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Sharpness in the k-Nearest-Neighbours Random Geometric Graph Model

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  04 January 2016

Victor Falgas-Ravry  and
Mark Walters
Victor Falgas-Ravry*
Affiliation:
Queen Mary, University of London
Mark Walters*
Affiliation:
Queen Mary, University of London
*
∗ Postal address: School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK.
∗ Postal address: School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK.
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Abstract

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Let Sn, 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 Sn, k is connected is at least ε, then the probability that Sn, 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 Sn,k(n) is connected tends to 1 as n → ∞, then, for any integer sequence s(n) with s(n) = o(logn), the probability Sn,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).

Keywords

Random geometric graphconnectivitysharp transition

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 617 - 634
Copyright
© Applied Probability Trust 

References

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