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(|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 $p_{c}^{\mathrm{site}} > p_{c}^{\mathrm{bond}}$p_{c}^{\mathrm{site}} > p_{c}^{\mathrm{bond}}. We also show that reducing the connection function f strictly increases the critical Poisson intensity.