arXiv subjects

Anna Gusakova

Publications and source records attributed to Anna Gusakova.

At least 19 recordsLinked to original sources

arXiv2026

Random hyperbolic polyhedra in horoballs

We study the geodesic convex hull of a stationary Poisson point process restricted to a horoball in $d$-dimensional hyperbolic space. The resulting random set is an unbounded hyperbolic polyhedron with a distinguished ideal direction. Projecting its boundary facets to the bounding horosphere yields a stationary Euclidean tessellation of $\mathbb{R}^{d-1}$, which we identify as a dual Poisson--Laguerre tessellation with an explicit height density. We derive an exact formula for its cell intensity and, in the critical regime where the intensity of the Poisson point process is matched with the height of the horoball, we prove local convergence of the projected tessellation to the classical Poisson--Delaunay tessellation in $\mathbb{R}^{d-1}$. As consequences, the typical cell converges in distribution and the intensities of all $k$-dimensional faces converge to their Poisson--Delaunay counterparts. We also study a localised volume functional of the hyperbolic convex hull, determine its limiting expectation in the same regime, and use an Efron-type identity to obtain a second-order asymptotic expansion for the vertex intensity. In addition, we derive an exact formula for the expected localised surface area and determine its critical asymptotics. In dimensions $d\ge3$ the expected unnormalised local surface area converges to a finite limit, whereas in dimension $d=2$ it exhibits logarithmic growth.

math.PR↗

arXiv2026

Limits of Poisson-Laguerre tessellations

For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(η_n)_{n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity measure of $η_n$ has density of the form $(v,h)\mapsto f_n(h)$ with respect to the Lebesgue measure, where $v\in \mathbb{R}^d$ and $h\in \mathbb{R}$. Identifying a tessellation with its skeleton (the union of the boundaries of all its cells) we provide verifiable conditions on $(f_n)_{n\in\mathbb{N}}$ that ensure convergence either to the classical Poisson-Voronoi/Poisson-Delaunay tessellation or to another Poisson-Laguerre tessellation. We also discuss convergence of the corresponding typical cells. As a corollary, we show that the Poisson-Voronoi and the Poisson-Delaunay tessellations arise as limits of the $β$-Voronoi and the $β$-Delaunay tessellations, respectively, as $β\to -1$.

math.PR↗

arXiv2025

Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells

We show that tessellations of hyperbolic space by isometry-invariant Poisson processes of $(d-1)$-dimensional hyperplanes do not have an unbounded cell at the critical intensity. This extends a result by Porret-Blanc for the hyperbolic plane (C. R. Acad. Sci. Paris, Ser. I, Vol. 344 (2007)) to dimensions $d\ge3$. We also show that for intensities strictly below the critical intensity, infinitely many unbounded cells exist, while for intensities larger than or equal to the critical intensity, no unbounded cell exists. This completely describes the basic phase transition of this continuum percolation model. Our proof uses a method from discrete percolation theory which we adapt to the continuum and combine with specific computations for Poisson hyperplane processes.

math.PR↗

arXiv2025

Gaussian approximation for Extreme Points in Laguerre tessellations

We consider Gaussian approximation in three particular models of Poisson-Laguerre tessellations, namely, the $β$-, $β'$- and Gaussian-Voronoi tessellations. The tessellations are constructed based on inhomogeneous Poisson point processes in space-time $\mathbb{R}^d \times \mathbb{R}$, where some of the points of the process give rise to a cell in $\mathbb{R}^d$, known as extreme points, while the other points produce an empty cell. Using the notion of region-stabilization, we derive quantitative central limit theorems with presumably optimal rates of convergence for the number of extreme points of $β$-, $β'$- and Gaussian-Voronoi tessellations in a growing window $W_n=[-n,n]^d$ as $n\to\infty$. Our bounds improve and extend previously known results by Schreiber and Yukich (2008) for the $β$-model, and are the first quantitative results for the $β'$- and Gaussian models.

math.PR↗

arXiv2025

On the moments of the volume for random convex chains

Let $T$ be the triangle in the plane with vertices $(0, 0)$, $(0,1)$ and $(0, 1)$. The convex hull $T_n$ of points $(0, 1)$, $(1, 0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain. In this paper we study the moments of the volume of random polytope $T_n$ and derive exact formulas for $k$-th moments for any integer $k\ge 0$. As an intermediate result, we find an explicit representation for the probability generating function of the number of vertices of $T_n$, from which an alternative formula for the probability that $T_n$ has $k$ vertices follows.

math.PR↗

arXiv2025

Sylvester's problem for beta-type distributions

Consider $d+2$ i.i.d. random points $X_1,\ldots, X_{d+2}$ in $\mathbb R^d$. In this note, we compute the probability that their convex hull is a simplex focusing on three specific distributional settings: (i) the distribution of $X_1$ is multivariate standard normal; (ii) the density of $X_1$ is proportional to $(1-\|x\|^2)^β$ on the unit ball (the beta distribution); (iii) the density of $X_1$ is proportional to $(1+\|x\|^2)^{-β}$ (the beta prime distribution). In the Gaussian case, we show that this probability equals twice the sum of the solid angles of a regular $(d+1)$-dimensional simplex.

math.PR↗

arXiv2025

Poisson-Laguerre tessellations

In this paper we introduce a family of Poisson-Laguerre tessellations in $\mathbb{R}^d$ generated by a Poisson point process in $\mathbb{R}^d\times \mathbb{R}$, whose intensity measure has a density of the form $(v,h)\mapsto f(h){\rm d} h {\rm d} v$, where $v\in\mathbb{R}^d$ and $h\in\mathbb{R}$, with respect to the Lebesgue measure. We study its sectional properties and show that the $\ell$-dimensional section of a Poisson-Laguerre tessellation corresponding to $f$ is an $\ell$-dimensional Poisson-Laguerre tessellation corresponding to $f_{\ell}$, which is up to a constant a fractional integral of $f$ of order $(d-\ell)/2$. Further we derive an explicit representation for the distribution of the volume weighted typical cell of the dual Poisson-Laguerre tessellation in terms of fractional integrals and derivatives of $f$.

math.PR↗

arXiv2024

Percolation of fat Poisson cylinders in hyperbolic space

In this paper we study Poisson processes of so-called "fat" cylinders in hyperbolic space. As our main result we show that this model undergoes a percolation phase transition. We prove this by establishing a novel link between the fat Poisson cylinder process and semi-scale invariant random fractal models on the unit sphere and in $\mathbb{R}^d.$ As a secondary result, we show that the semi-scale invariant fractal ball model in $\mathbb{R}^d$ has a non-empty sheet phase.

math.PR↗

arXiv2024

Random polytopes in convex bodies: Bridging the gap between extremal containers

We investigate the asymptotic properties of random polytopes arising as convex hulls of $n$ independent random points sampled from a family of block-beta distributions. Notably, this family includes the uniform distribution on a product of Euclidean balls of varying dimensions as a key example. As $n\to\infty$, we establish explicit growth rates for the expected number of facets, which depend in a subtle way on the the underlying model parameters. For the case of the uniform distribution, we further examine the expected number of faces of arbitrary dimensions as well as the volume difference. Our findings reveal that the family of random polytopes we introduce exhibits novel interpolative properties, bridging the gap between the classical extremal cases observed in the behavior of random polytopes within smooth versus polytopal convex containers.

math.PR↗

arXiv2024

Concentration inequalities for Poisson $U$-statistics

In this article we obtain concentration inequalities for Poisson $U$-statistics $F_m(f,η)$ of order $m\ge 1$ with kernels $f$ under general assumptions on $f$ and the intensity measure $γΛ$ of underlying Poisson point process $η$. The main result are new concentration bounds of the form \[ \mathbb{P}(|F_m ( f , η) -\mathbb{E} F_m ( f , η)| \ge t)\leq 2\exp(-I(γ,t)), \] where $I(γ,t)$ is of optimal order in $t$, namely it satisfies $I(γ,t)=Θ(t^{1\over m}\log t)$ as $t\to\infty$ and $γ$ is fixed. The function $I(γ,t)$ is given explicitly in terms of parameters of the assumptions satisfied by $f$ and $Λ$. One of the key ingredients of the proof is bounding the centred moments of $F_m(f,η)$. We discuss the optimality of obtained concentration bounds and consider a number of applications related to Gilbert graphs and Poisson hyperplane processes in constant curvature spaces.

math.PR↗

arXiv2023

The volume of random simplices from elliptical distributions in high dimension

Random simplices and more general random convex bodies of dimension $p$ in $\mathbb{R}^n$ with $p\leq n$ are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if $p\to\infty$ and $n\to\infty$ in such a way that $p/n\toγ\in(0,1)$, a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of $p\times n$ random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well.

math.PR↗

arXiv2023

Sectional Voronoi tessellations: Characterization and high-dimensional limits

The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi tessellations in $\mathbb{R}^d$ with $\ell$-dimensional affine subspaces, $1\leq \ell\leq d-1$, are shown to be random tessellations of the same type but with different model parameters. In particular, the intersection of a classical Poisson-Voronoi tessellation with an affine subspace is shown to have the same distribution as a certain beta-Voronoi tessellation. The geometric properties of the typical cell and, more generally, typical $k$-faces, of the sectional Poisson-Voronoi tessellation are studied in detail. It is proved that in high dimensions, that is as $d\to\infty$, the intersection of the $d$-dimensional Poison-Voronoi tessellation with an affine subspace of fixed dimension $\ell$ converges to the $\ell$-dimensional Gaussian-Voronoi tessellation.

math.PR↗

arXiv2022

Intrinsic volumes of ellipsoids

We deduce explicit formulae for the intrinsic volumes of an ellipsoid in $\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we show that \begin{align*} V_k({\mathcal E})=κ_k\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\int_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\,\rm{d}t \end{align*} for all $k=1,\ldots,d$, where $s_{k-1}$ is the $(k-1)$-th elementary symmetric polynomial and $κ_k$ is the volume of the $k$-dimensional unit ball. Some examples of the intrinsic volumes $V_k$ with low and high $k$ are given where our formulae look particularly simple. As an application we derive new formulae for the expected $k$-dimensional volume of random $k$-simplex in an ellipsoid and random Gaussian $k$-simplex.

math.MG↗

arXiv2022

Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon

Fix a container polygon $P$ in the plane and consider the convex hull $P_n$ of $n\geq 3$ independent and uniformly distributed in $P$ random points. In the focus of this paper is the vertex number of the random polygon $P_n$. The precise variance expansion for the vertex number is determined up to the constant-order term, a result which can be considered as a second-order analogue of the classical expansion for the expectation of Rényi and Sulanke (1963). Moreover, a sharp Berry-Esseen bound is derived for the vertex number of the random polygon $P_n$, which is of the same order as the square-root of the variance. The main idea behind the proof of both results is a decomposition of the boundary of the random polygon $P_n$ into random convex chains and a careful merging of the variance expansions and Berry-Esseen bounds for the vertex numbers of the individual chains.

math.PR↗

arXiv2022

Spherical convex hull of random points on a wedge

Consider two half-spaces $H_1^+$ and $H_2^+$ in $\mathbb{R}^{d+1}$ whose bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the origin. The intersection $\mathbb{S}_{2,+}^d:=\mathbb{S}^d\cap H_1^+\cap H_2^+$ is a spherical convex subset of the $d$-dimensional unit sphere $\mathbb{S}^d$, which contains a great subsphere of dimension $d-2$ and is called a spherical wedge. Choose $n$ independent random points uniformly at random on $\mathbb{S}_{2,+}^d$ and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of $\log n$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on $\mathbb{S}_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

math.PR↗

arXiv2021

The $β$-Delaunay tessellation I: Description of the model and geometry of typical cells

In this paper two new classes of stationary random simplicial tessellations, the so-called $β$- and $β'$-Delaunay tessellations, are introduced. Their construction is based on a space-time paraboloid hull process and generalizes that of the classical Poisson-Delaunay tessellation. The distribution of volume-power weighted typical cells is explicitly identified, establishing thereby a remarkable connection to the classes of $β$- and $β'$-polytopes. These representations are used to determine principal characteristics of such cells, including volume moments, expected angle sums and cell intensities.

math.PR↗

arXiv2021

The $β$-Delaunay tessellation IV: Mixing properties and central limit theorems

Various mixing properties of $β$-, $β'$- and Gaussian Delaunay tessellations in $\mathbb{R}^{d-1}$ are studied. It is shown that these tessellation models are absolutely regular, or $β$-mixing. In the $β$- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the $β'$-case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of $β$- and Gaussian Delaunay tessellations are established. This includes the number of $k$-dimensional faces and the $k$-volume of the $k$-sk

math.PR↗

arXiv2021

Sharp inequalities for the mean distance of random points in convex bodies

For a convex body $K\subset\mathbb{R}^d$ the mean distance $Δ(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for ratio between $Δ(K)$ and the first intrinsic volume $V_1(K)$ of $K$ (normalized mean width) are derived and degenerate extremal cases are discussed. The argument relies on Riesz's rearrangement inequality and the solution of an optimization problem for powers of concave functions. The relation with results known from the existing literature is reviewed in detail.

math.MG↗