author={Mathew D. Penrose},
  journal={Annals of Applied Probability},
  • M. Penrose
  • Published 15 November 2013
  • Mathematics
  • Annals of Applied Probability
Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n \to \infty$ the probability of full connectivity is governed by that of having no isolated vertices, itself governed by a Poisson approximation for the number of isolated vertices, uniformly over all choices of $p,r$. We determine the asymptotic probability of connectivity for all $(p_n,r_n)$ subject to$r_n = O… 

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