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 Rd, d≥2, with points x marked by i.i.d. variables Ex. Fixed symmetric function h(⋅,⋅), the vertexes of are given process, while edges pairs {x,y} x≠y and |x−y|≤h(Ex,Ey). call h-generalized Boolean model, as one recovers standard model taking h(a,b)≔a+b Ex≥0. Under general conditions, we show that in supercritical phase maximal number vertex-disjoint left–right crossings box size n is lower bounded Cnd−1 apart from an event exponentially small probability. As special applications, when marks non-negative, its generalization to h(a,b)=(a+b)γ γ>0, weight-dependent connection models max-kernel min-kernel graph obtained Miller–Abrahams resistor network which only filaments conductivity fixed positive constant kept.