Coverage of the unit cube by dynamic Boolean models

TL;DR

This paper analyzes how a dynamic Boolean model with randomly placed and dilated cylinders covers a unit cube, showing the coverage radius scales with the logarithm of the number of cylinders divided by that number.

Contribution

It introduces a new analysis of coverage in a dynamic Boolean model with cylinders generated by stochastic processes, including Brownian motion, and establishes the asymptotic behavior of the coverage radius.

Findings

Coverage radius scales as (log ρ)/ρ for large ρ

Generalized cylinders from stochastic processes have similar coverage behavior

High probability bounds for coverage radius as ρ tends to infinity

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 $\rho$ be the expected number of cylinders, the random variable of interest is the coverage radius $R_\rho$, which is the cylinder radius required to cover the $d$-dimensional unit cube. We show that $R_\rho^{d-1}$ is of the order $\log \rho / \rho$ with high probability as $\rho$ 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.