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Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold


 
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1. Title Title of document Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold
 
2. Creator Author's name, affiliation, country Victor Falgas-Ravry; Umeå Universitet; Sweden
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Random geometric graphs
 
3. Subject Subject classification 60K35
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) EPSRC, Kempe Stiftelse
 
7. Date (YYYY-MM-DD) 2013-09-18
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2465
 
10. Identifier Digital Object Identifier 10.1214/EJP.v18-2465
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 18
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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