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

We consider random graphs G built on a homogeneous Poisson point process on R d , d ≥ 2 , with points x marked by i.i.d. random variables E x . Fixed a symmetric function h ( ⋅ , ⋅ ) , the vertexes of G are given by points of the Poisson point process, while the edges are given by pairs { x , y } with x ≠ y and | x − y | ≤ h ( E x , E y ) . We call G Poisson h -generalized Boolean model, as one recovers the standard Poisson Boolean model by taking h ( a , b ) ≔ a + b and E x ≥ 0 . Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left–right crossings in a box of size n is lower bounded by C n d − 1 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 h ( a , b ) = ( a + b ) γ with γ > 0 , 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.