Colloquium in probability

We consider the random connection model with bounded edges which is generated by a Poisson point process with density $\lambda$ in $\mathbb{R}^d$. We prove that this model undergoes a sharp phase transition, i.e. 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 ($\lambda-\lambda_c)$, $\lambda_c$ being the critical density. This proof uses newly developed methods by Last, Peccati and Yogeshwaran in their recent work, in particular a continuous version of the OSSS inequality for Poisson functionals, relying on stopping sets and continuous-time decision trees. This approach simplifies an earlier result of Faggionato and Mimun, who proved sharp phase transition in the random connection model via the discrete OSSS inequality.