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

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 P(S n,k connected) > n −γ1 then the probability that S n,k contains a pair of 'small' components 'close' to each other is o(n −c1 ) (in a precise sense of 'small' and 'close'), for some absolute constants γ 1 > 0 and c 1 > 0. This answers a question of Walters [13]. (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.

Revisiting the cavity-method threshold for random 3-SAT

“…General coupon-collector problems have been studied, e.g., in Ref. [21] where k-SAT is also discussed, and for many such examples the type of monotonicity conjectured above can be proven. In fact we know of no natural examples where this type of monotonicity is known to fail, but there is no general monotonicity result which includes the case of k-SAT for fixed k.…”