arXiv subjects

Pierre Calka

Publications and source records attributed to Pierre Calka.

At least 19 recordsLinked to original sources

arXiv2025

Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell

We consider the typical Poisson-Voronoi cell in the Euclidean space R d and in particular the maximal distance D from a vertex of that cell to its nucleus. We provide a sharp asymptotics for the tail distribution of D. As a byproduct, we prove that the extremal index related to the sequence of such distances for all Voronoi cells included in a large box is equal to (2d) -1 . This confirms a conjecture formulated by Chenavier and Robert. The explicit constant appearing in the estimate of the tail probability of D is proved to be the mean volume of a random simplex formed by uniformly distributed points on the unit sphere conditioned on satisfying some spatial condition.

math.PR↗

arXiv2024

Fractal random sets associated with multitype Galton-Watson trees

In this paper, we consider a regular tessellation of the Euclidean plane and the sequence of its geometric scalings by negative powers of a fixed integer. We generate iteratively random sets as the union of adjacent tiles from these rescaled tessellations. We encode this geometric construction into a combinatorial object, namely a multitype Galton-Watson tree. Our main result concerns the geometric properties of the limiting planar set. In particular, we show that both box and Hausdorff dimensions coincide and we calculate them in function of the spectral radius of the reproduction matrix associated with this branching process. We then make that spectral radius explicit in several concrete examples when the regular tessellation is either hexagonal, square or triangular.

math.PR↗

arXiv2024

Limit theory for the first layers of the random convex hull peeling in a simple polytope

The convex hull peeling of a point set consists in taking the convex hull, then removing the extreme points and iterating that procedure until no point remains. The boundary of each hull is called a layer. Following on from [15], we study the first layers generated by the peeling procedure when the point set is chosen as a homogeneous Poisson point process inside a polytope when the intensity goes to infinity. We focus on some specific functionals, namely the number of k-dimensional faces and the outer defect volume. Since the early works of R{\'e}nyi and Sulanke, it is well known that both the techniques and the rates are completely different for the convex hull when the underlying convex body has a smooth boundary or when it is itself a polytope. We expect such dichotomy to extend to the further layers of the peeling. More precisely we provide asymptotic limits for their expectation and variance as well as a central limit theorem. In particular, as in the unit ball, the growth rates do not depend on the layer. The method builds upon previous constructions for the convex hull contained in [9] and [18] and requires the assumption that the underlying polytope is simple.

math.PR↗

arXiv2024

The support function of the high-dimensional Poisson polytope

Let $K_\lambda^d$ be the convex hull of the intersection of the homogeneous Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, $d \ge 2$, with the Euclidean unit ball $\mathbb{B}^d$. In this paper, we study the asymptotic behavior as $d\to\infty$ of the support function $h_\lambda^{(d)}(u) :=\sup_{x\in K_\lambda^d}\langle u,x\rangle$ in an arbitrary direction $u \in {\mathbb S}^{d-1}$ of the Poisson polytope $K_\lambda^d$. We identify three different regimes (subcritical, critical, and supercritical) in terms of the intensity $\lambda:=\lambda(d)$ and derive in each regime the precise distributional convergence of $h_\lambda^{(d)}$ after suitable scaling. We especially treat this question when the support function is considered over multiple directions at once. We finally deduce partial counterparts for the radius-vector function of the polytope.

math.PR↗

arXiv2022

Limit theory for the first layers of the random convex hull peeling in the unit ball

The convex hull peeling of a point set is obtained by taking the convex hull of the set and repeating iteratively the operation on the interior points until no point remains. The boundary of each hull is called a layer. We study the number of k-dimensional faces and the outer defect intrinsic volumes of the first layers of the convex hull peeling of a homogeneous Poisson point process in the unit ball whose intensity goes to infinity. More precisely we provide asymptotic limits for their expectation and variance as well as a central limit theorem. In particular, the growth rates do not depend on the layer.

math.PR↗

arXiv2022

Elongated Poisson-Voronoi cells in an empty half-plane

The Voronoi tessellation of a homogeneous Poisson point process in the lower half-plane gives rise to a family of vertical elongated cells in the upper half-plane. The set of edges of these cells is ruled by a Markovian branching mechanism which is asymptotically described by two sequences of iid variables which are respectively Beta and exponentially distributed. This leads to a precise description of the scaling limit of a so-called typical cell. The limit object is a random apeirogon that we name menhir in reference to the Gallic huge stones. We also deduce from the aforementioned branching mechanism that the number of vertices of a cell of height $\lambda$ is asymptotically equal to $\frac45\log\lambda$.

math.PR↗

arXiv2019

Convex hulls of perturbed random point sets

We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $\S^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin and of radius $n^{\alpha}, \alpha \in (-\infty, \infty)$. This model, inspired by the smoothed complexity analysis introduced in computational geometry \cite{DGGT,ST}, is a perturbation of the classical random polytope. We show that the perturbed point process, after rescaling, converges in the scaling limit to one of five Poisson point processes according to whether $\alpha$ belongs to one of five regimes. The intensity measure of the limit Poisson point process undergoes a transition at the values $\alpha = \frac{-2} {d -1}$ and $\alpha = \frac{2} {d + 1}$ and it gives rise to four rescalings for the $k$-face functional on perturbed data. These rescalings are used to establish explicit expectation asymptotics for the number of $k$-dimensional faces of the convex hull of either perturbed binomial or Poisson data. In the case of Poisson input, we establish explicit variance asymptotics and a central limit theorem for the number of $k$-dimensional faces. Finally it is shown that the rescaled boundary of the convex hull of the perturbed point process converges to the boundary of a parabolic hull process.

math.PR↗

arXiv2018

Mean asymptotics for a Poisson-Voronoi cell on a Riemannian manifold

In this paper, we consider a Riemannian manifold $M$ and the Poisson-Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $\lambda$ on $M$. We obtain asymptotic expansions up to the second order for the means of several characteristics of the Voronoi cell associated with $x_0$, including its volume and number of vertices. In each case, the first term of the estimate is equal to the mean characteristic in the Euclidean setting while the second term may contain a particular curvature of $M$ at $x_0$: the scalar curvature in the case of the mean number of vertices, the Ricci curvature in the case of the density of vertices and the sectional curvatures in the cases of the volume and number of vertices of a section of the Voronoi cell. Several explicit formulas are also derived in the particular case of constant curvature. The key tool for proving these results is a new change of variables formula of Blaschke-Petkantschin type in the Riemannian setting. Finally, a probabilistic proof of the Gauss-Bonnet Theorem is deduced from the asymptotic estimate of the total number of vertices of the tessellation in dimension two.

math.PR↗

arXiv2016

The Poisson-Voronoi cell around an isolated nucleus

Consider a planar random point process made of the union of a point (the origin) and of a Poisson point process with a uniform intensity outside a deterministic set surrounding the origin. When the intensity goes to infinity, we show that the Voronoi cell associated with the origin converges from above to a deterministic convex set. We describe this set and give the asymptotics of the expectation of its defect area, defect perimeter and number of vertices. On the way, two intermediary questions are treated. First, we describe the mean characteristics of the Poisson-Voronoi cell conditioned on containing a fixed convex body around the origin and secondly, we show that the nucleus of such cell converges to the Steiner point of the convex body. As in Rényi and Sulanke's seminal papers on random convex hulls, the regularity of the convex body has crucial importance. We deal with both the smooth and polygonal cases. Techniques are based notably on accurate estimates of the area of the Voronoi flower and of the support function of the cell containing the origin as well as on an Efron-type relation.

math.PR↗

arXiv2016

Cells with many facets in a Poisson hyperplane tessellation

Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellation in $\mathbb{R}^d$. The distribution of the number of facets $f(Z)$ of the typical cell is investigated. It is shown, that under a well-spread condition on the directional distribution, the quantity $n^{\frac{2}{d-1}}\sqrt[n]{\mathbb{P}(f(Z)=n)}$ is bounded from above and from below. When $f(Z)$ is large, the isoperimetric ratio of $Z$ is bounded away from zero with high probability. These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of $Z$ and on the other hand on several geometric estimates related to the approximation of polytopes by polytopes with fewer facets. From the asymptotics of the distribution of $f(Z)$, tail estimates for the so-called $Φ$ content of $Z$ are derived as well as results on the conditional distribution of $Z$ when its $Φ$ content is large.

math.PR↗

arXiv2016

Variance Asymptotics and Scaling Limits for Random Polytopes

Let K be a convex set in R d and let K $λ$ be the convex hull of a homogeneous Poisson point process P $λ$ of intensity $λ$ on K. When K is a simple polytope, we establish scaling limits as $λ$ $\rightarrow$ $\infty$ for the boundary of K $λ$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $λ$, k $\in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $λ$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $\times$ R having intensity $\sqrt$ de dh dhdv.

math.PR↗

arXiv2014

Variance asymptotics and scaling limits for Gaussian Polytopes

Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k \in \{0,1,...,d-1\}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $\R^{d-1} \times \R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n \to \infty$ converges to a festoon of parabolic surfaces, coinciding with that featuring in the geometric construction of the zero viscosity solution to Burgers' equation with random input.

math.PR↗

arXiv2013

Fractal random series generated by Poisson-Voronoi tessellations

In this paper, we construct a new family of random series defined on $\R^D$, indexed by one scaling parameter and two Hurst-like exponents. The model is close to Takagi-Knopp functions, save for the fact that the underlying partitions of $\R^D$ are not the usual dyadic meshes but random Voronoi tessellations generated by Poisson point processes. This approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and Hausdorff dimensions. The proof of this main result is based on several new distributional properties of the Poisson-Voronoi tessellation on the one hand, an estimate of the oscillations of the function coupled with an application of a Frostman-type lemma on the other hand. Finally, we introduce two related models and provide in particular a box-dimension calculation for a derived deterministic Takagi-Knopp series with hexagonal bases.

math.PR↗

arXiv2013

Brownian limits, local limits and variance asymptotics for convex hulls in the ball

Schreiber and Yukich [Ann. Probab. 36 (2008) 363-396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^d, d\geq2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.

math.PR↗

arXiv2013

Extreme values for characteristic radii of a Poisson-Voronoi tessellation

A homogeneous Poisson-Voronoi tessellation of intensity $γ$ is observed in a convex body $W$. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in $W$. We prove that when $γ\rightarrow\infty$, these four quantities converge to Gumbel or Weibull distributions up to a rescaling. Moreover, the contribution of boundary cells is shown to be negligible. Such approach is motivated by the analysis of the global regularity of the tessellation. In particular, consequences of our study include the convergence to the simplex shape of the cell with smallest circumscribed radius and an upper-bound for the Hausdorff distance between $W$ and its so-called Poisson-Voronoi approximation.

math.PR↗

arXiv2012

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \R^d$ be a smooth convex set and let $¶_\la$ be a Poisson point process on $\R^d$ of intensity $\la$. The convex hull of $¶_\la \cap K$ is a random convex polytope $K_\la$. As $\la \to \infty$, we show that the variance of the number of $k$-dimensional faces of $K_\la$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$.

math.PR↗

arXiv2011

Asymptotics of visibility in the hyperbolic plane

At each point of a Poisson point process of intensity $λ$ in the hyperbolic place, center a ball of bounded random radius. Consider the probability $P_r$ that from a fixed point, there is some direction in which one can reach distance $r$ without hitting any ball. It is known \cite{BJST} that if $λ$ is strictly smaller than a critical intensity $λ_{gv}$ then $P_r$ does not go to $0$ as $r\to \infty$. The main result in this note shows that in the case $λ=λ_{gv}$, the probability of reaching distance larger than $r$ decays essentially polynomial, while if $λ>λ_{gv}$, the decay is exponential. We also extend these results to various related models.

math.PR↗

arXiv2011

Asymptotics of the visibility function in the Boolean model

The aim of this paper is to give a precise estimate on the tail probability of the visibility function in a germ-grain model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle in a Boolean model. We proceed in two or more dimensions using coverage techniques. Moreover, convergence results involving a type I extreme value distribution are shown in the two particular cases of small obstacles or a large obstacle-free region.

math.PR↗