Random coverage of a manifold with boundary

Abstract

Let $A$ be a compact $d$-dimensional $C^2$ Riemannian manifold with boundary, embedded in ${\bf R}^m$ where $m \geq d \geq 2$, and let $B$ be a nice subset of $A$ (possibly $B=A$). Let $X_1,X_2, \ldots $ be independent random uniform points in $A$. Define the {\em coverage threshold} $R_n$ to be the smallest $r$ such that $B$ is covered by the geodetic 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 Riemannian volume 1 and its boundary has surface measure $|\partial A|$, and $B=A$, then if $d=3$ then ${\bf P}[n\pi R_n^3 - \log n - 2 \log (\log n) \leq x]$ 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, while if $d=2$ then ${\bf P}[n \pi R_n^2 - \log n - \log (\log n) \leq x]$ converges to $\exp(- e^{-x}- |\partial A|\pi^{-1/2} e^{-x/2})$. We generalize to allow for multiple coverage. For the strong laws of large numbers, we can relax the requirement that the underlying density on $A$ be uniform. For the limiting distribution, we have a similar result for Poisson samples. Our results still hold if we use Euclidean rather than geodetic balls.