated by critical
() Bernoulli site percolation converge in the scaling limit to a
-Liouville quantum gravity (LQG) surface (equivalently, a Brownian
surface) decorated by SLE in two different ways:
1. The triangulation, viewed as a curve-decorated metric measure space
equipped with its graph distance, the counting measure on vertices, and a
single percolation interface converges with respect to a version of the
Gromov-Hausdorff topology.
2. There is a bijective encoding of the site-percolated triangulation by
means of a two-dimensional random walk, and this walk converges to the
correlated two-dimensional Brownian motion which encodes SLE-decorated
-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield
(2014); this is sometimes called .
We prove that one in fact has convergence in both of these
two senses simultaneously. We also improve the metric convergence result by
showing that the map decorated by the full collection of percolation interfaces
(rather than just a single interface) converges to -LQG decorated
by CLE in the metric space sense.
This is the first work to prove simultaneous convergence of any random planar
map model in the metric and peanosphere senses. Moreover, this work is an
important step in an ongoing program to prove that random triangulations
embedded into via the so-called converge
to -LQG.Comment: 55 pages; 13 Figures. Minor revision according to a referee report.
Accepted for publication at EJ
A Quantitative Steinitz Theorem for Plane Triangulations
We give a new proof of Steinitz's classical theorem in the case of plane
triangulations, which allows us to obtain a new general bound on the grid size
of the simplicial polytope realizing a given triangulation, subexponential in a
number of special cases.
Formally, we prove that every plane triangulation with vertices can
be embedded in in such a way that it is the vertical projection
of a convex polyhedral surface. We show that the vertices of this surface may
be placed in a integer grid, where and denotes the shedding diameter of , a
quantity defined in the paper.Comment: 25 pages, 6 postscript figure
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