Left–right crossings in the Miller–Abrahams random resistor network and in generalized Boolean models

We consider random graphs built on a homogeneous Poisson point process on , , with points marked by i.i.d. random variables . Fixed a symmetric function , the vertexes of are given by points of the Poisson point process, while the edges are given by pairs with and . We call Poisson -generalized Boolean model, as one recovers the standard Poisson Boolean model by taking ≔ and . Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left–right crossings in a box of size is lower bounded by apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to with , the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller–Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept.