Coverage of the unit cube by dynamic Boolean models

2025/06/27 by Hanna Döring, Döring, Hanna, Lianne de Jonge +3
Mathematics · Physics and Astronomy · Computer Science · #Stochastic processes and statistical mechanics #Complex Network Analysis Techniques #Distributed Control Multi-Agent Systems

paper · pdf · doi:10.48550/arxiv.2506.22286

Abstract

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 unit cube and dilate them to define cylinders. Letting ρ be the expected number of cylinders, the random variable of interest is the coverage radius R_ρ, which is the cylinder radius required to cover the d-dimensional unit cube. We show that R_ρ^d-1 is of the order log ρ/ ρ with high probability as ρ tends to infinity. We also consider alternative dynamics resulting in generalized cylinders that are generated by dilating the trajectories of stochastic processes, in particular Brownian motions. This leads to a coverage radius of the same order.

Citations

• Asymptotics of a time bounded cylinder model
• Random Euclidean coverage from within
• Random cover times using the Poisson cylinder process
• Extremes for the inradius in the Poisson line tessellation
• Extreme values for characteristic radii of a Poisson-Voronoi tessellation

Related

• Asymptotics of a time bounded cylinder model
• Intersections of random sets
• Covering a compact space by fixed-radius or growing random balls
• Boolean models in hyperbolic space
• The mobile Boolean model: an overview and further results