arXiv subjects

Matthias Irlbeck

Publications and source records attributed to Matthias Irlbeck.

3 recordsLinked to original sources

arXiv2026

Non-vanishing uniqueness threshold for hyperbolic Poisson-Voronoi percolation in dimension at least three

We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space $\mathbb{H}^d$ for $d\geq 3$. By recent results of Greb\'ik and Recke and d'Achille et al., this "uniqueness threshold" $p_u(\lambda)$ tends to zero as the intensity $\lambda$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products $\mathbb{H}^{d_1}\times\dots\times\mathbb{H}^{d_k}$ with $k,d_1,\dots,d_k\geq 2$. In contrast, for Poisson-Voronoi percolation on the hyperbolic plane $\mathbb{H}^2$, Benjamini and Schramm have shown that $p_u(\lambda)$ tends to one as $\lambda$ tends to zero, and $p_u(\lambda)>1/2$ for all $\lambda>0$. An unpublished argument of D'Achille and Curien shows that for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$, the uniqueness threshold satisfies $p_u(\lambda)\leq 1/2$ for all $\lambda>0$. Here we will show that $\inf_{\lambda>0}p_u(\lambda)>0$ for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$. This answers a question of Greb\'ik and Recke.

math.PR↗

arXiv2026

Thresholds for colouring the random Borsuk graph

We consider the chromatic number of the random Borsuk graph. The random Borsuk graph is obtained by sampling $n$ points i.i.d. uniformly at random on the $d$-dimensional sphere $S^d$, and joining a pair of points by an edge whenever their geodesic distance is $>π-α$ where the parameter $α=α(n)$ may depend on $n$. Kahle and Martinez-Figueroa have shown that the switch from being $(d+1)$-colourable to needing $\geq d+2$ colours occurs in the regime where the average degree is of logarithmic order. We show that for each $2\leq k\leq d$, the switch from being $k$-colourable to needing $> k$ colours occurs in the regime when the average degree is constant. What is more, we show that for $k=2$ there is a sharp threshold of the form $α(n) = c \cdot n^{-1/d}$, where the constant $c$ can be expressed in terms of the critical intensity for continuum AB percolation on $\mathbb{R}^d$. For $k=3,\dots,d+1$ we show that there is a sharp threshold for "almost all $n$".

math.PR↗

arXiv2025

On the shape of the typical Poisson-Voronoi cell in high dimensions

We study the typical cell of the Poisson-Voronoi tessellation. We show that when divided by the $d$-th root of the intensity parameter $λ$ of the Poisson process times the volume of the unit ball, the inradius, outradius, diameter and mean width of the typical cell converge in probability to the constants $1/2, 1, 2, 2$ respectively, as the dimension $d\to\infty$. We also show that the width of the typical cell, when rescaled in the same way, is bounded between $2\sqrt{5}/(2+\sqrt{5})-o_d(1)$ and $3/2+o_d(1)$, with probability $1-o_d(1)$. These results in particular imply that, with probability $1-o_d(1)$, the Hausdorff distance between the typical cell and any ball is at least of the order of the diameter of the typical cell. In addition, we show that for all $k$ with $d-k\to\infty$, with probability $1-o_d(1)$, all faces of dimension $k$ have a diameter that is of a much smaller order than the diameter, inradius, etc., of the full typical cell. The same is true for ''almost all'' faces of dimension $d-k$ with $k$ fixed. And, we show that the number of such faces is $\left( (k+1)^{(k+1)/2} / k^{k/2} \pm o_d(1) \right)^d$ with probability $1-o_d(1)$.

math.PR↗