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2013 Distribution of components in the $k$-nearest neighbour random geometric graph for $k$ below the connectivity threshold
Victor Falgas-Ravry
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Electron. J. Probab. 18: 1-22 (2013). DOI: 10.1214/EJP.v18-2465

Abstract

Let $S_{n,k}$ denote the random geometric graph obtained by placing points inside a square of area $n$ according to a Poisson point process of intensity $1$ and joining each such point to the $k=k(n)$ points of the process nearest to it.

In this paper we show that if $\mathbb{P}(S_{n,k} \textrm{ connected})>n^{-\gamma_1}$ then the probability that $S_{n,k}$ contains a pair of `small' components `close' to each other is $o(n^{-c_1})$ (in a precise sense of `small' and 'close'), for some absolute constants $\gamma_1>0$ and $c_1 >0$. This answers a question of Walters. (A similar result was independently obtained by Balister.)

As an application of our result, we show that the distribution of the connected components of $S_{n,k}$ below the connectivity threshold is asymptotically Poisson.

Citation

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Victor Falgas-Ravry. "Distribution of components in the $k$-nearest neighbour random geometric graph for $k$ below the connectivity threshold." Electron. J. Probab. 18 1 - 22, 2013. https://doi.org/10.1214/EJP.v18-2465

Information

Accepted: 18 September 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1288.60128
MathSciNet: MR3109622
Digital Object Identifier: 10.1214/EJP.v18-2465

Subjects:
Primary: 60K35

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