19,018 research outputs found
Local limits of uniform triangulations in high genus
We prove a conjecture of Benjamini and Curien stating that the local limits
of uniform random triangulations whose genus is proportional to the number of
faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in
arXiv:1401.3297. The proof relies on a combinatorial argument and the
Goulden--Jackson recurrence relation to obtain tightness, and probabilistic
arguments showing the uniqueness of the limit. As a consequence, we obtain
asymptotics up to subexponential factors on the number of triangulations when
both the size and the genus go to infinity.
As a part of our proof, we also obtain the following result of independent
interest: if a random triangulation of the plane is weakly Markovian in the
sense that the probability to observe a finite triangulation around the
root only depends on the perimeter and volume of , then is a mixture of
PSHT.Comment: 36 pages, 10 figure
Evaluating the Differences of Gridding Techniques for Digital Elevation Models Generation and Their Influence on the Modeling of Stony Debris Flows Routing: A Case Study From Rovina di Cancia Basin (North-Eastern Italian Alps)
Debris \ufb02ows are among the most hazardous phenomena in mountain areas. To cope
with debris \ufb02ow hazard, it is common to delineate the risk-prone areas through
routing models. The most important input to debris \ufb02ow routing models are the
topographic data, usually in the form of Digital Elevation Models (DEMs). The quality
of DEMs depends on the accuracy, density, and spatial distribution of the sampled
points; on the characteristics of the surface; and on the applied gridding methodology.
Therefore, the choice of the interpolation method affects the realistic representation
of the channel and fan morphology, and thus potentially the debris \ufb02ow routing
modeling outcomes. In this paper, we initially investigate the performance of common
interpolation methods (i.e., linear triangulation, natural neighbor, nearest neighbor,
Inverse Distance to a Power, ANUDEM, Radial Basis Functions, and ordinary kriging)
in building DEMs with the complex topography of a debris \ufb02ow channel located
in the Venetian Dolomites (North-eastern Italian Alps), by using small footprint full-
waveform Light Detection And Ranging (LiDAR) data. The investigation is carried
out through a combination of statistical analysis of vertical accuracy, algorithm
robustness, and spatial clustering of vertical errors, and multi-criteria shape reliability
assessment. After that, we examine the in\ufb02uence of the tested interpolation algorithms
on the performance of a Geographic Information System (GIS)-based cell model for
simulating stony debris \ufb02ows routing. In detail, we investigate both the correlation
between the DEMs heights uncertainty resulting from the gridding procedure and
that on the corresponding simulated erosion/deposition depths, both the effect of
interpolation algorithms on simulated areas, erosion and deposition volumes, solid-liquid
discharges, and channel morphology after the event. The comparison among the tested
interpolation methods highlights that the ANUDEM and ordinary kriging algorithms
are not suitable for building DEMs with complex topography. Conversely, the linear
triangulation, the natural neighbor algorithm, and the thin-plate spline plus tension and completely regularized spline functions ensure the best trade-off among accuracy
and shape reliability. Anyway, the evaluation of the effects of gridding techniques on
debris \ufb02ow routing modeling reveals that the choice of the interpolation algorithm does
not signi\ufb01cantly affect the model outcomes
The Theory of Dynamical Random Surfaces with Extrinsic Curvature
We analyze numerically the critical properties of a two-dimensional
discretized random surface with extrinsic curvature embedded in a
three-dimensional space. The use of the toroidal topology enables us to enforce
the non-zero external extension without the necessity of defining a boundary
and allows us to measure directly the string tension. We show that a phase
transition from the crumpled phase to the smooth phase observed earlier for a
spherical topology appears also for a toroidal surface for the same finite
value of the coupling constant of the extrinsic curvature term. The phase
transition is characterized by the vanishing of the string tension. We discuss
the possible non-trivial continuum limit of the theory, when approaching the
critical point. Numerically we find a value of the critical exponent \n to be
between .38 and .42. The specific heat, related to the extrinsic curvature term
seems not to diverge (or diverge slower than logarithmically) at the critical
point.Comment: figures available as postscript files on request. 28 pages plus
figure
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
One machine, one minute, three billion tetrahedra
This paper presents a new scalable parallelization scheme to generate the 3D
Delaunay triangulation of a given set of points. Our first contribution is an
efficient serial implementation of the incremental Delaunay insertion
algorithm. A simple dedicated data structure, an efficient sorting of the
points and the optimization of the insertion algorithm have permitted to
accelerate reference implementations by a factor three. Our second contribution
is a multi-threaded version of the Delaunay kernel that is able to concurrently
insert vertices. Moore curve coordinates are used to partition the point set,
avoiding heavy synchronization overheads. Conflicts are managed by modifying
the partitions with a simple rescaling of the space-filling curve. The
performances of our implementation have been measured on three different
processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we
have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds
to a generation rate of over 55 million tetrahedra per second. We finally show
how this very efficient parallel Delaunay triangulation can be integrated in a
Delaunay refinement mesh generator which takes as input the triangulated
surface boundary of the volume to mesh
- …