## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 44, Number 3 (2012), 617-634.

### Sharpness in the *k*-nearest-neighbours random geometric graph model

Victor Falgas-Ravry and Mark Walters

#### Abstract

Let *S*_{n,k} denote the random graph obtained by
placing points in a square box of area *n* according to a Poisson process
of intensity 1 and joining each point to its *k* nearest neighbours.
Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for
every 0 < ε < 1 and all sufficiently large *n*, there
exists *C* = *C*(ε) such that, whenever the probability that
*S*_{n,k} is connected is at least ε, then
the probability that *S*_{n,k+C} is connected
is at least 1 - ε. In this paper we prove this conjecture. As a
corollary, we prove that there exists a constant *C*' such that, whenever
*k*(*n*) is a sequence of integers such that the probability
*S*_{n,k(n)} is connected tends to 1 as
*n* → ∞, then, for any integer sequence *s*(*n*)
with *s*(*n*) = *o*(log*n*), the probability
*S*_{n,k(n)+⌊C'slog logn⌋}
is *s*-connected (i.e. remains connected after the deletion of any
*s* - 1 vertices) tends to 1 as *n* → ∞. This proves
another conjecture given in Balister, Bollobás, Sarkar and Walters
(2009).

#### Article information

**Source**

Adv. in Appl. Probab., Volume 44, Number 3 (2012), 617-634.

**Dates**

First available in Project Euclid: 6 September 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1346955257

**Digital Object Identifier**

doi:10.1239/aap/1346955257

**Mathematical Reviews number (MathSciNet)**

MR3024602

**Zentralblatt MATH identifier**

1278.60142

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82B43: Percolation [See also 60K35]

**Keywords**

Random geometric graph connectivity sharp transition

#### Citation

Falgas-Ravry, Victor; Walters, Mark. Sharpness in the k -nearest-neighbours random geometric graph model. Adv. in Appl. Probab. 44 (2012), no. 3, 617--634. doi:10.1239/aap/1346955257. https://projecteuclid.org/euclid.aap/1346955257