Xiaochuan Yang

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math.PR2026

Random coverage of a manifold with boundary

Mathew D. Penrose, Xiaochuan Yang

Let $A$ be a compact $d$ -dimensional $C^{2}$ Riemannian manifold with boundary, embedded in $R^{m}$ where $m \geq d \geq 2$ , and let $B$ be a nice subset of $A$ (possibly $B = A$ ).…

math.PR2026

On the components of random geometric graphs in the dense limit

Mathew D. Penrose, Xiaochuan Yang

Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of $R^{d}, d \geq 2$ , with $C^{2}$ boundary, or in the unit square, with d…

math.PR2025

Coverage of the unit cube by dynamic Boolean models

Hanna DÃ¶ring, Lianne de Jonge, Xiaochuan Yang

Motivated by peer-to-peer telecommunication, we study a dynamic Boolean model. We define a Poisson number of random lines through the $(d - 1)$ -dimensional base of a $d$ -dimensional…

math.PR2025

On the rate of convergence in the Hall-Janson coverage theorem

Mathew D. Penrose, Xiaochuan Yang

Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$ -space with $d \geq 2$ , with Poisson intensity $t$ and radii distributed like $r Y$ with $r \geq 0$ a scaling paramete…

math.PR2025

On $k$ -clusters of high-intensity random geometric graphs

Mathew D. Penrose, Xiaochuan Yang

Let $k, d$ be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a $d$ -dimensional Poisson Boolean mod…

math.PR2024

Fluctuations of the connectivity threshold and largest nearest-neighbour link

Mathew D. Penrose, Xiaochuan Yang

Consider a random uniform sample of $n$ points in a compact region $A$ of Euclidean $d$ -space, $d \geq 2$ , with a smooth or (when $d = 2$ ) polygonal boundary. Fix $k \in N$ . Le…

Random coverage of a manifold with boundary

On the components of random geometric graphs in the dense limit

Coverage of the unit cube by dynamic Boolean models

On the rate of convergence in the Hall-Janson coverage theorem

On k-clusters of high-intensity random geometric graphs

Fluctuations of the connectivity threshold and largest nearest-neighbour link

On $k$ -clusters of high-intensity random geometric graphs