arXiv subjects

Tillmann Bühler

Publications and source records attributed to Tillmann Bühler.

6 recordsLinked to original sources

arXiv2026

The volume of hyperbolic Poisson zero cells: critical divergence and exact second moment

We investigate the second volume moment of the zero cell $Z_o$ of a Poisson hyperplane tessellation with intensity $γ$ in the $d$-dimensional hyperbolic space. We focus on the phase transition at the critical intensity $γ_c^{(d)}$, the minimum value for which $Z_o$ is almost surely bounded. In the critical regime $γ=γ_c^{(d)}$, we show that the second volume moment of the restricted zero cell $Z_o \cap B_R$, where $B_R$ is a hyperbolic ball of radius $R$ centred at $o$, diverges in any dimension at the universal rate $R^3$ as $R \to \infty$. In the supercritical case $γ> γ_c^{(d)}$, we prove that the full second volume moment is finite. Using tools from harmonic analysis in hyperbolic space, we derive an exact expression for this moment in terms of the Meijer $G$-function. Furthermore, we determine the asymptotic behaviour of the second moment as $γ\to \infty$ and as $γ\downarrow γ_c^{(d)}$, facilitating a direct comparison with the corresponding Euclidean values as well as the mean-field universality class of percolation theory.

math.PR↗

arXiv2025

Visibility and intersection density for Boolean models in hyperbolic space

For Poisson particle processes in hyperbolic space we introduce and study concepts analogous to the intersection density and the mean visible volume, which were originally considered in the analysis of Boolean models in Euclidean space. In particular, we determine a necessary and sufficient condition for the finiteness of the mean visible volume of a Boolean model in terms of the intensity and the mean surface area of the typical grain.

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

High-dimensional limits arising from hyperbolic Poisson k-plane processes

We consider a stationary Poisson process of $k$-planes in the $d$-dimensional hyperbolic space $\mathbb H^d$ of constant curvature $-1$, with $d \ge 4$ and $1 \le k \le d-1$. It is known that, after centring and normalization, the total $k$-volume of all intersections of $k$-planes with a geodesic ball of radius $R$ converges in distribution, as $R \to \infty$, to a non-Gaussian infinitely divisible random variable $Z_{d,k}$ whenever $2k > d+1$. We investigate the distributional behaviour of $Z_{d,k}$ in the high-dimensional regime $d \to \infty$ and depending on how fast $k$ grows in relation to $d$. We derive precise conditions for the variance normalized sequence to converge in law to a standard Gaussian random variable or to a degenerate law, respectively, and show that an alternative rescaling of the Lévy measures yields an explicit non-Gaussian infinitely divisible limit for fixed codimension $d-k$ and a standard Gaussian limit for $d-k \to \infty$.

math.PR↗

arXiv2025

High-intensity Voronoi percolation on manifolds

We study Voronoi percolation on a large class of $d$-dimensional Riemannian manifolds, which includes the hyperbolic spaces $\mathbb{H}^d$, $d\geq 2$. We prove that as the intensity $λ$ of the underlying Poisson point process tends to infinity, both critical parameters $p_c(M,λ)$ and $p_u(M,λ)$ converge to the Euclidean critical parameter $p_c(\mathbb{R}^d)$. This extends a recent result of Hansen & Müller in the special case $M=\mathbb{H}^2$ to a general class of manifolds of arbitrary dimension. A crucial step in our proof, which may be of independent interest, is to show that if $M$ is simply connected and one-ended, then embedded graphs induced by a general class of tessellations on $M$ have connected minimal cutsets. In particular, this result applies to $\varepsilon$-nets, allowing us to implement a "fine-graining" argument.

math.PR↗

arXiv2024

Intersections of Poisson k-flats in hyperbolic space: completing the picture

In recent years there has been a lot of interest in the study of isometry invariant Poisson processes of $k$-flats in $d$-dimensional hyperbolic space $\mathbb{H}^d$, for $0\le k\le d-1$. A phenomenon that has no counterpart in euclidean geometry arises in the investigation of the total $k$-dimensional volume $F_r$ of the process inside a spherical observation window $B_r$ of radius $r$ when one lets $r$ tend to infinity. While $F_r$ is asymptotically normally distributed for $2k\leq d+1$, it has been shown to obey a nonstandard central limit theorem for $2k>d+1$. The intersection process of order $m$, for $d-m(d-k) \geq 0$, of the original process $η$ consists of all intersections of distinct flats $E_1,\ldots,E_m \in η$ with $\dim(E_1\cap\ldots\cap E_m) = d-m(d-k)$. For this intersection process, the total $d-m(d-k)$-dimensional volume $F^{(m)}_r$ of the process in $B_r$, again as $r \to \infty$, is of particular interest. For $2k \leq d+1$ it has been shown that $F^{(m)}_r$ is again asymptotically normally distributed. For $m \geq 2$, the limit is so far unknown, although it has been shown for certain $d$ and $k$ that it cannot be a normal distribution. We determine the limit distribution for all values of $d,k,m$. In addition, we establish explicit rates of convergence in the Kolmogorov distance and discuss properties of the limit distribution. Furthermore we show that the asymptotic covariance matrix of the vector $(F^{(1)}_r,\ldots,F^{(m)}_r)^\top$ has full rank when $2k < d+1$ and rank one when $2k \geq d+1$.

math.PR↗