Strict inequalities of critical values in continuum percolation

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(vertical bar y-x vertical bar) 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(c)(site) > p(c)(bond). 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.