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 Rd, d≥2, with points x marked by i.i.d. random variables Ex. 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(Ex,Ey). We call GPoissonh-generalized Boolean model, as one recovers the standard Poisson Boolean model by taking h(a,b)≔a+b and Ex≥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 Cnd−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.