The random connection model (RCM) built on a Poisson process of intensity $\lambda$ can be constructed by site percolation with parameter $1/N$ on an RCM of intensity $N\lambda$. We will see that as $N \to \infty$, the RCM constructed in this way behaves more and more like a site percolation cluster on a vertex transitive graph, in a way we will make precise.
We call this method "asymptotic transitivity" and we will apply it to extend the recent exploration methods of Vanneuville to the RCM. In particular, for the subcritical RCM with any connection function, we will see that the probability the cluster of the origin has size at least $n$ decays exponentially as $n$ increases. To our knowledge this is the first proof of the exponential decay of the volume for any long-range percolation model.