We consider the supercritical finite-range random connection model where the points x, y of a homogeneous planar Poisson process are connected with probability f (|y − x|) for a given f . Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality p site c > p bond c . We also show that reducing the connection function f strictly increases the critical Poisson intensity.Finally, we deduce that performing a spreading transformation on f (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) strictly reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.

International audience We provide normal approximation error bounds for sums of the form $\sum_x \xi_x$, indexed by the points $x$ of a Poisson process (not necessarily homogeneous) in the unit $d$-cube, with each term $\xi_x$ determined by the configuration of Poisson points near to $x$ in some sense. We consider geometric graphs and coverage processes as examples of our general results.

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