2024/11/01 by Holger Sambale, Sambale, Holger, Christoph Thäle +3 · 1 citation
Mathematics · #60D05 #60F05 #60G55 #FOS: Mathematics #Graph theory and applications #Mathematics and Applications #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2411.00748
openalex publication_date 2024/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a stationary Poisson process η in the d-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set η as follows. First, each point x∈η is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until x is contained in the convex hull of the points already connected to x. The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered.