Connection probabilities in Poisson random graphs with uniformly bounded edges

We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and Miller-Abrahams random resistor network with lower-bounded conductances. The latter is relevant for the analysis of conductivity by Mott variable range hopping in strongly disordered systems. By using the method of randomized algorithms developed by Duminil-Copin et al. we prove that in the subcritical phase the probability that the origin is connected to some point at distance $n$ decays exponentially in $n$, while in the supercritical phase the probability that the origin is connected to infinity is strictly positive and bounded from below by a term proportional to $(\lambda-\lambda_c)$, $\lambda$ being the density of the Poisson point process and $\lambda_c$ being the critical density.