Largest nearest-neighbour link and connectivity threshold in a polytopal random sample

TL;DR

This paper establishes a strong law of large numbers for the largest nearest-neighbour link and the connectivity threshold in a random sample within a convex polytopal domain, revealing their asymptotic behavior as the sample size grows.

Contribution

It provides the first rigorous asymptotic analysis of the largest nearest-neighbour link and connectivity threshold in polytopal domains, linking these thresholds to the domain's geometry.

Findings

Almost sure convergence of scaled thresholds to a domain-dependent limit

Asymptotic equivalence of the largest nearest-neighbour link and connectivity threshold

Explicit limit involving the domain's geometric properties

Abstract

Let $X_1,X_2, \ldots $ be independent identically distributed random points in a convex polytopal domain $A \subset \mathbb{R}^d$. Define the largest nearest neighbour link $L_n$ to be the smallest $r$ such that every point of $\mathcal X_n:=\{X_1,\ldots,X_n\}$ has another such point within distance $r$. We obtain a strong law of large numbers for $L_n$ in the large-$n$ limit. A related threshold, the connectivity threshold $M_n$, is the smallest $r$ such that the random geometric graph $G(\mathcal X_n, r)$ is connected. We show that as $n \to \infty$, almost surely $nL_n^d/\log n$ tends to a limit that depends on the geometry of $A$, and $nM_n^d/\log n$ tends to the same limit.