  We study a general class of percolation models in Euclidean space including
long-range percolation, scale-free percolation, the weight-dependent random
connection model and several other previously investigated models. Our focus is
on the weak decay regime, in which inter-cluster long-range connection
probabilities fall off polynomially with small exponent, and for which we
establish several structural properties. Chief among them are the continuity of
the bond percolation function and the transience of infinite clusters.
