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

We consider random graphs $\mathcal{G}$ built on a homogeneous Poisson point process on $\mathbb{R}^d$, $d\geq 2$, with points $x$ marked by i.i.d. random variables $E_x$. Fixed a symmetric function $h(\cdot, \cdot)$, the vertexes of $\mathcal{G}$ are given by points of the Poisson point process, while the edges are given by pairs $\{x,y\}$ with $x\not =y$ and $|x-y|\leq h(E_x,E_y)$. We call $\mathcal{G}$ Poisson $h$-generalized Boolean model, as one recovers the standard Poisson Boolean model by taking $h(a,b):=a+b$ and $E_x\geq 0$. 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 $Cn^{d-1}$ 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 $h(a,b)=(a+b)^\gamma$ with $\gamma>0$, 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.