Random Euclidean coverage from within

Abstract

Let $X_1,X_2, \ldots $ be independent random uniform points in a bounded domain $A \subset \mathbb{R}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_n$ and also a strong law of large numbers for $R_n$ in the large-$n$ limit. For example, if $A$ has volume 1 and perimeter $|\partial A|$, if $d=3$ then $\Pr[n\pi R_n^3 - \log n - 2 \log (\log n) \leq x]$ converges to $\exp(-2^{-4}\pi^{5/3} |\partial A| e^{-2 x/3})$ and $(n \pi R_n^3)/(\log n) \to 1$ almost surely, and if $d=2$ then $\Pr[n \pi R_n^2 - \log n - \log (\log n) \leq x]$ converges to $\exp(- e^{-x}- |\partial A|\pi^{-1/2} e^{-x/2})$. We give similar results for general $d$, and also for the case where $A$ is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on $A$ be uniform.