444 research outputs found
Asymptotic statistics of the n-sided planar Voronoi cell: II. Heuristics
We develop a set of heuristic arguments to explain several results on planar
Poisson-Voronoi tessellations that were derived earlier at the cost of
considerable mathematical effort. The results concern Voronoi cells having a
large number n of sides. The arguments start from an entropy balance applied to
the arrangement of n neighbors around a central cell. It is followed by a
simplified evaluation of the phase space integral for the probability p_n that
an arbitrary cell be n-sided. The limitations of the arguments are indicated.
As a new application we calculate the expected number of Gabriel (or full)
neighbors of an n-sided cell in the large-n limit.Comment: 22 pages, 10 figure
Sylvester's question and the Random Acceleration Process
Let n points be chosen randomly and independently in the unit disk.
"Sylvester's question" concerns the probability p_n that they are the vertices
of a convex n-sided polygon. Here we establish the link with another problem.
We show that for large n this polygon, when suitably parametrized by a function
r(phi) of the polar angle phi, satisfies the equation of the random
acceleration process (RAP), d^2 r/d phi^2 = f(phi), where f is Gaussian noise.
On the basis of this relation we derive the asymptotic expansion log p_n = -2n
log n + n log(2 pi^2 e^2) - c_0 n^{1/5} + ..., of which the first two terms
agree with a rigorous result due to Barany. The nonanalyticity in n of the
third term is a new result. The value 1/5 of the exponent follows from recent
work on the RAP due to Gyorgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show
that the n-sided polygon is effectively contained in an annulus of width \sim
n^{-4/5} along the edge of the disk. The distance delta_n of closest approach
to the edge is exponentially distributed with average 1/(2n).Comment: 29 pages, 4 figures; references added and minor change
New Monte Carlo method for planar Poisson-Voronoi cells
By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a
planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is
developed on the basis of earlier theoretical work; it exploits, in particular,
the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have
between four and six significant digits. Accurate n dependent averages, second
moments, and variances are obtained for the cell area and the cell perimeter.
The numerical large n behavior of these quantities is analyzed in terms of
asymptotic power series in 1/n. Snapshots are shown of typical occurrences of
extremely rare events implicating cells of up to n=1600 sides embedded in an
ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic
features of such many-sided cells and their immediate environment. Their
relevance for observable properties is stressed.Comment: 35 pages including 10 figures and 4 table
Large-n conditional facedness m_n of 3D Poisson-Voronoi cells
We consider the three-dimensional Poisson-Voronoi tessellation and study the
average facedness m_n of a cell known to neighbor an n-faced cell. Whereas
Aboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptotic
expansion m_n = 8 + k_1 n^{-1/6} +.... Recent new Monte Carlo data due to Lazar
et al., based on a very large data set, now clearly rule out Aboav's law. In
this work we determine the numerical value of k_1 and compare the expansion to
the Monte Carlo data. The calculation of k_1 involves an auxiliary planar
cellular structure composed of circular arcs, that we will call the
Poisson-Moebius diagram. It is a special case of more general Moebius diagrams
(or multiplicatively weighted power diagrams) and is of interest for its own
sake. We obtain exact results for the total edge length per unit area, which is
a prerequisite for the coefficient k_1, and a few other quantities in this
diagram.Comment: 18 pages, 5 figure
Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results
We achieve a detailed understanding of the -sided planar Poisson-Voronoi
cell in the limit of large . Let be the probability for a cell to
have sides. We construct the asymptotic expansion of up to
terms that vanish as . We obtain the statistics of the lengths of
the perimeter segments and of the angles between adjoining segments: to leading
order as , and after appropriate scaling, these become independent
random variables whose laws we determine; and to next order in they have
nontrivial long range correlations whose expressions we provide. The -sided
cell tends towards a circle of radius (n/4\pi\lambda)^{\half}, where
is the cell density; hence Lewis' law for the average area of
the -sided cell behaves as with . For
the cell perimeter, expressed as a function of the polar
angle , satisfies , where is known Gaussian
noise; we deduce from it the probability law for the perimeter's long
wavelength deviations from circularity. Many other quantities related to the
asymptotic cell shape become accessible to calculation.Comment: 54 pages, 3 figure
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