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

Connection probabilities in Poisson random graphs with uniformly bounded edges