#### Document Type

Article

#### Publication Date

2009

#### Abstract

Let 𝓟 be a Poisson process of intensity 1 in a square *S*_{n} of area *n*. For a fixed integer *k*, join every point of 𝓟 to its *k* nearest neighbours, creating an undirected random geometric graph *G*_{n,k}. We prove that there exists a critical constant *c*_{crit} such that, for *c*‹*c*_{crit}, *G*_{n,⌊clogn⌋} is disconnected with probability tending to 1 as *n* →∞ and, for *c*‹*c*_{crit}, *G*_{n,⌊clogn⌋} is connected with probability tending to 1 as *n* →∞. This answers a question posed in Balister *et al*. (2005).

#### Publication Title

Advances in Applied Probability

#### Volume

41

#### Issue

1

#### First Page

1

#### Last Page

12

#### Required Publisher's Statement

Published by Project Euclid

DOI:10.1239/aap/1240319574

#### Recommended Citation

Balister, Paul; Bollobás, Béla; Sarkar, Amites; Walters, Mark. A critical constant for the k nearest-neighbour model. Adv. in Appl. Probab. 41 (2009), no. 1, 001--012. doi:10.1239/aap/1240319574. http://projecteuclid.org/euclid.aap/1240319574.

## Comments

This is the authors' version of the article. The publisher version is at: http://projecteuclid.org/euclid.aap/1240319574