M. Penrose

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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

Largest component and sharpness in continuum percolation

Niclas KÃ¼pper, Mathew D. Penrose

We investigate the behavior of large connected components in the Poisson Random Connection model in non-critical regimes with any bounded connection function. We show that the asym…

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

Random coverage from within with variable radii, and Johnson-Mehl cover times

Mathew D. Penrose, Frankie Higgs

Given a compact planar region $A$ , let $I ¨ _{A}$ be the (random) time it takes for the Johnson-Mehl tessellation of $A$ to be complete, i.e. the time it takes for $A$ to be fully cove…

math.PR2025

Supercritical phase of the random connection model

Mathew D. Penrose

Given $d \in N, I^» > 0$ , the random connection model in a region $A \subseteq R^{d}$ is a graph with vertex set given by a homogeneous Poisson point process of intensity $Î…

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…