  We consider the soft Boolean model, a model that interpolates between the
Boolean model and long-range percolation, where vertices are given via a
stationary Poisson point process. Each vertex carries an independent
Pareto-distributed radius and each pair of vertices is assigned another
independent Pareto weight with a potentially different tail exponent. Two
vertices are now connected if they are within distance of the larger radius
multiplied by the edge weight. We determine the tail behaviour of the Euclidean
diameter and the number of points of a typical maximally connected component in
a subcritical percolation phase. For this, we present a sharp criterion in
terms of the tail exponents of the edge-weight and radius distributions that
distinguish a regime where the tail behaviour is controlled only by the edge
exponent from a regime in which both exponents are relevant. Our proofs rely on
fine path-counting arguments identifying the precise order of decay of the
probability that far-away vertices are connected.
