Related papers: Faces in random great hypersphere tessellations
We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models,…
We investigate the problem of density estimation on the unit circle and the unit sphere from a computational perspective. Our primary goal is to develop new density estimators that are both rate-optimal and computationally efficient for…
In the past decades, the central limit theorem (CLT) has been generalized to non-Euclidean data spaces. Some years ago, it was found that for some random variables on the circle, the sample Fr\'echet mean fluctuates around the population…
The isostatic jamming limit of frictionless spherical particles from Edwards' statistical mechanics [Song \emph{et al.}, Nature (London) {\bf 453}, 629 (2008)] is generalized to arbitrary dimension $d$ using a liquid-state description. The…
In the process of protostar formation, astrophysical gas clouds undergo thermodynamically irreversible processes and emit heat and radiation to their surroundings. Due the emission of this energy one can envision an idealized situation in…
Let M be a compact smooth manifold of dimension n with or without boundary, and f : M $\rightarrow$ R be a smooth Gaussian random field. It is very natural to suppose that for a large positive real u, the random excursion set {f $\ge$ u} is…
Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic…
This work studies the chord length distribution, in the case where both ends lie on a $N$-dimensional hypersphere ($N \geq 2$). Actually, after connecting this distribution to the recently estimated surface of a hyperspherical cap…
We examine the effects of introducing a wall or edge into a directed percolation process. Scaling ansatzes are presented for the density and survival probability of a cluster in these geometries, and we make the connection to surface…
We present the first study of the isotropy of the distribution of morphological types of galaxies in the Local Universe out to around 200 Mpc using more than 60,000 galaxies from the HyperLeda database. We divide the sky into two opposite…
Stochastic separation theorems play important role in high-dimensional data analysis and machine learning. It turns out that in high dimension any point of a random set of points can be separated from other points by a hyperplane with high…
We consider random walks on the surface of the sphere $S_{n-1}$ ($n \geq 2$) of the $n$-dimensional Euclidean space $E_n$, in short a hypersphere. By solving the diffusion equation in $S_{n-1}$ we show that the usual law $<r^2 > \varpropto…
In this article, we introduce and study the concept of $\textit{spherical-vectors}$, which can be perceived as a natural extension of the arguments of complex numbers in the context of quaternions. We initially establish foundational…
The purpose of this note is to clarify the effect of the finite size of spherical particles upon the characteristics of their spatial distribution through a random Poisson process (RPP). This information is of special interest when using…
This note treats several problems for the fractional perimeter or $s$-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps.…
In this paper we consider the convex hull of a spherically symmetric sample in $R^d$. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex…
The distribution function of particles over clusters is proposed for a system of identical intersecting spheres, the centres of which are uniformly distributed in space. Consideration is based on the concept of the rank number of clusters,…
We study the problem of generating a hyperplane tessellation of an arbitrary set $T$ in $\mathbb{R}^n$, ensuring that the Euclidean distance between any two points corresponds to the fraction of hyperplanes separating them up to a…
The intent of this paper is to describe the large scale asymptotic geometry of iteration stable (STIT) tessellations in $\mathbb{R}^d$, which form a rather new, rich and flexible class of random tessellations considered in stochastic…
Assuming spherical symmetry and dynamical equilibrium within a given gravitational potential, a dwarf spheroidal (dSph) galaxy's globally averaged stellar velocity dispersion depends entirely on the shape of its stellar density profile.…