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We study a general class of percolation models in Euclidean space includinglong-range percolation, scale-free percolation, the weight-dependent randomconnection model and several other previously investigated models. Our focus ison the weak decay regime, in which inter-cluster long-range connectionprobabilities fall off polynomially with small exponent, and for which weestablish several structural properties. Chief among them are the continuity ofthe bond percolation function and the transience of infinite clusters.

Inhomogeneous long-range percolation in the weak decay regime