Exponential decay for the random connection model using asymptotic transitivity

We prove that the probability the cluster of the origin in a subcritical Poisson random connection model (RCM) has size at least $n$ decays exponentially as $n$ increases, under minimal assumptions. We extend a recent method of Vanneuville (arXiv:2304.12110) from Bernoulli percolation on vertex-transitive graphs to the RCM. The key idea is that the subcritical RCM can be constructed by site percolation on a very high-intensity RCM. The latter RCM becomes ``almost vertex-transitive'' in a certain sense at very high intensities, which is a new method that we expect to be useful for other problems. We obtain the result for connection functions with unbounded support, a setting in which it was not previously known.