  We study the signal-to-interference ratio (SINR) percolation model for a
stationary Cox point process in two or higher dimensions, in case of a bounded
and integrable path-loss function. We show that if this function has compact
support or if the stationary intensity measure evaluated at a unit box has some
exponential moments, then the SINR graph has an infinite connected component in
case the spatial density of points is large enough and the interferences are
sufficiently reduced (without vanishing). This holds under suitable
stabilization and connectivity assumptions on the intensity measure. We also
provide estimates on the critical interference cancellation factor.
