Largest component and sharpness in continuum percolation

We investigate the behavior of large connected components in the Poisson Random Connection model in non-critical regimes with any bounded connection function. We show that the asymptotic size of the largest component restricted to a window grows logarithmically in the volume of that window in the subcritical case, and linearly in the supercritical case. We also prove a sharpness result saying that the order of the cluster at the origin has an exponentially decaying tail in the subcritical regime.