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Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space

Holger Sambale, Christoph Thäle, Tara Trauthwein

Abstract

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 $x\inη$ 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.

Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space

Abstract

Consider a stationary Poisson process in the -dimensional Euclidean or hyperbolic space and construct a random graph with vertex set as follows. First, each point is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until is contained in the convex hull of the points already connected to . 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.

Paper Structure

This paper contains 8 sections, 11 theorems, 99 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Poisson functionals
  4. Geometric ingredients
  5. The radius of stabilization
  6. Proofs of the main results
  7. Proof of Theorem \ref{['thm:main']}
  8. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1.1

Let $N$ be a standard Gaussian random variable and $\diamondsuit\in\{K,W\}$. Then, for any $\alpha \ge 0$ and any $t \ge 2$, we have where $c>0$ is a constant only depending on the choice of $\mathbb{X}^d$ and on $\alpha$.

Figures (4)

  • Figure 1: A realization of the NNE graph in $2$-dimensional Euclidean space.
  • Figure 2: A realization of the NNE graph in $2$-dimensional hyperbolic space.
  • Figure 3: The construction of the variance lower bound for $F^{(\alpha)}_t$, in both the Euclidean and the hyperbolic case.
  • Figure 4: Illustration of the construction in the Euclidean case. Here $k=6$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 11 more