Random Euclidean coverage from within

DOI10.1007/S00440-022-01182-5arXiv2101.06306MaRDI QIDQ6358385

Mathew D. Penrose

Publication date: 15 January 2021

Abstract: Let $X_1,X_2,ldots$ be independent random uniform points in a bounded domain $Asubsetmathbb{R}^d$ 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 $|partialA|$, if $d=3$ then $Pr[npi{R}_n^3-logn-2log(logn)leqx]$ converges to $exp(-2^{-4}p{i}^{5/3}|partialA|e^{-2x/3})$ and $(npi{R}_n^3)/(logn)o1$ almost surely, and if $d=2$ then $Pr[npi{R}_n^2-logn-log(logn)leqx]$ converges to $exp(-e^{-x}-|partialA|p{i}^{-1/2}{e}^{-x/2})$. 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.

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) Central limit and other weak theorems (60F05) Strong limit theorems (60F15) Point processes (e.g., Poisson, Cox, Hawkes processes) (60G55)

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