## Sharpness in the $$k$$-nearest-neighbours random geometric graph model.(English)Zbl 1278.60142

Summary: 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. P. Balister et al. [Adv. Appl. Probab. 37, No. 1, 1–24 (2005; Zbl 1079.05086)] conjectured that, for every $$0 < \varepsilon < 1$$ and all sufficiently large $$n$$, there exists $$C = C(\varepsilon )$$ such that, whenever the probability that $$S_{n,k}$$ is connected is at least $$\varepsilon$$, then the probability that $$S_{n,k+C}$$ is connected is at least $$1 - \varepsilon$$.
In this paper, we prove this conjecture. As a corollary, we prove that there exists a constant $$C^{\prime}$$ such that, whenever $$k(n)$$ is a sequence of integers such that the probability $$S_{n,k(n)}$$ is connected tends to 1 as $$n \rightarrow \infty$$; then, for any integer sequence $$s(n)$$ with $$s(n) = o(\log n)$$, the probability $$S_{n,k(n)+\lfloor C's \log \log n\rfloor }$$ is $$s$$-connected (i.e., remains connected after the deletion of any $$s - 1$$ vertices) tends to 1 as $$n \rightarrow \infty$$. This proves another conjecture given in [Paul Balister et al., Discrete Appl. Math. 157, No. 2, 309–320 (2009; Zbl 1156.05054)].

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation

### Keywords:

random geometric graph; connectivity; sharp transition

### Citations:

Zbl 1079.05086; Zbl 1156.05054
Full Text:

### References:

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