Random Euclidean coverage from within

Abstract

AbstractLet $$X_1,X_2, \ldots $$ X 1 , X 2 , … be independent random uniform points in a bounded domain $$A \subset \mathbb {R}^d$$ A ⊂ R d with smooth boundary. Define the coverage threshold$$R_n$$ R n to be the smallest r such that A is covered by the balls of radius r centred on $$X_1,\ldots ,X_n$$ X 1 , … , X n . We obtain the limiting distribution of $$R_n$$ R n and also a strong law of large numbers for $$R_n$$ R n in the large-n limit. For example, if A has volume 1 and perimeter $$|\partial A|$$ | ∂ A | , if $$d=3$$ d = 3 then $$\mathbb {P}[n\pi R_n^3 - \log n - 2 \log (\log n) \le x]$$ P [ n π R n 3 - log n - 2 log ( log n ) ≤ x ] converges to $$\exp (-2^{-4}\pi ^{5/3} |\partial A| e^{-2 x/3})$$ exp ( - 2 - 4 π 5 / 3 | ∂ A | e - 2 x / 3 ) and $$(n \pi R_n^3)/(\log n) \rightarrow 1$$ ( n π R n 3 ) / ( log n ) → 1 almost surely, and if $$d=2$$ d = 2 then $$\mathbb {P}[n \pi R_n^2 - \log n - \log (\log n) \le x]$$ P [ n π R n 2 - log n - log ( log n ) ≤ x ] converges to $$\exp (- e^{-x}- |\partial A|\pi ^{-1/2} e^{-x/2})$$ exp ( - e - x - | ∂ A | π - 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.